1 /* 2 * Double-precision x^y function. 3 * 4 * Copyright (c) 2018-2024, 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 #include "test_defs.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 29 top12 (double x) 30 { 31 return asuint64 (x) >> 52; 32 } 33 34 /* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about 35 additional 15 bits precision. IX is the bit representation of x, but 36 normalized in the subnormal range using the sign bit for the exponent. */ 37 static inline double_t 38 log_inline (uint64_t ix, double_t *tail) 39 { 40 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 41 double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; 42 uint64_t iz, tmp; 43 int k, i; 44 45 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 46 The range is split into N subintervals. 47 The ith subinterval contains z and c is near its center. */ 48 tmp = ix - OFF; 49 i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; 50 k = (int64_t) tmp >> 52; /* arithmetic shift */ 51 iz = ix - (tmp & 0xfffULL << 52); 52 z = asdouble (iz); 53 kd = (double_t) k; 54 55 /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ 56 invc = T[i].invc; 57 logc = T[i].logc; 58 logctail = T[i].logctail; 59 60 /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and 61 |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */ 62 #if HAVE_FAST_FMA 63 r = fma (z, invc, -1.0); 64 #else 65 /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ 66 double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32)); 67 double_t zlo = z - zhi; 68 double_t rhi = zhi * invc - 1.0; 69 double_t rlo = zlo * invc; 70 r = rhi + rlo; 71 #endif 72 73 /* k*Ln2 + log(c) + r. */ 74 t1 = kd * Ln2hi + logc; 75 t2 = t1 + r; 76 lo1 = kd * Ln2lo + logctail; 77 lo2 = t1 - t2 + r; 78 79 /* Evaluation is optimized assuming superscalar pipelined execution. */ 80 double_t ar, ar2, ar3, lo3, lo4; 81 ar = A[0] * r; /* A[0] = -0.5. */ 82 ar2 = r * ar; 83 ar3 = r * ar2; 84 /* k*Ln2 + log(c) + r + A[0]*r*r. */ 85 #if HAVE_FAST_FMA 86 hi = t2 + ar2; 87 lo3 = fma (ar, r, -ar2); 88 lo4 = t2 - hi + ar2; 89 #else 90 double_t arhi = A[0] * rhi; 91 double_t arhi2 = rhi * arhi; 92 hi = t2 + arhi2; 93 lo3 = rlo * (ar + arhi); 94 lo4 = t2 - hi + arhi2; 95 #endif 96 /* p = log1p(r) - r - A[0]*r*r. */ 97 #if POW_LOG_POLY_ORDER == 8 98 p = (ar3 99 * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); 100 #endif 101 lo = lo1 + lo2 + lo3 + lo4 + p; 102 y = hi + lo; 103 *tail = hi - y + lo; 104 return y; 105 } 106 107 #undef N 108 #undef T 109 #define N (1 << EXP_TABLE_BITS) 110 #define InvLn2N __exp_data.invln2N 111 #define NegLn2hiN __exp_data.negln2hiN 112 #define NegLn2loN __exp_data.negln2loN 113 #define Shift __exp_data.shift 114 #define T __exp_data.tab 115 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER] 116 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER] 117 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER] 118 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER] 119 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER] 120 121 /* Handle cases that may overflow or underflow when computing the result that 122 is scale*(1+TMP) without intermediate rounding. The bit representation of 123 scale is in SBITS, however it has a computed exponent that may have 124 overflown into the sign bit so that needs to be adjusted before using it as 125 a double. (int32_t)KI is the k used in the argument reduction and exponent 126 adjustment of scale, positive k here means the result may overflow and 127 negative k means the result may underflow. */ 128 static inline double 129 specialcase (double_t tmp, uint64_t sbits, uint64_t ki) 130 { 131 double_t scale, y; 132 133 if ((ki & 0x80000000) == 0) 134 { 135 /* k > 0, the exponent of scale might have overflowed by <= 460. */ 136 sbits -= 1009ull << 52; 137 scale = asdouble (sbits); 138 y = 0x1p1009 * (scale + scale * tmp); 139 return check_oflow (eval_as_double (y)); 140 } 141 /* k < 0, need special care in the subnormal range. */ 142 sbits += 1022ull << 52; 143 /* Note: sbits is signed scale. */ 144 scale = asdouble (sbits); 145 y = scale + scale * tmp; 146 if (fabs (y) < 1.0) 147 { 148 /* Round y to the right precision before scaling it into the subnormal 149 range to avoid double rounding that can cause 0.5+E/2 ulp error where 150 E is the worst-case ulp error outside the subnormal range. So this 151 is only useful if the goal is better than 1 ulp worst-case error. */ 152 double_t hi, lo, one = 1.0; 153 if (y < 0.0) 154 one = -1.0; 155 lo = scale - y + scale * tmp; 156 hi = one + y; 157 lo = one - hi + y + lo; 158 y = eval_as_double (hi + lo) - one; 159 /* Fix the sign of 0. */ 160 if (y == 0.0) 161 y = asdouble (sbits & 0x8000000000000000); 162 /* The underflow exception needs to be signaled explicitly. */ 163 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); 164 } 165 y = 0x1p-1022 * y; 166 return check_uflow (eval_as_double (y)); 167 } 168 169 #define SIGN_BIAS (0x800 << EXP_TABLE_BITS) 170 171 /* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. 172 The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ 173 static inline double 174 exp_inline (double_t x, double_t xtail, uint32_t sign_bias) 175 { 176 uint32_t abstop; 177 uint64_t ki, idx, top, sbits; 178 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 179 double_t kd, z, r, r2, scale, tail, tmp; 180 181 abstop = top12 (x) & 0x7ff; 182 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 183 { 184 if (abstop - top12 (0x1p-54) >= 0x80000000) 185 { 186 /* Avoid spurious underflow for tiny x. */ 187 /* Note: 0 is common input. */ 188 double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; 189 return sign_bias ? -one : one; 190 } 191 if (abstop >= top12 (1024.0)) 192 { 193 /* Note: inf and nan are already handled. */ 194 if (asuint64 (x) >> 63) 195 return __math_uflow (sign_bias); 196 else 197 return __math_oflow (sign_bias); 198 } 199 /* Large x is special cased below. */ 200 abstop = 0; 201 } 202 203 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ 204 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ 205 z = InvLn2N * x; 206 #if TOINT_INTRINSICS 207 kd = roundtoint (z); 208 ki = converttoint (z); 209 #elif EXP_USE_TOINT_NARROW 210 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ 211 kd = eval_as_double (z + Shift); 212 ki = asuint64 (kd) >> 16; 213 kd = (double_t) (int32_t) ki; 214 #else 215 /* z - kd is in [-1, 1] in non-nearest rounding modes. */ 216 kd = eval_as_double (z + Shift); 217 ki = asuint64 (kd); 218 kd -= Shift; 219 #endif 220 r = x + kd * NegLn2hiN + kd * NegLn2loN; 221 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ 222 r += xtail; 223 /* 2^(k/N) ~= scale * (1 + tail). */ 224 idx = 2 * (ki % N); 225 top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); 226 tail = asdouble (T[idx]); 227 /* This is only a valid scale when -1023*N < k < 1024*N. */ 228 sbits = T[idx + 1] + top; 229 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ 230 /* Evaluation is optimized assuming superscalar pipelined execution. */ 231 r2 = r * r; 232 /* Without fma the worst case error is 0.25/N ulp larger. */ 233 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ 234 #if EXP_POLY_ORDER == 4 235 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); 236 #elif EXP_POLY_ORDER == 5 237 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 238 #elif EXP_POLY_ORDER == 6 239 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 240 #endif 241 if (unlikely (abstop == 0)) 242 return specialcase (tmp, sbits, ki); 243 scale = asdouble (sbits); 244 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there 245 is no spurious underflow here even without fma. */ 246 return eval_as_double (scale + scale * tmp); 247 } 248 249 /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is 250 the bit representation of a non-zero finite floating-point value. */ 251 static inline int 252 checkint (uint64_t iy) 253 { 254 int e = iy >> 52 & 0x7ff; 255 if (e < 0x3ff) 256 return 0; 257 if (e > 0x3ff + 52) 258 return 2; 259 if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) 260 return 0; 261 if (iy & (1ULL << (0x3ff + 52 - e))) 262 return 1; 263 return 2; 264 } 265 266 /* Returns 1 if input is the bit representation of 0, infinity or nan. */ 267 static inline int 268 zeroinfnan (uint64_t i) 269 { 270 return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1; 271 } 272 273 double 274 pow (double x, double y) 275 { 276 uint32_t sign_bias = 0; 277 uint64_t ix, iy; 278 uint32_t topx, topy; 279 280 ix = asuint64 (x); 281 iy = asuint64 (y); 282 topx = top12 (x); 283 topy = top12 (y); 284 if (unlikely (topx - 0x001 >= 0x7ff - 0x001 285 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) 286 { 287 /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 288 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ 289 /* Special cases: (x < 0x1p-126 or inf or nan) or 290 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ 291 if (unlikely (zeroinfnan (iy))) 292 { 293 if (2 * iy == 0) 294 return issignaling_inline (x) ? x + y : 1.0; 295 if (ix == asuint64 (1.0)) 296 return issignaling_inline (y) ? x + y : 1.0; 297 if (2 * ix > 2 * asuint64 (INFINITY) 298 || 2 * iy > 2 * asuint64 (INFINITY)) 299 return x + y; 300 if (2 * ix == 2 * asuint64 (1.0)) 301 return 1.0; 302 if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63)) 303 return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ 304 return y * y; 305 } 306 if (unlikely (zeroinfnan (ix))) 307 { 308 double_t x2 = x * x; 309 if (ix >> 63 && checkint (iy) == 1) 310 { 311 x2 = -x2; 312 sign_bias = 1; 313 } 314 if (WANT_ERRNO && 2 * ix == 0 && iy >> 63) 315 return __math_divzero (sign_bias); 316 /* Without the barrier some versions of clang hoist the 1/x2 and 317 thus division by zero exception can be signaled spuriously. */ 318 return iy >> 63 ? opt_barrier_double (1 / x2) : x2; 319 } 320 /* Here x and y are non-zero finite. */ 321 if (ix >> 63) 322 { 323 /* Finite x < 0. */ 324 int yint = checkint (iy); 325 if (yint == 0) 326 return __math_invalid (x); 327 if (yint == 1) 328 sign_bias = SIGN_BIAS; 329 ix &= 0x7fffffffffffffff; 330 topx &= 0x7ff; 331 } 332 if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) 333 { 334 /* Note: sign_bias == 0 here because y is not odd. */ 335 if (ix == asuint64 (1.0)) 336 return 1.0; 337 if ((topy & 0x7ff) < 0x3be) 338 { 339 /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ 340 if (WANT_ROUNDING) 341 return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y; 342 else 343 return 1.0; 344 } 345 return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0) 346 : __math_uflow (0); 347 } 348 if (topx == 0) 349 { 350 /* Normalize subnormal x so exponent becomes negative. */ 351 /* Without the barrier some versions of clang evalutate the mul 352 unconditionally causing spurious overflow exceptions. */ 353 ix = asuint64 (opt_barrier_double (x) * 0x1p52); 354 ix &= 0x7fffffffffffffff; 355 ix -= 52ULL << 52; 356 } 357 } 358 359 double_t lo; 360 double_t hi = log_inline (ix, &lo); 361 double_t ehi, elo; 362 #if HAVE_FAST_FMA 363 ehi = y * hi; 364 elo = y * lo + fma (y, hi, -ehi); 365 #else 366 double_t yhi = asdouble (iy & -1ULL << 27); 367 double_t ylo = y - yhi; 368 double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27); 369 double_t llo = hi - lhi + lo; 370 ehi = yhi * lhi; 371 elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ 372 #endif 373 return exp_inline (ehi, elo, sign_bias); 374 } 375 #if USE_GLIBC_ABI 376 strong_alias (pow, __pow_finite) 377 hidden_alias (pow, __ieee754_pow) 378 # if LDBL_MANT_DIG == 53 379 long double powl (long double x, long double y) { return pow (x, y); } 380 # endif 381 #endif 382 383 TEST_ULP (pow, 0.05) 384 TEST_ULP_NONNEAREST (pow, 0.5) 385 TEST_INTERVAL2 (pow, 0.5, 2.0, 0, inf, 20000) 386 TEST_INTERVAL2 (pow, -0.5, -2.0, 0, inf, 20000) 387 TEST_INTERVAL2 (pow, 0.5, 2.0, -0, -inf, 20000) 388 TEST_INTERVAL2 (pow, -0.5, -2.0, -0, -inf, 20000) 389 TEST_INTERVAL2 (pow, 0.5, 2.0, 0x1p-10, 0x1p10, 40000) 390 TEST_INTERVAL2 (pow, 0.5, 2.0, -0x1p-10, -0x1p10, 40000) 391 TEST_INTERVAL2 (pow, 0, inf, 0.5, 2.0, 80000) 392 TEST_INTERVAL2 (pow, 0, inf, -0.5, -2.0, 80000) 393 TEST_INTERVAL2 (pow, 0x1.fp-1, 0x1.08p0, 0x1p8, 0x1p17, 80000) 394 TEST_INTERVAL2 (pow, 0x1.fp-1, 0x1.08p0, -0x1p8, -0x1p17, 80000) 395 TEST_INTERVAL2 (pow, 0, 0x1p-1000, 0, 1.0, 50000) 396 TEST_INTERVAL2 (pow, 0x1p1000, inf, 0, 1.0, 50000) 397 TEST_INTERVAL2 (pow, 0x1.ffffffffffff0p-1, 0x1.0000000000008p0, 0x1p60, 0x1p68, 398 50000) 399 TEST_INTERVAL2 (pow, 0x1.ffffffffff000p-1, 0x1p0, 0x1p50, 0x1p52, 50000) 400 TEST_INTERVAL2 (pow, -0x1.ffffffffff000p-1, -0x1p0, 0x1p50, 0x1p52, 50000) 401