1 /*
2 * Double-precision x^y function.
3 *
4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
5 * See https://llvm.org/LICENSE.txt for license information.
6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
7 */
8
9 #include <float.h>
10 #include <math.h>
11 #include <stdint.h>
12 #include "math_config.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 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
log_inline(uint64_t ix,double_t * tail)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 representable. */
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
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)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
exp_inline(double_t x,double_t xtail,uint32_t sign_bias)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
checkint(uint64_t iy)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
zeroinfnan(uint64_t i)268 zeroinfnan (uint64_t i)
269 {
270 return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
271 }
272
273 double
pow(double x,double y)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 evaluate 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
strong_alias(pow,__pow_finite)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