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 }
344