1/* 2 * Double-precision x^y function. 3 * 4 * Copyright (c) 2018-2020, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8#include <float.h> 9#include <math.h> 10#include <stdint.h> 11#include "math_config.h" 12 13/* 14Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53) 15relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma) 16ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma) 17*/ 18 19#define T __pow_log_data.tab 20#define A __pow_log_data.poly 21#define Ln2hi __pow_log_data.ln2hi 22#define Ln2lo __pow_log_data.ln2lo 23#define N (1 << POW_LOG_TABLE_BITS) 24#define OFF 0x3fe6955500000000 25 26/* Top 12 bits of a double (sign and exponent bits). */ 27static inline uint32_t 28top12 (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. */ 36static inline double_t 37log_inline (uint64_t ix, double_t *tail) 38{ 39 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 40 double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; 41 uint64_t iz, tmp; 42 int k, i; 43 44 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 45 The range is split into N subintervals. 46 The ith subinterval contains z and c is near its center. */ 47 tmp = ix - OFF; 48 i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; 49 k = (int64_t) tmp >> 52; /* arithmetic shift */ 50 iz = ix - (tmp & 0xfffULL << 52); 51 z = asdouble (iz); 52 kd = (double_t) k; 53 54 /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ 55 invc = T[i].invc; 56 logc = T[i].logc; 57 logctail = T[i].logctail; 58 59 /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and 60 |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */ 61#if HAVE_FAST_FMA 62 r = fma (z, invc, -1.0); 63#else 64 /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ 65 double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32)); 66 double_t zlo = z - zhi; 67 double_t rhi = zhi * invc - 1.0; 68 double_t rlo = zlo * invc; 69 r = rhi + rlo; 70#endif 71 72 /* k*Ln2 + log(c) + r. */ 73 t1 = kd * Ln2hi + logc; 74 t2 = t1 + r; 75 lo1 = kd * Ln2lo + logctail; 76 lo2 = t1 - t2 + r; 77 78 /* Evaluation is optimized assuming superscalar pipelined execution. */ 79 double_t ar, ar2, ar3, lo3, lo4; 80 ar = A[0] * r; /* A[0] = -0.5. */ 81 ar2 = r * ar; 82 ar3 = r * ar2; 83 /* k*Ln2 + log(c) + r + A[0]*r*r. */ 84#if HAVE_FAST_FMA 85 hi = t2 + ar2; 86 lo3 = fma (ar, r, -ar2); 87 lo4 = t2 - hi + ar2; 88#else 89 double_t arhi = A[0] * rhi; 90 double_t arhi2 = rhi * arhi; 91 hi = t2 + arhi2; 92 lo3 = rlo * (ar + arhi); 93 lo4 = t2 - hi + arhi2; 94#endif 95 /* p = log1p(r) - r - A[0]*r*r. */ 96#if POW_LOG_POLY_ORDER == 8 97 p = (ar3 98 * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); 99#endif 100 lo = lo1 + lo2 + lo3 + lo4 + p; 101 y = hi + lo; 102 *tail = hi - y + lo; 103 return y; 104} 105 106#undef N 107#undef T 108#define N (1 << EXP_TABLE_BITS) 109#define InvLn2N __exp_data.invln2N 110#define NegLn2hiN __exp_data.negln2hiN 111#define NegLn2loN __exp_data.negln2loN 112#define Shift __exp_data.shift 113#define T __exp_data.tab 114#define C2 __exp_data.poly[5 - EXP_POLY_ORDER] 115#define C3 __exp_data.poly[6 - EXP_POLY_ORDER] 116#define C4 __exp_data.poly[7 - EXP_POLY_ORDER] 117#define C5 __exp_data.poly[8 - EXP_POLY_ORDER] 118#define C6 __exp_data.poly[9 - EXP_POLY_ORDER] 119 120/* Handle cases that may overflow or underflow when computing the result that 121 is scale*(1+TMP) without intermediate rounding. The bit representation of 122 scale is in SBITS, however it has a computed exponent that may have 123 overflown into the sign bit so that needs to be adjusted before using it as 124 a double. (int32_t)KI is the k used in the argument reduction and exponent 125 adjustment of scale, positive k here means the result may overflow and 126 negative k means the result may underflow. */ 127static inline double 128specialcase (double_t tmp, uint64_t sbits, uint64_t ki) 129{ 130 double_t scale, y; 131 132 if ((ki & 0x80000000) == 0) 133 { 134 /* k > 0, the exponent of scale might have overflowed by <= 460. */ 135 sbits -= 1009ull << 52; 136 scale = asdouble (sbits); 137 y = 0x1p1009 * (scale + scale * tmp); 138 return check_oflow (eval_as_double (y)); 139 } 140 /* k < 0, need special care in the subnormal range. */ 141 sbits += 1022ull << 52; 142 /* Note: sbits is signed scale. */ 143 scale = asdouble (sbits); 144 y = scale + scale * tmp; 145 if (fabs (y) < 1.0) 146 { 147 /* Round y to the right precision before scaling it into the subnormal 148 range to avoid double rounding that can cause 0.5+E/2 ulp error where 149 E is the worst-case ulp error outside the subnormal range. So this 150 is only useful if the goal is better than 1 ulp worst-case error. */ 151 double_t hi, lo, one = 1.0; 152 if (y < 0.0) 153 one = -1.0; 154 lo = scale - y + scale * tmp; 155 hi = one + y; 156 lo = one - hi + y + lo; 157 y = eval_as_double (hi + lo) - one; 158 /* Fix the sign of 0. */ 159 if (y == 0.0) 160 y = asdouble (sbits & 0x8000000000000000); 161 /* The underflow exception needs to be signaled explicitly. */ 162 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); 163 } 164 y = 0x1p-1022 * y; 165 return check_uflow (eval_as_double (y)); 166} 167 168#define SIGN_BIAS (0x800 << EXP_TABLE_BITS) 169 170/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. 171 The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ 172static inline double 173exp_inline (double_t x, double_t xtail, uint32_t sign_bias) 174{ 175 uint32_t abstop; 176 uint64_t ki, idx, top, sbits; 177 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 178 double_t kd, z, r, r2, scale, tail, tmp; 179 180 abstop = top12 (x) & 0x7ff; 181 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 182 { 183 if (abstop - top12 (0x1p-54) >= 0x80000000) 184 { 185 /* Avoid spurious underflow for tiny x. */ 186 /* Note: 0 is common input. */ 187 double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; 188 return sign_bias ? -one : one; 189 } 190 if (abstop >= top12 (1024.0)) 191 { 192 /* Note: inf and nan are already handled. */ 193 if (asuint64 (x) >> 63) 194 return __math_uflow (sign_bias); 195 else 196 return __math_oflow (sign_bias); 197 } 198 /* Large x is special cased below. */ 199 abstop = 0; 200 } 201 202 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ 203 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ 204 z = InvLn2N * x; 205#if TOINT_INTRINSICS 206 kd = roundtoint (z); 207 ki = converttoint (z); 208#elif EXP_USE_TOINT_NARROW 209 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ 210 kd = eval_as_double (z + Shift); 211 ki = asuint64 (kd) >> 16; 212 kd = (double_t) (int32_t) ki; 213#else 214 /* z - kd is in [-1, 1] in non-nearest rounding modes. */ 215 kd = eval_as_double (z + Shift); 216 ki = asuint64 (kd); 217 kd -= Shift; 218#endif 219 r = x + kd * NegLn2hiN + kd * NegLn2loN; 220 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ 221 r += xtail; 222 /* 2^(k/N) ~= scale * (1 + tail). */ 223 idx = 2 * (ki % N); 224 top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); 225 tail = asdouble (T[idx]); 226 /* This is only a valid scale when -1023*N < k < 1024*N. */ 227 sbits = T[idx + 1] + top; 228 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ 229 /* Evaluation is optimized assuming superscalar pipelined execution. */ 230 r2 = r * r; 231 /* Without fma the worst case error is 0.25/N ulp larger. */ 232 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ 233#if EXP_POLY_ORDER == 4 234 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); 235#elif EXP_POLY_ORDER == 5 236 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 237#elif EXP_POLY_ORDER == 6 238 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 239#endif 240 if (unlikely (abstop == 0)) 241 return specialcase (tmp, sbits, ki); 242 scale = asdouble (sbits); 243 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there 244 is no spurious underflow here even without fma. */ 245 return eval_as_double (scale + scale * tmp); 246} 247 248/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is 249 the bit representation of a non-zero finite floating-point value. */ 250static inline int 251checkint (uint64_t iy) 252{ 253 int e = iy >> 52 & 0x7ff; 254 if (e < 0x3ff) 255 return 0; 256 if (e > 0x3ff + 52) 257 return 2; 258 if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) 259 return 0; 260 if (iy & (1ULL << (0x3ff + 52 - e))) 261 return 1; 262 return 2; 263} 264 265/* Returns 1 if input is the bit representation of 0, infinity or nan. */ 266static inline int 267zeroinfnan (uint64_t i) 268{ 269 return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1; 270} 271 272double 273pow (double x, double y) 274{ 275 uint32_t sign_bias = 0; 276 uint64_t ix, iy; 277 uint32_t topx, topy; 278 279 ix = asuint64 (x); 280 iy = asuint64 (y); 281 topx = top12 (x); 282 topy = top12 (y); 283 if (unlikely (topx - 0x001 >= 0x7ff - 0x001 284 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) 285 { 286 /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 287 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ 288 /* Special cases: (x < 0x1p-126 or inf or nan) or 289 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ 290 if (unlikely (zeroinfnan (iy))) 291 { 292 if (2 * iy == 0) 293 return issignaling_inline (x) ? x + y : 1.0; 294 if (ix == asuint64 (1.0)) 295 return issignaling_inline (y) ? x + y : 1.0; 296 if (2 * ix > 2 * asuint64 (INFINITY) 297 || 2 * iy > 2 * asuint64 (INFINITY)) 298 return x + y; 299 if (2 * ix == 2 * asuint64 (1.0)) 300 return 1.0; 301 if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63)) 302 return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ 303 return y * y; 304 } 305 if (unlikely (zeroinfnan (ix))) 306 { 307 double_t x2 = x * x; 308 if (ix >> 63 && checkint (iy) == 1) 309 { 310 x2 = -x2; 311 sign_bias = 1; 312 } 313 if (WANT_ERRNO && 2 * ix == 0 && iy >> 63) 314 return __math_divzero (sign_bias); 315 /* Without the barrier some versions of clang hoist the 1/x2 and 316 thus division by zero exception can be signaled spuriously. */ 317 return iy >> 63 ? opt_barrier_double (1 / x2) : x2; 318 } 319 /* Here x and y are non-zero finite. */ 320 if (ix >> 63) 321 { 322 /* Finite x < 0. */ 323 int yint = checkint (iy); 324 if (yint == 0) 325 return __math_invalid (x); 326 if (yint == 1) 327 sign_bias = SIGN_BIAS; 328 ix &= 0x7fffffffffffffff; 329 topx &= 0x7ff; 330 } 331 if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) 332 { 333 /* Note: sign_bias == 0 here because y is not odd. */ 334 if (ix == asuint64 (1.0)) 335 return 1.0; 336 if ((topy & 0x7ff) < 0x3be) 337 { 338 /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ 339 if (WANT_ROUNDING) 340 return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y; 341 else 342 return 1.0; 343 } 344 return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0) 345 : __math_uflow (0); 346 } 347 if (topx == 0) 348 { 349 /* Normalize subnormal x so exponent becomes negative. */ 350 /* Without the barrier some versions of clang evalutate the mul 351 unconditionally causing spurious overflow exceptions. */ 352 ix = asuint64 (opt_barrier_double (x) * 0x1p52); 353 ix &= 0x7fffffffffffffff; 354 ix -= 52ULL << 52; 355 } 356 } 357 358 double_t lo; 359 double_t hi = log_inline (ix, &lo); 360 double_t ehi, elo; 361#if HAVE_FAST_FMA 362 ehi = y * hi; 363 elo = y * lo + fma (y, hi, -ehi); 364#else 365 double_t yhi = asdouble (iy & -1ULL << 27); 366 double_t ylo = y - yhi; 367 double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27); 368 double_t llo = hi - lhi + lo; 369 ehi = yhi * lhi; 370 elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ 371#endif 372 return exp_inline (ehi, elo, sign_bias); 373} 374#if USE_GLIBC_ABI 375strong_alias (pow, __pow_finite) 376hidden_alias (pow, __ieee754_pow) 377# if LDBL_MANT_DIG == 53 378long double powl (long double x, long double y) { return pow (x, y); } 379# endif 380#endif 381