1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */
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2 | /*-
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3 | * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
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4 | * All rights reserved.
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5 | *
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6 | * Redistribution and use in source and binary forms, with or without
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7 | * modification, are permitted provided that the following conditions
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8 | * are met:
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9 | * 1. Redistributions of source code must retain the above copyright
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10 | * notice, this list of conditions and the following disclaimer.
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11 | * 2. Redistributions in binary form must reproduce the above copyright
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12 | * notice, this list of conditions and the following disclaimer in the
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13 | * documentation and/or other materials provided with the distribution.
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14 | *
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15 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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16 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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17 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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18 | * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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19 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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20 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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21 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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22 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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23 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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24 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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25 | * SUCH DAMAGE.
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26 | */
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27 |
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28 |
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29 | #include "libm.h"
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30 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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31 | long double fmal(long double x, long double y, long double z)
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32 | {
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33 | return fma(x, y, z);
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34 | }
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35 | #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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36 | #include <fenv.h>
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37 | #if LDBL_MANT_DIG == 64
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38 | #define LASTBIT(u) (u.i.m & 1)
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39 | #define SPLIT (0x1p32L + 1)
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40 | #elif LDBL_MANT_DIG == 113
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41 | #define LASTBIT(u) (u.i.lo & 1)
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42 | #define SPLIT (0x1p57L + 1)
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43 | #endif
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44 |
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45 | /*
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46 | * A struct dd represents a floating-point number with twice the precision
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47 | * of a long double. We maintain the invariant that "hi" stores the high-order
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48 | * bits of the result.
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49 | */
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50 | struct dd {
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51 | long double hi;
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52 | long double lo;
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53 | };
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54 |
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55 | /*
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56 | * Compute a+b exactly, returning the exact result in a struct dd. We assume
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57 | * that both a and b are finite, but make no assumptions about their relative
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58 | * magnitudes.
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59 | */
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60 | static inline struct dd dd_add(long double a, long double b)
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61 | {
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62 | struct dd ret;
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63 | long double s;
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64 |
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65 | ret.hi = a + b;
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66 | s = ret.hi - a;
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67 | ret.lo = (a - (ret.hi - s)) + (b - s);
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68 | return (ret);
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69 | }
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70 |
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71 | /*
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72 | * Compute a+b, with a small tweak: The least significant bit of the
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73 | * result is adjusted into a sticky bit summarizing all the bits that
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74 | * were lost to rounding. This adjustment negates the effects of double
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75 | * rounding when the result is added to another number with a higher
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76 | * exponent. For an explanation of round and sticky bits, see any reference
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77 | * on FPU design, e.g.,
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78 | *
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79 | * J. Coonen. An Implementation Guide to a Proposed Standard for
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80 | * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
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81 | */
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82 | static inline long double add_adjusted(long double a, long double b)
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83 | {
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84 | struct dd sum;
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85 | union ldshape u;
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86 |
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87 | sum = dd_add(a, b);
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88 | if (sum.lo != 0) {
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89 | u.f = sum.hi;
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90 | if (!LASTBIT(u))
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91 | sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
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92 | }
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93 | return (sum.hi);
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94 | }
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95 |
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96 | /*
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97 | * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
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98 | * that the result will be subnormal, and care is taken to ensure that
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99 | * double rounding does not occur.
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100 | */
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101 | static inline long double add_and_denormalize(long double a, long double b, int scale)
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102 | {
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103 | struct dd sum;
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104 | int bits_lost;
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105 | union ldshape u;
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106 |
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107 | sum = dd_add(a, b);
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108 |
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109 | /*
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110 | * If we are losing at least two bits of accuracy to denormalization,
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111 | * then the first lost bit becomes a round bit, and we adjust the
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112 | * lowest bit of sum.hi to make it a sticky bit summarizing all the
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113 | * bits in sum.lo. With the sticky bit adjusted, the hardware will
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114 | * break any ties in the correct direction.
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115 | *
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116 | * If we are losing only one bit to denormalization, however, we must
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117 | * break the ties manually.
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118 | */
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119 | if (sum.lo != 0) {
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120 | u.f = sum.hi;
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121 | bits_lost = -u.i.se - scale + 1;
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122 | if ((bits_lost != 1) ^ LASTBIT(u))
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123 | sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
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124 | }
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125 | return scalbnl(sum.hi, scale);
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126 | }
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127 |
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128 | /*
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129 | * Compute a*b exactly, returning the exact result in a struct dd. We assume
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130 | * that both a and b are normalized, so no underflow or overflow will occur.
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131 | * The current rounding mode must be round-to-nearest.
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132 | */
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133 | static inline struct dd dd_mul(long double a, long double b)
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134 | {
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135 | struct dd ret;
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136 | long double ha, hb, la, lb, p, q;
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137 |
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138 | p = a * SPLIT;
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139 | ha = a - p;
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140 | ha += p;
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141 | la = a - ha;
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142 |
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143 | p = b * SPLIT;
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144 | hb = b - p;
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145 | hb += p;
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146 | lb = b - hb;
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147 |
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148 | p = ha * hb;
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149 | q = ha * lb + la * hb;
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150 |
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151 | ret.hi = p + q;
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152 | ret.lo = p - ret.hi + q + la * lb;
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153 | return (ret);
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154 | }
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155 |
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156 | /*
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157 | * Fused multiply-add: Compute x * y + z with a single rounding error.
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158 | *
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159 | * We use scaling to avoid overflow/underflow, along with the
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160 | * canonical precision-doubling technique adapted from:
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161 | *
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162 | * Dekker, T. A Floating-Point Technique for Extending the
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163 | * Available Precision. Numer. Math. 18, 224-242 (1971).
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164 | */
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165 | long double fmal(long double x, long double y, long double z)
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166 | {
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167 | #pragma STDC FENV_ACCESS ON
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168 | long double xs, ys, zs, adj;
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169 | struct dd xy, r;
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170 | int oround;
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171 | int ex, ey, ez;
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172 | int spread;
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173 |
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174 | /*
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175 | * Handle special cases. The order of operations and the particular
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176 | * return values here are crucial in handling special cases involving
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177 | * infinities, NaNs, overflows, and signed zeroes correctly.
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178 | */
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179 | if (!isfinite(x) || !isfinite(y))
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180 | return (x * y + z);
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181 | if (!isfinite(z))
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182 | return (z);
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183 | if (x == 0.0 || y == 0.0)
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184 | return (x * y + z);
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185 | if (z == 0.0)
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186 | return (x * y);
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187 |
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188 | xs = frexpl(x, &ex);
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189 | ys = frexpl(y, &ey);
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190 | zs = frexpl(z, &ez);
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191 | oround = fegetround();
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192 | spread = ex + ey - ez;
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193 |
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194 | /*
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195 | * If x * y and z are many orders of magnitude apart, the scaling
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196 | * will overflow, so we handle these cases specially. Rounding
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197 | * modes other than FE_TONEAREST are painful.
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198 | */
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199 | if (spread < -LDBL_MANT_DIG) {
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200 | #ifdef FE_INEXACT
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201 | feraiseexcept(FE_INEXACT);
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202 | #endif
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203 | #ifdef FE_UNDERFLOW
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204 | if (!isnormal(z))
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205 | feraiseexcept(FE_UNDERFLOW);
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206 | #endif
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207 | switch (oround) {
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208 | default: /* FE_TONEAREST */
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209 | return (z);
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210 | #ifdef FE_TOWARDZERO
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211 | case FE_TOWARDZERO:
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212 | if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
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213 | return (z);
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214 | else
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215 | return (nextafterl(z, 0));
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216 | #endif
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217 | #ifdef FE_DOWNWARD
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218 | case FE_DOWNWARD:
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219 | if (x > 0.0 ^ y < 0.0)
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220 | return (z);
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221 | else
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222 | return (nextafterl(z, -INFINITY));
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223 | #endif
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224 | #ifdef FE_UPWARD
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225 | case FE_UPWARD:
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226 | if (x > 0.0 ^ y < 0.0)
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227 | return (nextafterl(z, INFINITY));
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228 | else
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229 | return (z);
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230 | #endif
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231 | }
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232 | }
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233 | if (spread <= LDBL_MANT_DIG * 2)
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234 | zs = scalbnl(zs, -spread);
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235 | else
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236 | zs = copysignl(LDBL_MIN, zs);
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237 |
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238 | fesetround(FE_TONEAREST);
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239 |
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240 | /*
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241 | * Basic approach for round-to-nearest:
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242 | *
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243 | * (xy.hi, xy.lo) = x * y (exact)
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244 | * (r.hi, r.lo) = xy.hi + z (exact)
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245 | * adj = xy.lo + r.lo (inexact; low bit is sticky)
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246 | * result = r.hi + adj (correctly rounded)
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247 | */
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248 | xy = dd_mul(xs, ys);
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249 | r = dd_add(xy.hi, zs);
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250 |
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251 | spread = ex + ey;
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252 |
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253 | if (r.hi == 0.0) {
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254 | /*
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255 | * When the addends cancel to 0, ensure that the result has
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256 | * the correct sign.
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257 | */
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258 | fesetround(oround);
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259 | volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
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260 | return xy.hi + vzs + scalbnl(xy.lo, spread);
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261 | }
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262 |
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263 | if (oround != FE_TONEAREST) {
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264 | /*
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265 | * There is no need to worry about double rounding in directed
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266 | * rounding modes.
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267 | * But underflow may not be raised correctly, example in downward rounding:
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268 | * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L)
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269 | */
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270 | long double ret;
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271 | #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
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272 | int e = fetestexcept(FE_INEXACT);
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273 | feclearexcept(FE_INEXACT);
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274 | #endif
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275 | fesetround(oround);
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276 | adj = r.lo + xy.lo;
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277 | ret = scalbnl(r.hi + adj, spread);
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278 | #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
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279 | if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT))
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280 | feraiseexcept(FE_UNDERFLOW);
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281 | else if (e)
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282 | feraiseexcept(FE_INEXACT);
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283 | #endif
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284 | return ret;
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285 | }
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286 |
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287 | adj = add_adjusted(r.lo, xy.lo);
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288 | if (spread + ilogbl(r.hi) > -16383)
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289 | return scalbnl(r.hi + adj, spread);
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290 | else
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291 | return add_and_denormalize(r.hi, adj, spread);
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292 | }
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293 | #endif
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