1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
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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 SunSoft, 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 | * Optimized by Bruce D. Evans.
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13 | */
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14 | /* __rem_pio2(x,y)
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15 | *
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16 | * return the remainder of x rem pi/2 in y[0]+y[1]
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17 | * use __rem_pio2_large() for large x
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18 | */
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19 |
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20 | #include "libm.h"
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21 |
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22 | #if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
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23 | #define EPS DBL_EPSILON
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24 | #elif FLT_EVAL_METHOD==2
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25 | #define EPS LDBL_EPSILON
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26 | #endif
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27 |
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28 | /*
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29 | * invpio2: 53 bits of 2/pi
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30 | * pio2_1: first 33 bit of pi/2
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31 | * pio2_1t: pi/2 - pio2_1
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32 | * pio2_2: second 33 bit of pi/2
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33 | * pio2_2t: pi/2 - (pio2_1+pio2_2)
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34 | * pio2_3: third 33 bit of pi/2
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35 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
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36 | */
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37 | static const double
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38 | toint = 1.5/EPS,
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39 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
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40 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
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41 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
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42 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
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43 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
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44 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
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45 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
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46 |
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47 | /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
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48 | int __rem_pio2(double x, double *y)
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49 | {
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50 | union {double f; uint64_t i;} u = {x};
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51 | double_t z,w,t,r,fn;
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52 | double tx[3],ty[2];
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53 | uint32_t ix;
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54 | int sign, n, ex, ey, i;
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55 |
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56 | sign = u.i>>63;
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57 | ix = u.i>>32 & 0x7fffffff;
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58 | if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
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59 | if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
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60 | goto medium; /* cancellation -- use medium case */
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61 | if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
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62 | if (!sign) {
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63 | z = x - pio2_1; /* one round good to 85 bits */
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64 | y[0] = z - pio2_1t;
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65 | y[1] = (z-y[0]) - pio2_1t;
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66 | return 1;
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67 | } else {
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68 | z = x + pio2_1;
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69 | y[0] = z + pio2_1t;
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70 | y[1] = (z-y[0]) + pio2_1t;
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71 | return -1;
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72 | }
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73 | } else {
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74 | if (!sign) {
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75 | z = x - 2*pio2_1;
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76 | y[0] = z - 2*pio2_1t;
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77 | y[1] = (z-y[0]) - 2*pio2_1t;
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78 | return 2;
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79 | } else {
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80 | z = x + 2*pio2_1;
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81 | y[0] = z + 2*pio2_1t;
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82 | y[1] = (z-y[0]) + 2*pio2_1t;
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83 | return -2;
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84 | }
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85 | }
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86 | }
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87 | if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
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88 | if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
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89 | if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
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90 | goto medium;
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91 | if (!sign) {
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92 | z = x - 3*pio2_1;
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93 | y[0] = z - 3*pio2_1t;
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94 | y[1] = (z-y[0]) - 3*pio2_1t;
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95 | return 3;
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96 | } else {
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97 | z = x + 3*pio2_1;
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98 | y[0] = z + 3*pio2_1t;
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99 | y[1] = (z-y[0]) + 3*pio2_1t;
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100 | return -3;
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101 | }
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102 | } else {
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103 | if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
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104 | goto medium;
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105 | if (!sign) {
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106 | z = x - 4*pio2_1;
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107 | y[0] = z - 4*pio2_1t;
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108 | y[1] = (z-y[0]) - 4*pio2_1t;
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109 | return 4;
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110 | } else {
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111 | z = x + 4*pio2_1;
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112 | y[0] = z + 4*pio2_1t;
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113 | y[1] = (z-y[0]) + 4*pio2_1t;
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114 | return -4;
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115 | }
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116 | }
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117 | }
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118 | if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
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119 | medium:
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120 | /* rint(x/(pi/2)), Assume round-to-nearest. */
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121 | fn = (double_t)x*invpio2 + toint - toint;
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122 | n = (int32_t)fn;
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123 | r = x - fn*pio2_1;
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124 | w = fn*pio2_1t; /* 1st round, good to 85 bits */
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125 | y[0] = r - w;
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126 | u.f = y[0];
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127 | ey = u.i>>52 & 0x7ff;
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128 | ex = ix>>20;
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129 | if (ex - ey > 16) { /* 2nd round, good to 118 bits */
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130 | t = r;
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131 | w = fn*pio2_2;
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132 | r = t - w;
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133 | w = fn*pio2_2t - ((t-r)-w);
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134 | y[0] = r - w;
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135 | u.f = y[0];
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136 | ey = u.i>>52 & 0x7ff;
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137 | if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
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138 | t = r;
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139 | w = fn*pio2_3;
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140 | r = t - w;
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141 | w = fn*pio2_3t - ((t-r)-w);
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142 | y[0] = r - w;
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143 | }
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144 | }
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145 | y[1] = (r - y[0]) - w;
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146 | return n;
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147 | }
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148 | /*
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149 | * all other (large) arguments
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150 | */
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151 | if (ix >= 0x7ff00000) { /* x is inf or NaN */
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152 | y[0] = y[1] = x - x;
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153 | return 0;
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154 | }
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155 | /* set z = scalbn(|x|,-ilogb(x)+23) */
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156 | u.f = x;
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157 | u.i &= (uint64_t)-1>>12;
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158 | u.i |= (uint64_t)(0x3ff + 23)<<52;
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159 | z = u.f;
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160 | for (i=0; i < 2; i++) {
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161 | tx[i] = (double)(int32_t)z;
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162 | z = (z-tx[i])*0x1p24;
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163 | }
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164 | tx[i] = z;
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165 | /* skip zero terms, first term is non-zero */
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166 | while (tx[i] == 0.0)
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167 | i--;
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168 | n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
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169 | if (sign) {
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170 | y[0] = -ty[0];
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171 | y[1] = -ty[1];
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172 | return -n;
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173 | }
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174 | y[0] = ty[0];
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175 | y[1] = ty[1];
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176 | return n;
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177 | }
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