sdl-ios-xcode: src/video/e

annotate src/video/e_sqrt.h @ 2435:775e061e9440 gsoc2008_iphone

changed #ifndef statment to _SDL_config_iphoneos_h instead of _SDL_config_minimal_h.

author	Holmes Futrell <hfutrell@umail.ucsb.edu>
date	Fri, 15 Aug 2008 18:22:12 +0000
parents	c121d94672cb
children

rev	line source
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	1 /* @(#)e_sqrt.c 5.1 93/09/24 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	2 /*
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	3 * ====================================================
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	5 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	6 * Developed at SunPro, a Sun Microsystems, Inc. business.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	7 * Permission to use, copy, modify, and distribute this
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	8 * software is freely granted, provided that this notice
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	9 * is preserved.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	10 * ====================================================
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	11 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	12
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	13 #if defined(LIBM_SCCS) && !defined(lint)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	14 static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	15 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	16
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	17 /* __ieee754_sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	18 * Return correctly rounded sqrt.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	19 * ------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	20 * \| Use the hardware sqrt if you have one \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	21 * ------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	22 * Method:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	23 * Bit by bit method using integer arithmetic. (Slow, but portable)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	24 * 1. Normalization
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	25 * Scale x to y in [1,4) with even powers of 2:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	26 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	27 * sqrt(x) = 2^k * sqrt(y)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	28 * 2. Bit by bit computation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	29 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	30 * i 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	31 * i+1 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	32 * s = 2q , and y = 2 ( y - q ). (1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	33 * i i i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	34 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	35 * To compute q from q , one checks whether
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	36 * i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	37 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	38 * -(i+1) 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	39 * (q + 2 ) <= y. (2)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	40 * i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	41 * -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	42 * If (2) is false, then q = q ; otherwise q = q + 2 .
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	43 * i+1 i i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	44 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	45 * With some algebric manipulation, it is not difficult to see
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	46 * that (2) is equivalent to
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	47 * -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	48 * s + 2 <= y (3)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	49 * i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	50 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	51 * The advantage of (3) is that s and y can be computed by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	52 * i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	53 * the following recurrence formula:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	54 * if (3) is false
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	55 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	56 * s = s , y = y ; (4)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	57 * i+1 i i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	58 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	59 * otherwise,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	60 * -i -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	61 * s = s + 2 , y = y - s - 2 (5)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	62 * i+1 i i+1 i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	63 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	64 * One may easily use induction to prove (4) and (5).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	65 * Note. Since the left hand side of (3) contain only i+2 bits,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	66 * it does not necessary to do a full (53-bit) comparison
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	67 * in (3).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	68 * 3. Final rounding
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	69 * After generating the 53 bits result, we compute one more bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	70 * Together with the remainder, we can decide whether the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	71 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	72 * (it will never equal to 1/2ulp).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	73 * The rounding mode can be detected by checking whether
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	74 * huge + tiny is equal to huge, and whether huge - tiny is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	75 * equal to huge for some floating point number "huge" and "tiny".
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	76 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	77 * Special cases:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	78 * sqrt(+-0) = +-0 ... exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	79 * sqrt(inf) = inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	80 * sqrt(-ve) = NaN ... with invalid signal
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	81 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	82 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	83 * Other methods : see the appended file at the end of the program below.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	84 *---------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	85 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	86
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	87 /#include "math.h"/
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	88 #include "math_private.h"
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	89
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	90 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	91 double SDL_NAME(copysign) (double x, double y)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	92 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	93 double SDL_NAME(copysign) (x, y)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	94 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	95 x,
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	96 y;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	97 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	98 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	99 u_int32_t hx, hy;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	100 GET_HIGH_WORD(hx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	101 GET_HIGH_WORD(hy, y);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	102 SET_HIGH_WORD(x, (hx & 0x7fffffff) \| (hy & 0x80000000));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	103 return x;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	104 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	105
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	106 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	107 double SDL_NAME(scalbn) (double x, int n)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	108 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	109 double SDL_NAME(scalbn) (x, n)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	110 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	111 x;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	112 int
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	113 n;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	114 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	115 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	116 int32_t k, hx, lx;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	117 EXTRACT_WORDS(hx, lx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	118 k = (hx & 0x7ff00000) >> 20; /* extract exponent */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	119 if (k == 0) { /* 0 or subnormal x */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	120 if ((lx \| (hx & 0x7fffffff)) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	121 return x; /* +-0 */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	122 x *= two54;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	123 GET_HIGH_WORD(hx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	124 k = ((hx & 0x7ff00000) >> 20) - 54;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	125 if (n < -50000)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	126 return tiny * x; /underflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	127 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	128 if (k == 0x7ff)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	129 return x + x; /* NaN or Inf */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	130 k = k + n;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	131 if (k > 0x7fe)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	132 return huge * SDL_NAME(copysign) (huge, x); /* overflow */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	133 if (k > 0) { /* normal result */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	134 SET_HIGH_WORD(x, (hx & 0x800fffff) \| (k << 20));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	135 return x;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	136 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	137 if (k <= -54) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	138 if (n > 50000) /* in case integer overflow in n+k */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	139 return huge * SDL_NAME(copysign) (huge, x); /overflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	140 else
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	141 return tiny * SDL_NAME(copysign) (tiny, x); /underflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	142 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	143 k += 54; /* subnormal result */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	144 SET_HIGH_WORD(x, (hx & 0x800fffff) \| (k << 20));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	145 return x * twom54;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	146 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	147
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	148 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	149 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	150 __ieee754_sqrt(double x)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	151 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	152 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	153 __ieee754_sqrt(x)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	154 double x;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	155 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	156 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	157 double z;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	158 int32_t sign = (int) 0x80000000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	159 int32_t ix0, s0, q, m, t, i;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	160 u_int32_t r, t1, s1, ix1, q1;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	161
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	162 EXTRACT_WORDS(ix0, ix1, x);
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	163
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	164 /* take care of Inf and NaN */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	165 if ((ix0 & 0x7ff00000) == 0x7ff00000) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	166 return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	167 sqrt(-inf)=sNaN */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	168 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	169 /* take care of zero */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	170 if (ix0 <= 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	171 if (((ix0 & (~sign)) \| ix1) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	172 return x; /* sqrt(+-0) = +-0 */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	173 else if (ix0 < 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	174 return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	175 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	176 /* normalize x */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	177 m = (ix0 >> 20);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	178 if (m == 0) { /* subnormal x */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	179 while (ix0 == 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	180 m -= 21;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	181 ix0 \|= (ix1 >> 11);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	182 ix1 <<= 21;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	183 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	184 for (i = 0; (ix0 & 0x00100000) == 0; i++)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	185 ix0 <<= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	186 m -= i - 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	187 ix0 \|= (ix1 >> (32 - i));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	188 ix1 <<= i;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	189 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	190 m -= 1023; /* unbias exponent */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	191 ix0 = (ix0 & 0x000fffff) \| 0x00100000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	192 if (m & 1) { /* odd m, double x to make it even */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	193 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	194 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	195 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	196 m >>= 1; /* m = [m/2] */
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	197
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	198 /* generate sqrt(x) bit by bit */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	199 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	200 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	201 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	202 r = 0x00200000; /* r = moving bit from right to left */
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	203
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	204 while (r != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	205 t = s0 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	206 if (t <= ix0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	207 s0 = t + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	208 ix0 -= t;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	209 q += r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	210 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	211 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	212 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	213 r >>= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	214 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	215
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	216 r = sign;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	217 while (r != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	218 t1 = s1 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	219 t = s0;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	220 if ((t < ix0) \|\| ((t == ix0) && (t1 <= ix1))) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	221 s1 = t1 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	222 if (((int32_t) (t1 & sign) == sign) && (s1 & sign) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	223 s0 += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	224 ix0 -= t;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	225 if (ix1 < t1)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	226 ix0 -= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	227 ix1 -= t1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	228 q1 += r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	229 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	230 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	231 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	232 r >>= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	233 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	234
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	235 /* use floating add to find out rounding direction */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	236 if ((ix0 \| ix1) != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	237 z = one - tiny; /* trigger inexact flag */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	238 if (z >= one) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	239 z = one + tiny;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	240 if (q1 == (u_int32_t) 0xffffffff) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	241 q1 = 0;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	242 q += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	243 } else if (z > one) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	244 if (q1 == (u_int32_t) 0xfffffffe)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	245 q += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	246 q1 += 2;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	247 } else
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	248 q1 += (q1 & 1);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	249 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	250 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	251 ix0 = (q >> 1) + 0x3fe00000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	252 ix1 = q1 >> 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	253 if ((q & 1) == 1)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	254 ix1 \|= sign;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	255 ix0 += (m << 20);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	256 INSERT_WORDS(z, ix0, ix1);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	257 return z;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	258 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	259
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	260 /*
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	261 Other methods (use floating-point arithmetic)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	262 -------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	263 (This is a copy of a drafted paper by Prof W. Kahan
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	264 and K.C. Ng, written in May, 1986)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	265
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	266 Two algorithms are given here to implement sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	267 (IEEE double precision arithmetic) in software.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	268 Both supply sqrt(x) correctly rounded. The first algorithm (in
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	269 Section A) uses newton iterations and involves four divisions.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	270 The second one uses reciproot iterations to avoid division, but
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	271 requires more multiplications. Both algorithms need the ability
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	272 to chop results of arithmetic operations instead of round them,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	273 and the INEXACT flag to indicate when an arithmetic operation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	274 is executed exactly with no roundoff error, all part of the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	275 standard (IEEE 754-1985). The ability to perform shift, add,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	276 subtract and logical AND operations upon 32-bit words is needed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	277 too, though not part of the standard.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	278
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	279 A. sqrt(x) by Newton Iteration
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	280
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	281 (1) Initial approximation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	282
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	283 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	284 a floating point number x (in IEEE double format) respectively
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	285
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	286 1 11 52 ...widths
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	287 ------------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	288 x: \|s\| e \| f \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	289 ------------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	290 msb lsb msb lsb ...order
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	291
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	292
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	293 ------------------------ ------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	294 x0: \|s\| e \| f1 \| x1: \| f2 \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	295 ------------------------ ------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	296
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	297 By performing shifts and subtracts on x0 and x1 (both regarded
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	298 as integers), we obtain an 8-bit approximation of sqrt(x) as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	299 follows.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	300
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	301 k := (x0>>1) + 0x1ff80000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	302 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	303 Here k is a 32-bit integer and T1[] is an integer array containing
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	304 correction terms. Now magically the floating value of y (y's
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	305 leading 32-bit word is y0, the value of its trailing word is 0)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	306 approximates sqrt(x) to almost 8-bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	307
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	308 Value of T1:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	309 static int T1[32]= {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	310 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	311 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	312 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	313 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	314
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	315 (2) Iterative refinement
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	316
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	317 Apply Heron's rule three times to y, we have y approximates
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	318 sqrt(x) to within 1 ulp (Unit in the Last Place):
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	319
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	320 y := (y+x/y)/2 ... almost 17 sig. bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	321 y := (y+x/y)/2 ... almost 35 sig. bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	322 y := y-(y-x/y)/2 ... within 1 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	323
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	324
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	325 Remark 1.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	326 Another way to improve y to within 1 ulp is:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	327
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	328 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	329 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	330
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	331 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	332 (x-y )*y
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	333 y := y + 2* ---------- ...within 1 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	334 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	335 3y + x
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	336
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	337
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	338 This formula has one division fewer than the one above; however,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	339 it requires more multiplications and additions. Also x must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	340 scaled in advance to avoid spurious overflow in evaluating the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	341 expression 3y*y+x. Hence it is not recommended uless division
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	342 is slow. If division is very slow, then one should use the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	343 reciproot algorithm given in section B.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	344
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	345 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	346
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	347 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	348 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	349 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	350 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	351 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	352 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	353 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	354 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	355
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	356 I := FALSE; ... reset INEXACT flag I
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	357 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	358 z := x/y; ... chopped quotient, possibly inexact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	359 If(not I) then { ... if the quotient is exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	360 if(z=y) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	361 I := i; ... restore inexact flag
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	362 R := r; ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	363 return sqrt(x):=y.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	364 } else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	365 z := z - ulp; ... special rounding
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	366 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	367 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	368 i := TRUE; ... sqrt(x) is inexact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	369 If (r=RN) then z=z+ulp ... rounded-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	370 If (r=RP) then { ... round-toward-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	371 y = y+ulp; z=z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	372 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	373 y := y+z; ... chopped sum
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	374 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	375 I := i; ... restore inexact flag
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	376 R := r; ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	377 return sqrt(x):=y.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	378
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	379 (4) Special cases
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	380
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	381 Square root of +inf, +-0, or NaN is itself;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	382 Square root of a negative number is NaN with invalid signal.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	383
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	384
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	385 B. sqrt(x) by Reciproot Iteration
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	386
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	387 (1) Initial approximation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	388
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	389 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	390 a floating point number x (in IEEE double format) respectively
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	391 (see section A). By performing shifs and subtracts on x0 and y0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	392 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	393
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	394 k := 0x5fe80000 - (x0>>1);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	395 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	396
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	397 Here k is a 32-bit integer and T2[] is an integer array
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	398 containing correction terms. Now magically the floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	399 value of y (y's leading 32-bit word is y0, the value of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	400 its trailing word y1 is set to zero) approximates 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	401 to almost 7.8-bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	402
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	403 Value of T2:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	404 static int T2[64]= {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	405 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	406 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	407 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	408 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	409 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	410 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	411 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	412 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	413
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	414 (2) Iterative refinement
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	415
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	416 Apply Reciproot iteration three times to y and multiply the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	417 result by x to get an approximation z that matches sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	418 to about 1 ulp. To be exact, we will have
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	419 -1ulp < sqrt(x)-z<1.0625ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	420
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	421 ... set rounding mode to Round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	422 y := y(1.5-0.5xyy) ... almost 15 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	423 y := y((1.5-2^-30)+0.5xyy)... about 29 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	424 ... special arrangement for better accuracy
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	425 z := xy ... 29 bits to sqrt(x), with zy<1
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	426 z := z + 0.5z(1-z*y) ... about 1 ulp to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	427
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	428 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	429 (a) the term z*y in the final iteration is always less than 1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	430 (b) the error in the final result is biased upward so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	431 -1 ulp < sqrt(x) - z < 1.0625 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	432 instead of \|sqrt(x)-z\|<1.03125ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	433
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	434 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	435
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	436 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	437 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	438 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	439 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	440 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	441 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	442 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	443 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	444
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	445 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	446 switch(r) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	447 case RN: ... round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	448 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	449 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	450 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	451 case RZ:case RM: ... round-to-zero or round-to--inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	452 R:=RP; ... reset rounding mod to round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	453 if(x<z*z ... rounded up) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	454 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	455 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	456 case RP: ... round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	457 if(x>(z+ulp)(z+ulp)...chopped) z = z+2ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	458 if(x>z*z ...chopped) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	459 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	460 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	461
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	462 Remark 3. The above comparisons can be done in fixed point. For
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	463 example, to compare x and w=z*z chopped, it suffices to compare
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	464 x1 and w1 (the trailing parts of x and w), regarding them as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	465 two's complement integers.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	466
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	467 ...Is z an exact square root?
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	468 To determine whether z is an exact square root of x, let z1 be the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	469 trailing part of z, and also let x0 and x1 be the leading and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	470 trailing parts of x.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	471
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	472 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	473 I := 1; ... Raise Inexact flag: z is not exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	474 else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	475 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	476 k := z1 >> 26; ... get z's 25-th and 26-th
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	477 fraction bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	478 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	479 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	480 R:= r ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	481 return sqrt(x):=z.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	482
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	483 If multiplication is cheaper then the foregoing red tape, the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	484 Inexact flag can be evaluated by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	485
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	486 I := i;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	487 I := (z*z!=x) or I.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	488
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	489 Note that z*z can overwrite I; this value must be sensed if it is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	490 True.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	491
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	492 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	493 zero.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	494
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	495 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	496 z1: \| f2 \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	497 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	498 bit 31 bit 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	499
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	500 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	501 or even of logb(x) have the following relations:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	502
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	503 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	504 bit 27,26 of z1 bit 1,0 of x1 logb(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	505 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	506 00 00 odd and even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	507 01 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	508 10 10 odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	509 10 00 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	510 11 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	511 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	512
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	513 (4) Special cases (see (4) of Section A).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	514
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	515 */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	516 /* vi: set ts=4 sw=4 expandtab: */

Mercurial > sdl-ios-xcode

annotate src/video/e_sqrt.h @ 2435:775e061e9440 gsoc2008_iphone