1 /*
2 * Double-precision x^y function.
3 *
4 * Copyright (c) 2018-2020, Arm Limited.
5 * SPDX-License-Identifier: MIT
6 */
7
8 #include <float.h>
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.h"
12
13 /*
14 Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
15 relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
16 ulperr_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). */
27 static inline uint32_t
top12(double x)28 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 log_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. */
127 static inline double
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)128 specialcase (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. */
172 static inline double
exp_inline(double_t x,double_t xtail,uint32_t sign_bias)173 exp_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. */
250 static inline int
checkint(uint64_t iy)251 checkint (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. */
266 static inline int
zeroinfnan(uint64_t i)267 zeroinfnan (uint64_t i)
268 {
269 return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
270 }
271
272 double
pow(double x,double y)273 pow (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
strong_alias(pow,__pow_finite)375 strong_alias (pow, __pow_finite)
376 hidden_alias (pow, __ieee754_pow)
377 # if LDBL_MANT_DIG == 53
378 long double powl (long double x, long double y) { return pow (x, y); }
379 # endif
380 #endif
381