diff options
Diffstat (limited to 'sysdeps/ieee754')
| -rw-r--r-- | sysdeps/ieee754/flt-32/e_exp10f.c | 198 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/math_config.h | 2 |
2 files changed, 199 insertions, 1 deletions
diff --git a/sysdeps/ieee754/flt-32/e_exp10f.c b/sysdeps/ieee754/flt-32/e_exp10f.c new file mode 100644 index 0000000000..034a9e364a --- /dev/null +++ b/sysdeps/ieee754/flt-32/e_exp10f.c @@ -0,0 +1,198 @@ +/* Single-precision 10^x function. + Copyright (C) 2020 Free Software Foundation, Inc. + This file is part of the GNU C Library. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, see + <https://www.gnu.org/licenses/>. */ + +#include <math.h> +#include <math-narrow-eval.h> +#include <stdint.h> +#include <libm-alias-finite.h> +#include <libm-alias-float.h> +#include "math_config.h" + +/* + EXP2F_TABLE_BITS 5 + EXP2F_POLY_ORDER 3 + + Max. ULP error: 0.502 (normal range, nearest rounding). + Max. relative error: 2^-33.240 (before rounding to float). + Wrong count: 169839. + Non-nearest ULP error: 1 (rounded ULP error). + + Detailed error analysis (for EXP2F_TABLE_BITS=3 thus N=32): + + - We first compute z = RN(InvLn10N * x) where + + InvLn10N = N*log(10)/log(2) * (1 + theta1) with |theta1| < 2^-54.150 + + since z is rounded to nearest: + + z = InvLn10N * x * (1 + theta2) with |theta2| < 2^-53 + + thus z = N*log(10)/log(2) * x * (1 + theta3) with |theta3| < 2^-52.463 + + - Since |x| < 38 in the main branch, we deduce: + + z = N*log(10)/log(2) * x + theta4 with |theta4| < 2^-40.483 + + - We then write z = k + r where k is an integer and |r| <= 0.5 (exact) + + - We thus have + + x = z*log(2)/(N*log(10)) - theta4*log(2)/(N*log(10)) + = z*log(2)/(N*log(10)) + theta5 with |theta5| < 2^-47.215 + + 10^x = 2^(k/N) * 2^(r/N) * 10^theta5 + = 2^(k/N) * 2^(r/N) * (1 + theta6) with |theta6| < 2^-46.011 + + - Then 2^(k/N) is approximated by table lookup, the maximal relative error + is for (k%N) = 5, with + + s = 2^(i/N) * (1 + theta7) with |theta7| < 2^-53.249 + + - 2^(r/N) is approximated by a degree-3 polynomial, where the maximal + mathematical relative error is 2^-33.243. + + - For C[0] * r + C[1], assuming no FMA is used, since |r| <= 0.5 and + |C[0]| < 1.694e-6, |C[0] * r| < 8.47e-7, and the absolute error on + C[0] * r is bounded by 1/2*ulp(8.47e-7) = 2^-74. Then we add C[1] with + |C[1]| < 0.000235, thus the absolute error on C[0] * r + C[1] is bounded + by 2^-65.994 (z is bounded by 0.000236). + + - For r2 = r * r, since |r| <= 0.5, we have |r2| <= 0.25 and the absolute + error is bounded by 1/4*ulp(0.25) = 2^-56 (the factor 1/4 is because |r2| + cannot exceed 1/4, and on the left of 1/4 the distance between two + consecutive numbers is 1/4*ulp(1/4)). + + - For y = C[2] * r + 1, assuming no FMA is used, since |r| <= 0.5 and + |C[2]| < 0.0217, the absolute error on C[2] * r is bounded by 2^-60, + and thus the absolute error on C[2] * r + 1 is bounded by 1/2*ulp(1)+2^60 + < 2^-52.988, and |y| < 1.01085 (the error bound is better if a FMA is + used). + + - for z * r2 + y: the absolute error on z is bounded by 2^-65.994, with + |z| < 0.000236, and the absolute error on r2 is bounded by 2^-56, with + r2 < 0.25, thus |z*r2| < 0.000059, and the absolute error on z * r2 + (including the rounding error) is bounded by: + + 2^-65.994 * 0.25 + 0.000236 * 2^-56 + 1/2*ulp(0.000059) < 2^-66.429. + + Now we add y, with |y| < 1.01085, and error on y bounded by 2^-52.988, + thus the absolute error is bounded by: + + 2^-66.429 + 2^-52.988 + 1/2*ulp(1.01085) < 2^-51.993 + + - Now we convert the error on y into relative error. Recall that y + approximates 2^(r/N), for |r| <= 0.5 and N=32. Thus 2^(-0.5/32) <= y, + and the relative error on y is bounded by + + 2^-51.993/2^(-0.5/32) < 2^-51.977 + + - Taking into account the mathematical relative error of 2^-33.243 we have: + + y = 2^(r/N) * (1 + theta8) with |theta8| < 2^-33.242 + + - Since we had s = 2^(k/N) * (1 + theta7) with |theta7| < 2^-53.249, + after y = y * s we get y = 2^(k/N) * 2^(r/N) * (1 + theta9) + with |theta9| < 2^-33.241 + + - Finally, taking into account the error theta6 due to the rounding error on + InvLn10N, and when multiplying InvLn10N * x, we get: + + y = 10^x * (1 + theta10) with |theta10| < 2^-33.240 + + - Converting into binary64 ulps, since y < 2^53*ulp(y), the error is at most + 2^19.76 ulp(y) + + - If the result is a binary32 value in the normal range (i.e., >= 2^-126), + then the error is at most 2^-9.24 ulps, i.e., less than 0.00166 (in the + subnormal range, the error in ulps might be larger). + + Note that this bound is tight, since for x=-0x25.54ac0p0 the final value of + y (before conversion to float) differs from 879853 ulps from the correctly + rounded value, and 879853 ~ 2^19.7469. */ + +#define N (1 << EXP2F_TABLE_BITS) + +#define InvLn10N (0x3.5269e12f346e2p0 * N) /* log(10)/log(2) to nearest */ +#define T __exp2f_data.tab +#define C __exp2f_data.poly_scaled +#define SHIFT __exp2f_data.shift + +static inline uint32_t +top13 (float x) +{ + return asuint (x) >> 19; +} + +float +__ieee754_exp10f (float x) +{ + uint32_t abstop; + uint64_t ki, t; + double kd, xd, z, r, r2, y, s; + + xd = (double) x; + abstop = top13 (x) & 0xfff; /* Ignore sign. */ + if (__glibc_unlikely (abstop >= top13 (38.0f))) + { + /* |x| >= 38 or x is nan. */ + if (asuint (x) == asuint (-INFINITY)) + return 0.0f; + if (abstop >= top13 (INFINITY)) + return x + x; + /* 0x26.8826ap0 is the largest value such that 10^x < 2^128. */ + if (x > 0x26.8826ap0f) + return __math_oflowf (0); + /* -0x2d.278d4p0 is the smallest value such that 10^x > 2^-150. */ + if (x < -0x2d.278d4p0f) + return __math_uflowf (0); +#if WANT_ERRNO_UFLOW + if (x < -0x2c.da7cfp0) + return __math_may_uflowf (0); +#endif + /* the smallest value such that 10^x >= 2^-126 (normal range) + is x = -0x25.ee060p0 */ + /* we go through here for 2014929 values out of 2060451840 + (not counting NaN and infinities, i.e., about 0.1% */ + } + + /* x*N*Ln10/Ln2 = k + r with r in [-1/2, 1/2] and int k. */ + z = InvLn10N * xd; + /* |xd| < 38 thus |z| < 1216 */ +#if TOINT_INTRINSICS + kd = roundtoint (z); + ki = converttoint (z); +#else +# define SHIFT __exp2f_data.shift + kd = math_narrow_eval ((double) (z + SHIFT)); /* Needs to be double. */ + ki = asuint64 (kd); + kd -= SHIFT; +#endif + r = z - kd; + + /* 10^x = 10^(k/N) * 10^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ + t = T[ki % N]; + t += ki << (52 - EXP2F_TABLE_BITS); + s = asdouble (t); + z = C[0] * r + C[1]; + r2 = r * r; + y = C[2] * r + 1; + y = z * r2 + y; + y = y * s; + return (float) y; +} +libm_alias_finite (__ieee754_exp10f, __exp10f) diff --git a/sysdeps/ieee754/flt-32/math_config.h b/sysdeps/ieee754/flt-32/math_config.h index bf79274ce7..4817e500e1 100644 --- a/sysdeps/ieee754/flt-32/math_config.h +++ b/sysdeps/ieee754/flt-32/math_config.h @@ -109,7 +109,7 @@ attribute_hidden float __math_may_uflowf (uint32_t); attribute_hidden float __math_divzerof (uint32_t); attribute_hidden float __math_invalidf (float); -/* Shared between expf, exp2f and powf. */ +/* Shared between expf, exp2f, exp10f, and powf. */ #define EXP2F_TABLE_BITS 5 #define EXP2F_POLY_ORDER 3 extern const struct exp2f_data |