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annotate src/video/e_pow.h @ 1604:bfba1ea4452a

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author Sam Lantinga <slouken@libsdl.org>
date Thu, 23 Mar 2006 21:39:02 +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,
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parents:
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1 /* @(#)e_pow.c 5.1 93/09/24 */
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2 /*
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3 * ====================================================
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4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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5 *
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6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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7 * Permission to use, copy, modify, and distribute this
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8 * software is freely granted, provided that this notice
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9 * is preserved.
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10 * ====================================================
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11 */
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12
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13 #if defined(LIBM_SCCS) && !defined(lint)
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14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
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15 #endif
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16
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17 /* __ieee754_pow(x,y) return x**y
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18 *
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19 * n
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20 * Method: Let x = 2 * (1+f)
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21 * 1. Compute and return log2(x) in two pieces:
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22 * log2(x) = w1 + w2,
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23 * where w1 has 53-24 = 29 bit trailing zeros.
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24 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
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25 * arithmetic, where |y'|<=0.5.
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26 * 3. Return x**y = 2**n*exp(y'*log2)
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27 *
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28 * Special cases:
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29 * 1. (anything) ** 0 is 1
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30 * 2. (anything) ** 1 is itself
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31 * 3. (anything) ** NAN is NAN
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32 * 4. NAN ** (anything except 0) is NAN
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33 * 5. +-(|x| > 1) ** +INF is +INF
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34 * 6. +-(|x| > 1) ** -INF is +0
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35 * 7. +-(|x| < 1) ** +INF is +0
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36 * 8. +-(|x| < 1) ** -INF is +INF
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37 * 9. +-1 ** +-INF is NAN
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38 * 10. +0 ** (+anything except 0, NAN) is +0
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39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
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40 * 12. +0 ** (-anything except 0, NAN) is +INF
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41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
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42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
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43 * 15. +INF ** (+anything except 0,NAN) is +INF
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44 * 16. +INF ** (-anything except 0,NAN) is +0
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45 * 17. -INF ** (anything) = -0 ** (-anything)
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46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
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47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
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48 *
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49 * Accuracy:
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50 * pow(x,y) returns x**y nearly rounded. In particular
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51 * pow(integer,integer)
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52 * always returns the correct integer provided it is
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53 * representable.
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54 *
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55 * Constants :
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56 * The hexadecimal values are the intended ones for the following
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57 * constants. The decimal values may be used, provided that the
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58 * compiler will convert from decimal to binary accurately enough
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59 * to produce the hexadecimal values shown.
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60 */
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61
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62 /*#include "math.h"*/
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63 #include "math_private.h"
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64
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65 #ifdef __STDC__
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66 static const double
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67 #else
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68 static double
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69 #endif
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70 bp[] = {1.0, 1.5,},
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71 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
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72 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
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73 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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74 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
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75 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
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76 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
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77 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
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78 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
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79 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
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80 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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81 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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82 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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83 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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84 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
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85 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
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86 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
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87 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
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88 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
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89 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
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90 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
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91 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
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92 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
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93 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
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94 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
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95
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96 #ifdef __STDC__
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97 double __ieee754_pow(double x, double y)
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98 #else
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99 double __ieee754_pow(x,y)
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100 double x, y;
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101 #endif
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102 {
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103 double z,ax,z_h,z_l,p_h,p_l;
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104 double y1,t1,t2,r,s,t,u,v,w;
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105 int32_t i,j,k,yisint,n;
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106 int32_t hx,hy,ix,iy;
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107 u_int32_t lx,ly;
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108
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109 EXTRACT_WORDS(hx,lx,x);
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110 EXTRACT_WORDS(hy,ly,y);
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111 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
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112
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113 /* y==zero: x**0 = 1 */
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114 if((iy|ly)==0) return one;
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115
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116 /* +-NaN return x+y */
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117 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
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118 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
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119 return x+y;
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diff changeset
120
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diff changeset
121 /* determine if y is an odd int when x < 0
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diff changeset
122 * yisint = 0 ... y is not an integer
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diff changeset
123 * yisint = 1 ... y is an odd int
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diff changeset
124 * yisint = 2 ... y is an even int
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diff changeset
125 */
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diff changeset
126 yisint = 0;
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diff changeset
127 if(hx<0) {
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diff changeset
128 if(iy>=0x43400000) yisint = 2; /* even integer y */
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129 else if(iy>=0x3ff00000) {
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diff changeset
130 k = (iy>>20)-0x3ff; /* exponent */
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131 if(k>20) {
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diff changeset
132 j = ly>>(52-k);
1345
7f32b9bede06 Fixes for Visual C++ 6.0
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133 if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
1330
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diff changeset
134 } else if(ly==0) {
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135 j = iy>>(20-k);
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diff changeset
136 if((j<<(20-k))==iy) yisint = 2-(j&1);
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137 }
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diff changeset
138 }
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139 }
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diff changeset
140
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diff changeset
141 /* special value of y */
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diff changeset
142 if(ly==0) {
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diff changeset
143 if (iy==0x7ff00000) { /* y is +-inf */
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diff changeset
144 if(((ix-0x3ff00000)|lx)==0)
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diff changeset
145 return y - y; /* inf**+-1 is NaN */
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parents:
diff changeset
146 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
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diff changeset
147 return (hy>=0)? y: zero;
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diff changeset
148 else /* (|x|<1)**-,+inf = inf,0 */
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diff changeset
149 return (hy<0)?-y: zero;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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150 }
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diff changeset
151 if(iy==0x3ff00000) { /* y is +-1 */
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diff changeset
152 if(hy<0) return one/x; else return x;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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153 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
154 if(hy==0x40000000) return x*x; /* y is 2 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
155 if(hy==0x3fe00000) { /* y is 0.5 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
156 if(hx>=0) /* x >= +0 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
157 return __ieee754_sqrt(x);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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158 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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159 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
160
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
161 ax = x < 0 ? -x : x; /*fabs(x);*/
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diff changeset
162 /* special value of x */
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diff changeset
163 if(lx==0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
164 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
165 z = ax; /*x is +-0,+-inf,+-1*/
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parents:
diff changeset
166 if(hy<0) z = one/z; /* z = (1/|x|) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
167 if(hx<0) {
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diff changeset
168 if(((ix-0x3ff00000)|yisint)==0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
169 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
170 } else if(yisint==1)
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diff changeset
171 z = -z; /* (x<0)**odd = -(|x|**odd) */
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diff changeset
172 }
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diff changeset
173 return z;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
174 }
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diff changeset
175 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
176
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
177 /* (x<0)**(non-int) is NaN */
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diff changeset
178 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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diff changeset
179
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diff changeset
180 /* |y| is huge */
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diff changeset
181 if(iy>0x41e00000) { /* if |y| > 2**31 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
182 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
183 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
184 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
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185 }
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diff changeset
186 /* over/underflow if x is not close to one */
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diff changeset
187 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
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diff changeset
188 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
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diff changeset
189 /* now |1-x| is tiny <= 2**-20, suffice to compute
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parents:
diff changeset
190 log(x) by x-x^2/2+x^3/3-x^4/4 */
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diff changeset
191 t = x-1; /* t has 20 trailing zeros */
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diff changeset
192 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
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diff changeset
193 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
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diff changeset
194 v = t*ivln2_l-w*ivln2;
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diff changeset
195 t1 = u+v;
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196 SET_LOW_WORD(t1,0);
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197 t2 = v-(t1-u);
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diff changeset
198 } else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
199 double s2,s_h,s_l,t_h,t_l;
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diff changeset
200 n = 0;
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diff changeset
201 /* take care subnormal number */
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diff changeset
202 if(ix<0x00100000)
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diff changeset
203 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
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diff changeset
204 n += ((ix)>>20)-0x3ff;
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diff changeset
205 j = ix&0x000fffff;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
206 /* determine interval */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
207 ix = j|0x3ff00000; /* normalize ix */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
208 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
209 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
210 else {k=0;n+=1;ix -= 0x00100000;}
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diff changeset
211 SET_HIGH_WORD(ax,ix);
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diff changeset
212
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diff changeset
213 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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parents:
diff changeset
214 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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diff changeset
215 v = one/(ax+bp[k]);
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diff changeset
216 s = u*v;
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diff changeset
217 s_h = s;
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diff changeset
218 SET_LOW_WORD(s_h,0);
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diff changeset
219 /* t_h=ax+bp[k] High */
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diff changeset
220 t_h = zero;
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diff changeset
221 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
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diff changeset
222 t_l = ax - (t_h-bp[k]);
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diff changeset
223 s_l = v*((u-s_h*t_h)-s_h*t_l);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
224 /* compute log(ax) */
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diff changeset
225 s2 = s*s;
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diff changeset
226 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
227 r += s_l*(s_h+s);
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diff changeset
228 s2 = s_h*s_h;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
229 t_h = 3.0+s2+r;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
230 SET_LOW_WORD(t_h,0);
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diff changeset
231 t_l = r-((t_h-3.0)-s2);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
232 /* u+v = s*(1+...) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
233 u = s_h*t_h;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
234 v = s_l*t_h+t_l*s;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
235 /* 2/(3log2)*(s+...) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
236 p_h = u+v;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
237 SET_LOW_WORD(p_h,0);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
238 p_l = v-(p_h-u);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
239 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
240 z_l = cp_l*p_h+p_l*cp+dp_l[k];
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
241 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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 t = (double)n;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
243 t1 = (((z_h+z_l)+dp_h[k])+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
244 SET_LOW_WORD(t1,0);
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245 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
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246 }
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247
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248 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
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249 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
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250 s = -one;/* (-ve)**(odd int) */
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251
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252 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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253 y1 = y;
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254 SET_LOW_WORD(y1,0);
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255 p_l = (y-y1)*t1+y*t2;
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256 p_h = y1*t1;
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257 z = p_l+p_h;
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258 EXTRACT_WORDS(j,i,z);
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259 if (j>=0x40900000) { /* z >= 1024 */
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260 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
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261 return s*huge*huge; /* overflow */
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262 else {
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263 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
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264 }
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265 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
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266 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
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267 return s*tiny*tiny; /* underflow */
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268 else {
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269 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
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270 }
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271 }
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272 /*
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273 * compute 2**(p_h+p_l)
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274 */
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275 i = j&0x7fffffff;
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276 k = (i>>20)-0x3ff;
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277 n = 0;
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278 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
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279 n = j+(0x00100000>>(k+1));
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280 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
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281 t = zero;
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282 SET_HIGH_WORD(t,n&~(0x000fffff>>k));
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283 n = ((n&0x000fffff)|0x00100000)>>(20-k);
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284 if(j<0) n = -n;
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285 p_h -= t;
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286 }
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287 t = p_l+p_h;
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288 SET_LOW_WORD(t,0);
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289 u = t*lg2_h;
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290 v = (p_l-(t-p_h))*lg2+t*lg2_l;
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291 z = u+v;
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292 w = v-(z-u);
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293 t = z*z;
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294 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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295 r = (z*t1)/(t1-two)-(w+z*w);
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296 z = one-(r-z);
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297 GET_HIGH_WORD(j,z);
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298 j += (n<<20);
1424
7a610f25c12f Updated MacOS Classic MPW build
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299 if((j>>20)<=0) z = SDL_NAME(scalbn)(z,n); /* subnormal output */
1330
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300 else SET_HIGH_WORD(z,j);
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301 return s*z;
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302 }