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| 1 | | /* |
| 2 | | * Double-precision x^y function. |
| 3 | | * |
| 4 | | * Copyright (c) 2018, Arm Limited. |
| 5 | | * SPDX-License-Identifier: MIT |
| 6 | | */ |
| 7 | | |
| 8 | | #include <math.h> |
| 9 | | #include <stdint.h> |
| 10 | | #include "libm.h" |
| 11 | | #include "exp_data.h" |
| 12 | | #include "pow_data.h" |
| 13 | | |
| 14 | | /* |
| 15 | | Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53) |
| 16 | | relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma) |
| 17 | | ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma) |
| 18 | | */ |
| 19 | | |
| 20 | | #define T __pow_log_data.tab |
| 21 | | #define A __pow_log_data.poly |
| 22 | | #define Ln2hi __pow_log_data.ln2hi |
| 23 | | #define Ln2lo __pow_log_data.ln2lo |
| 24 | | #define N (1 << POW_LOG_TABLE_BITS) |
| 25 | | #define OFF 0x3fe6955500000000 |
| 26 | | |
| 27 | | /* Top 12 bits of a double (sign and exponent bits). */ |
| 28 | | static inline uint32_t top12(double x) |
| 29 | | { |
| 30 | | return asuint64(x) >> 52; |
| 31 | | } |
| 32 | | |
| 33 | | /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about |
| 34 | | additional 15 bits precision. IX is the bit representation of x, but |
| 35 | | normalized in the subnormal range using the sign bit for the exponent. */ |
| 36 | | static inline double_t log_inline(uint64_t ix, double_t *tail) |
| 37 | | { |
| 38 | | /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
| 39 | | double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; |
| 40 | | uint64_t iz, tmp; |
| 41 | | int k, i; |
| 42 | | |
| 43 | | /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. |
| 44 | | The range is split into N subintervals. |
| 45 | | The ith subinterval contains z and c is near its center. */ |
| 46 | | tmp = ix - OFF; |
| 47 | | i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; |
| 48 | | k = (int64_t)tmp >> 52; /* arithmetic shift */ |
| 49 | | iz = ix - (tmp & 0xfffULL << 52); |
| 50 | | z = asdouble(iz); |
| 51 | | kd = (double_t)k; |
| 52 | | |
| 53 | | /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ |
| 54 | | invc = T[i].invc; |
| 55 | | logc = T[i].logc; |
| 56 | | logctail = T[i].logctail; |
| 57 | | |
| 58 | | /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and |
| 59 | | |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */ |
| 60 | | #if __FP_FAST_FMA |
| 61 | | r = __builtin_fma(z, invc, -1.0); |
| 62 | | #else |
| 63 | | /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ |
| 64 | | double_t zhi = asdouble((iz + (1ULL << 31)) & (-1ULL << 32)); |
| 65 | | double_t zlo = z - zhi; |
| 66 | | double_t rhi = zhi * invc - 1.0; |
| 67 | | double_t rlo = zlo * invc; |
| 68 | | r = rhi + rlo; |
| 69 | | #endif |
| 70 | | |
| 71 | | /* k*Ln2 + log(c) + r. */ |
| 72 | | t1 = kd * Ln2hi + logc; |
| 73 | | t2 = t1 + r; |
| 74 | | lo1 = kd * Ln2lo + logctail; |
| 75 | | lo2 = t1 - t2 + r; |
| 76 | | |
| 77 | | /* Evaluation is optimized assuming superscalar pipelined execution. */ |
| 78 | | double_t ar, ar2, ar3, lo3, lo4; |
| 79 | | ar = A[0] * r; /* A[0] = -0.5. */ |
| 80 | | ar2 = r * ar; |
| 81 | | ar3 = r * ar2; |
| 82 | | /* k*Ln2 + log(c) + r + A[0]*r*r. */ |
| 83 | | #if __FP_FAST_FMA |
| 84 | | hi = t2 + ar2; |
| 85 | | lo3 = __builtin_fma(ar, r, -ar2); |
| 86 | | lo4 = t2 - hi + ar2; |
| 87 | | #else |
| 88 | | double_t arhi = A[0] * rhi; |
| 89 | | double_t arhi2 = rhi * arhi; |
| 90 | | hi = t2 + arhi2; |
| 91 | | lo3 = rlo * (ar + arhi); |
| 92 | | lo4 = t2 - hi + arhi2; |
| 93 | | #endif |
| 94 | | /* p = log1p(r) - r - A[0]*r*r. */ |
| 95 | | p = (ar3 * (A[1] + r * A[2] + |
| 96 | | ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); |
| 97 | | lo = lo1 + lo2 + lo3 + lo4 + p; |
| 98 | | y = hi + lo; |
| 99 | | *tail = hi - y + lo; |
| 100 | | return y; |
| 101 | | } |
| 102 | | |
| 103 | | #undef N |
| 104 | | #undef T |
| 105 | | #define N (1 << EXP_TABLE_BITS) |
| 106 | | #define InvLn2N __exp_data.invln2N |
| 107 | | #define NegLn2hiN __exp_data.negln2hiN |
| 108 | | #define NegLn2loN __exp_data.negln2loN |
| 109 | | #define Shift __exp_data.shift |
| 110 | | #define T __exp_data.tab |
| 111 | | #define C2 __exp_data.poly[5 - EXP_POLY_ORDER] |
| 112 | | #define C3 __exp_data.poly[6 - EXP_POLY_ORDER] |
| 113 | | #define C4 __exp_data.poly[7 - EXP_POLY_ORDER] |
| 114 | | #define C5 __exp_data.poly[8 - EXP_POLY_ORDER] |
| 115 | | #define C6 __exp_data.poly[9 - EXP_POLY_ORDER] |
| 116 | | |
| 117 | | /* Handle cases that may overflow or underflow when computing the result that |
| 118 | | is scale*(1+TMP) without intermediate rounding. The bit representation of |
| 119 | | scale is in SBITS, however it has a computed exponent that may have |
| 120 | | overflown into the sign bit so that needs to be adjusted before using it as |
| 121 | | a double. (int32_t)KI is the k used in the argument reduction and exponent |
| 122 | | adjustment of scale, positive k here means the result may overflow and |
| 123 | | negative k means the result may underflow. */ |
| 124 | | static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki) |
| 125 | | { |
| 126 | | double_t scale, y; |
| 127 | | |
| 128 | | if ((ki & 0x80000000) == 0) { |
| 129 | | /* k > 0, the exponent of scale might have overflowed by <= 460. */ |
| 130 | | sbits -= 1009ull << 52; |
| 131 | | scale = asdouble(sbits); |
| 132 | | y = 0x1p1009 * (scale + scale * tmp); |
| 133 | | return eval_as_double(y); |
| 134 | | } |
| 135 | | /* k < 0, need special care in the subnormal range. */ |
| 136 | | sbits += 1022ull << 52; |
| 137 | | /* Note: sbits is signed scale. */ |
| 138 | | scale = asdouble(sbits); |
| 139 | | y = scale + scale * tmp; |
| 140 | | if (fabs(y) < 1.0) { |
| 141 | | /* Round y to the right precision before scaling it into the subnormal |
| 142 | | range to avoid double rounding that can cause 0.5+E/2 ulp error where |
| 143 | | E is the worst-case ulp error outside the subnormal range. So this |
| 144 | | is only useful if the goal is better than 1 ulp worst-case error. */ |
| 145 | | double_t hi, lo, one = 1.0; |
| 146 | | if (y < 0.0) |
| 147 | | one = -1.0; |
| 148 | | lo = scale - y + scale * tmp; |
| 149 | | hi = one + y; |
| 150 | | lo = one - hi + y + lo; |
| 151 | | y = eval_as_double(hi + lo) - one; |
| 152 | | /* Fix the sign of 0. */ |
| 153 | | if (y == 0.0) |
| 154 | | y = asdouble(sbits & 0x8000000000000000); |
| 155 | | /* The underflow exception needs to be signaled explicitly. */ |
| 156 | | fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); |
| 157 | | } |
| 158 | | y = 0x1p-1022 * y; |
| 159 | | return eval_as_double(y); |
| 160 | | } |
| 161 | | |
| 162 | | #define SIGN_BIAS (0x800 << EXP_TABLE_BITS) |
| 163 | | |
| 164 | | /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. |
| 165 | | The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ |
| 166 | | static inline double exp_inline(double_t x, double_t xtail, uint32_t sign_bias) |
| 167 | | { |
| 168 | | uint32_t abstop; |
| 169 | | uint64_t ki, idx, top, sbits; |
| 170 | | /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
| 171 | | double_t kd, z, r, r2, scale, tail, tmp; |
| 172 | | |
| 173 | | abstop = top12(x) & 0x7ff; |
| 174 | | if (predict_false(abstop - top12(0x1p-54) >= |
| 175 | | top12(512.0) - top12(0x1p-54))) { |
| 176 | | if (abstop - top12(0x1p-54) >= 0x80000000) { |
| 177 | | /* Avoid spurious underflow for tiny x. */ |
| 178 | | /* Note: 0 is common input. */ |
| 179 | | double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; |
| 180 | | return sign_bias ? -one : one; |
| 181 | | } |
| 182 | | if (abstop >= top12(1024.0)) { |
| 183 | | /* Note: inf and nan are already handled. */ |
| 184 | | if (asuint64(x) >> 63) |
| 185 | | return __math_uflow(sign_bias); |
| 186 | | else |
| 187 | | return __math_oflow(sign_bias); |
| 188 | | } |
| 189 | | /* Large x is special cased below. */ |
| 190 | | abstop = 0; |
| 191 | | } |
| 192 | | |
| 193 | | /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ |
| 194 | | /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ |
| 195 | | z = InvLn2N * x; |
| 196 | | #if TOINT_INTRINSICS |
| 197 | | kd = roundtoint(z); |
| 198 | | ki = converttoint(z); |
| 199 | | #elif EXP_USE_TOINT_NARROW |
| 200 | | /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ |
| 201 | | kd = eval_as_double(z + Shift); |
| 202 | | ki = asuint64(kd) >> 16; |
| 203 | | kd = (double_t)(int32_t)ki; |
| 204 | | #else |
| 205 | | /* z - kd is in [-1, 1] in non-nearest rounding modes. */ |
| 206 | | kd = eval_as_double(z + Shift); |
| 207 | | ki = asuint64(kd); |
| 208 | | kd -= Shift; |
| 209 | | #endif |
| 210 | | r = x + kd * NegLn2hiN + kd * NegLn2loN; |
| 211 | | /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ |
| 212 | | r += xtail; |
| 213 | | /* 2^(k/N) ~= scale * (1 + tail). */ |
| 214 | | idx = 2 * (ki % N); |
| 215 | | top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); |
| 216 | | tail = asdouble(T[idx]); |
| 217 | | /* This is only a valid scale when -1023*N < k < 1024*N. */ |
| 218 | | sbits = T[idx + 1] + top; |
| 219 | | /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ |
| 220 | | /* Evaluation is optimized assuming superscalar pipelined execution. */ |
| 221 | | r2 = r * r; |
| 222 | | /* Without fma the worst case error is 0.25/N ulp larger. */ |
| 223 | | /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ |
| 224 | | tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); |
| 225 | | if (predict_false(abstop == 0)) |
| 226 | | return specialcase(tmp, sbits, ki); |
| 227 | | scale = asdouble(sbits); |
| 228 | | /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there |
| 229 | | is no spurious underflow here even without fma. */ |
| 230 | | return eval_as_double(scale + scale * tmp); |
| 231 | | } |
| 232 | | |
| 233 | | /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is |
| 234 | | the bit representation of a non-zero finite floating-point value. */ |
| 235 | | static inline int checkint(uint64_t iy) |
| 236 | | { |
| 237 | | int e = iy >> 52 & 0x7ff; |
| 238 | | if (e < 0x3ff) |
| 239 | | return 0; |
| 240 | | if (e > 0x3ff + 52) |
| 241 | | return 2; |
| 242 | | if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) |
| 243 | | return 0; |
| 244 | | if (iy & (1ULL << (0x3ff + 52 - e))) |
| 245 | | return 1; |
| 246 | | return 2; |
| 247 | | } |
| 248 | | |
| 249 | | /* Returns 1 if input is the bit representation of 0, infinity or nan. */ |
| 250 | | static inline int zeroinfnan(uint64_t i) |
| 251 | | { |
| 252 | | return 2 * i - 1 >= 2 * asuint64(INFINITY) - 1; |
| 253 | | } |
| 254 | | |
| 255 | | double pow(double x, double y) |
| 256 | | { |
| 257 | | uint32_t sign_bias = 0; |
| 258 | | uint64_t ix, iy; |
| 259 | | uint32_t topx, topy; |
| 260 | | |
| 261 | | ix = asuint64(x); |
| 262 | | iy = asuint64(y); |
| 263 | | topx = top12(x); |
| 264 | | topy = top12(y); |
| 265 | | if (predict_false(topx - 0x001 >= 0x7ff - 0x001 || |
| 266 | | (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) { |
| 267 | | /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 |
| 268 | | and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ |
| 269 | | /* Special cases: (x < 0x1p-126 or inf or nan) or |
| 270 | | (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ |
| 271 | | if (predict_false(zeroinfnan(iy))) { |
| 272 | | if (2 * iy == 0) |
| 273 | | return issignaling_inline(x) ? x + y : 1.0; |
| 274 | | if (ix == asuint64(1.0)) |
| 275 | | return issignaling_inline(y) ? x + y : 1.0; |
| 276 | | if (2 * ix > 2 * asuint64(INFINITY) || |
| 277 | | 2 * iy > 2 * asuint64(INFINITY)) |
| 278 | | return x + y; |
| 279 | | if (2 * ix == 2 * asuint64(1.0)) |
| 280 | | return 1.0; |
| 281 | | if ((2 * ix < 2 * asuint64(1.0)) == !(iy >> 63)) |
| 282 | | return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ |
| 283 | | return y * y; |
| 284 | | } |
| 285 | | if (predict_false(zeroinfnan(ix))) { |
| 286 | | double_t x2 = x * x; |
| 287 | | if (ix >> 63 && checkint(iy) == 1) |
| 288 | | x2 = -x2; |
| 289 | | /* Without the barrier some versions of clang hoist the 1/x2 and |
| 290 | | thus division by zero exception can be signaled spuriously. */ |
| 291 | | return iy >> 63 ? fp_barrier(1 / x2) : x2; |
| 292 | | } |
| 293 | | /* Here x and y are non-zero finite. */ |
| 294 | | if (ix >> 63) { |
| 295 | | /* Finite x < 0. */ |
| 296 | | int yint = checkint(iy); |
| 297 | | if (yint == 0) |
| 298 | | return __math_invalid(x); |
| 299 | | if (yint == 1) |
| 300 | | sign_bias = SIGN_BIAS; |
| 301 | | ix &= 0x7fffffffffffffff; |
| 302 | | topx &= 0x7ff; |
| 303 | | } |
| 304 | | if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) { |
| 305 | | /* Note: sign_bias == 0 here because y is not odd. */ |
| 306 | | if (ix == asuint64(1.0)) |
| 307 | | return 1.0; |
| 308 | | if ((topy & 0x7ff) < 0x3be) { |
| 309 | | /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ |
| 310 | | if (WANT_ROUNDING) |
| 311 | | return ix > asuint64(1.0) ? 1.0 + y : |
| 312 | | 1.0 - y; |
| 313 | | else |
| 314 | | return 1.0; |
| 315 | | } |
| 316 | | return (ix > asuint64(1.0)) == (topy < 0x800) ? |
| 317 | | __math_oflow(0) : |
| 318 | | __math_uflow(0); |
| 319 | | } |
| 320 | | if (topx == 0) { |
| 321 | | /* Normalize subnormal x so exponent becomes negative. */ |
| 322 | | ix = asuint64(x * 0x1p52); |
| 323 | | ix &= 0x7fffffffffffffff; |
| 324 | | ix -= 52ULL << 52; |
| 325 | | } |
| 326 | | } |
| 327 | | |
| 328 | | double_t lo; |
| 329 | | double_t hi = log_inline(ix, &lo); |
| 330 | | double_t ehi, elo; |
| 331 | | #if __FP_FAST_FMA |
| 332 | | ehi = y * hi; |
| 333 | | elo = y * lo + __builtin_fma(y, hi, -ehi); |
| 334 | | #else |
| 335 | | double_t yhi = asdouble(iy & -1ULL << 27); |
| 336 | | double_t ylo = y - yhi; |
| 337 | | double_t lhi = asdouble(asuint64(hi) & -1ULL << 27); |
| 338 | | double_t llo = hi - lhi + lo; |
| 339 | | ehi = yhi * lhi; |
| 340 | | elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ |
| 341 | | #endif |
| 342 | | return exp_inline(ehi, elo, sign_bias); |
| 343 | | } |
Ivel <ivel.santos@proton.me> 2026-02-03 00:53:58-03:00
Andrew Kelley <andrew@ziglang.org> 2026-02-03 20:14:22+01:00
src