sdl-ios-xcode: src/video/e

annotate src/video/e_sqrt.h @ 4301:5007e6a66814 SDL-1.2

Debian patch: 209_alsa_priority.diff

author	Sam Lantinga <slouken@libsdl.org>
date	Sat, 10 Oct 2009 09:07:51 +0000
parents	7a610f25c12f
children	782fd950bd46 c121d94672cb

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__
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 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
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	93 double SDL_NAME(copysign)(x,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	94 double 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	95 #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	96 {
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 u_int32_t hx,hy;
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 GET_HIGH_WORD(hx,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	99 GET_HIGH_WORD(hy,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	100 SET_HIGH_WORD(x,(hx&0x7fffffff)\|(hy&0x80000000));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	101 return 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	102 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	103
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 #ifdef __STDC__
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	105 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	106 #else
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	107 double SDL_NAME(scalbn) (x,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 double x; int n;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	109 #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	110 {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	111 int32_t k,hx,lx;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	112 EXTRACT_WORDS(hx,lx,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	113 k = (hx&0x7ff00000)>>20; /* extract exponent */
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 if (k==0) { /* 0 or subnormal 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	115 if ((lx\|(hx&0x7fffffff))==0) return x; /* +-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	116 x *= two54;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	117 GET_HIGH_WORD(hx,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	118 k = ((hx&0x7ff00000)>>20) - 54;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	119 if (n< -50000) return tinyx; /underflow*/
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	120 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	121 if (k==0x7ff) return x+x; /* NaN or 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	122 k = k+n;
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	123 if (k > 0x7fe) return hugeSDL_NAME(copysign)(huge,x); / overflow */
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	124 if (k > 0) /* normal result */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	125 {SET_HIGH_WORD(x,(hx&0x800fffff)\|(k<<20)); return 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	126 if (k <= -54) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	127 if (n > 50000) /* in case integer overflow in n+k */
1424 7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	128 return hugeSDL_NAME(copysign)(huge,x); /overflow*/
7a610f25c12f Updated MacOS Classic MPW build Sam Lantinga <slouken@libsdl.org> parents: 1345 diff changeset	129 else return tinySDL_NAME(copysign)(tiny,x); /underflow*/
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	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	131 k += 54; /* subnormal result */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	132 SET_HIGH_WORD(x,(hx&0x800fffff)\|(k<<20));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	133 return x*twom54;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	134 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	135
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	136 #ifdef __STDC__
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	137 double __ieee754_sqrt(double 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	138 #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	139 double __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	140 double 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	141 #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	142 {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	143 double 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	144 int32_t sign = (int)0x80000000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	145 int32_t ix0,s0,q,m,t,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	146 u_int32_t r,t1,s1,ix1,q1;
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 EXTRACT_WORDS(ix0,ix1,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	149
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	150 /* take care of Inf and 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	151 if((ix0&0x7ff00000)==0x7ff00000) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	152 return xx+x; / sqrt(NaN)=NaN, 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	153 sqrt(-inf)=sNaN */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	154 }
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 /* take care of 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	156 if(ix0<=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	157 if(((ix0&(~sign))\|ix1)==0) return x;/* sqrt(+-0) = +-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	158 else if(ix0<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	159 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	160 }
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 /* normalize 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	162 m = (ix0>>20);
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 if(m==0) { /* subnormal 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	164 while(ix0==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	165 m -= 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	166 ix0 \|= (ix1>>11); ix1 <<= 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	167 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	168 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=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	169 m -= 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	170 ix0 \|= (ix1>>(32-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	171 ix1 <<= 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	172 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	173 m -= 1023; /* unbias exponent */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	174 ix0 = (ix0&0x000fffff)\|0x00100000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	175 if(m&1){ /* odd m, double x to make it 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	176 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	177 ix1 += ix1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	178 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	179 m >>= 1; /* m = [m/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	180
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	181 /* generate sqrt(x) bit by 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	182 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	183 ix1 += ix1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	184 q = q1 = s0 = s1 = 0; /* [q,q1] = 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	185 r = 0x00200000; /* r = moving bit from right to left */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	186
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	187 while(r!=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	188 t = s0+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	189 if(t<=ix0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	190 s0 = t+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	191 ix0 -= t;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	192 q += 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	193 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	194 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	195 ix1 += ix1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	196 r>>=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	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
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	199 r = sign;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	200 while(r!=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	201 t1 = s1+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	202 t = s0;
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 if((t<ix0)\|\|((t==ix0)&&(t1<=ix1))) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	204 s1 = t1+r;
1345 7f32b9bede06 Fixes for Visual C++ 6.0 Sam Lantinga <slouken@libsdl.org> parents: 1330 diff changeset	205 if(((int32_t)(t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
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	206 ix0 -= t;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	207 if (ix1 < t1) ix0 -= 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	208 ix1 -= 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	209 q1 += 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	210 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	211 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	212 ix1 += ix1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	213 r>>=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	214 }
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
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	216 /* use floating add to find out rounding direction */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	217 if((ix0\|ix1)!=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	218 z = one-tiny; /* trigger 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	219 if (z>=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	220 z = one+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	221 if (q1==(u_int32_t)0xffffffff) { q1=0; 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	222 else if (z>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	223 if (q1==(u_int32_t)0xfffffffe) 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	224 q1+=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	225 } 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	226 q1 += (q1&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	227 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	228 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	229 ix0 = (q>>1)+0x3fe00000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	230 ix1 = q1>>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	231 if ((q&1)==1) ix1 \|= sign;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	232 ix0 += (m <<20);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	233 INSERT_WORDS(z,ix0,ix1);
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 return 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	235 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	236
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	237 /*
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	238 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	239 -------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	240 (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	241 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	242
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	243 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	244 (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	245 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	246 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	247 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	248 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	249 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	250 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	251 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	252 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	253 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	254 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	255
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	256 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	257
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 (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	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 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	261 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	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 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	264 ------------------------------------------------------
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 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	266 ------------------------------------------------------
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 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	268
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
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 ------------------------ ------------------------
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 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	272 ------------------------ ------------------------
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
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 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	275 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	276 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	277
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 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	279 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	280 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	281 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	282 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	283 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	284
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 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	286 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	287 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	288 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	289 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	290 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	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 (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	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 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	295 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	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 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	298 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	299 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	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
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 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	303 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	304
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 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	306 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	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 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	309 (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	310 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	311 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	312 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	313
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 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	316 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	317 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	318 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	319 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	320 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	321
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 (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	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 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	325 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	326 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	327 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	328 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	329 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	330 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	331 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	332
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 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	334 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	335 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	336 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	337 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	338 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	339 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	340 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	341 } 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	342 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	343 }
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 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	346 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	347 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	348 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	349 }
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 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	351 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	352 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	353 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	354 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	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 (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	357
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 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	359 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	360
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
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 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	363
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 (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	365
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 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	367 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	368 (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	369 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	370
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 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	372 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	373
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 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	375 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	376 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	377 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	378 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	379
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 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	381 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	382 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	383 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	384 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	385 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	386 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	387 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	388 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	389 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	390
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 (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	392
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 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	394 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	395 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	396 -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	397
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 ... 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	399 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	400 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	401 ... 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	402 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	403 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	404
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 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	406 (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	407 (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	408 -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	409 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	410
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 (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	412
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 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	414 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	415 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	416 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	417 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	418 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	419 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	420 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	421
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 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	423 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	424 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	425 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	426 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	427 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	428 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	429 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	430 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	431 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	432 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	433 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	434 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	435 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	436 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	437 }
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
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 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	440 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	441 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	442 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	443
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 ...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	445 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	446 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	447 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	448
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 ((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	450 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	451 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	452 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	453 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	454 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	455 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	456 }
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 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	458 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	459
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 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	461 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	462
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 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	464 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	465
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 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	467 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	468
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 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	470 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	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 --------------------
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 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	474 --------------------
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 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	476
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 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	478 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	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 -------------------------------------------------
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 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	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 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	484 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	485 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	486 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	487 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	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
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 (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	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 */
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

Mercurial > sdl-ios-xcode

annotate src/video/e_sqrt.h @ 4301:5007e6a66814 SDL-1.2