1 /* 2 * Single-precision tanh(x) function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 #include "math_config.h" 8 #include "pl_sig.h" 9 #include "pl_test.h" 10 11 #define BoringBound \ 12 0x41102cb3 /* 0x1.205966p+3, above which tanhf rounds to 1 (or -1 for \ 13 negative). */ 14 #define AbsMask 0x7fffffff 15 #define One 0x3f800000 16 17 #define Shift (0x1.8p23f) 18 #define InvLn2 (0x1.715476p+0f) 19 #define Ln2hi (0x1.62e4p-1f) 20 #define Ln2lo (0x1.7f7d1cp-20f) 21 22 #define C(i) __expm1f_poly[i] 23 24 static inline float 25 expm1f_inline (float x) 26 { 27 /* Helper routine for calculating exp(x) - 1. 28 Copied from expm1f_1u6.c, with several simplifications: 29 - No special-case handling for tiny or special values, instead return early 30 from the main routine. 31 - No special handling for large values: 32 - No early return for infinity. 33 - Simpler combination of p and t in final stage of algorithm. 34 - |i| < 27, so can calculate t by simpler shift-and-add, instead of 35 ldexpf (same as vector algorithm). */ 36 37 /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */ 38 float j = fmaf (InvLn2, x, Shift) - Shift; 39 int32_t i = j; 40 float f = fmaf (j, -Ln2hi, x); 41 f = fmaf (j, -Ln2lo, f); 42 43 /* Approximate expm1(f) with polynomial P, expm1(f) ~= f + f^2 * P(f). 44 Uses Estrin scheme, where the main expm1f routine uses Horner. */ 45 float f2 = f * f; 46 float p_01 = fmaf (f, C (1), C (0)); 47 float p_23 = fmaf (f, C (3), C (2)); 48 float p = fmaf (f2, p_23, p_01); 49 p = fmaf (f2 * f2, C (4), p); 50 p = fmaf (f2, p, f); 51 52 /* t = 2^i. */ 53 float t = asfloat ((uint32_t) (i + 127) << 23); 54 /* expm1(x) ~= p * t + (t - 1). */ 55 return fmaf (p, t, t - 1); 56 } 57 58 /* Approximation for single-precision tanh(x), using a simplified version of 59 expm1f. The maximum error is 2.58 ULP: 60 tanhf(0x1.fa5eep-5) got 0x1.f9ba02p-5 61 want 0x1.f9ba08p-5. */ 62 float 63 tanhf (float x) 64 { 65 uint32_t ix = asuint (x); 66 uint32_t iax = ix & AbsMask; 67 uint32_t sign = ix & ~AbsMask; 68 69 if (unlikely (iax > BoringBound)) 70 { 71 if (iax > 0x7f800000) 72 return __math_invalidf (x); 73 return asfloat (One | sign); 74 } 75 76 if (unlikely (iax < 0x34000000)) 77 return x; 78 79 /* tanh(x) = (e^2x - 1) / (e^2x + 1). */ 80 float q = expm1f_inline (2 * x); 81 return q / (q + 2); 82 } 83 84 PL_SIG (S, F, 1, tanh, -10.0, 10.0) 85 PL_TEST_ULP (tanhf, 2.09) 86 PL_TEST_SYM_INTERVAL (tanhf, 0, 0x1p-23, 1000) 87 PL_TEST_SYM_INTERVAL (tanhf, 0x1p-23, 0x1.205966p+3, 100000) 88 PL_TEST_SYM_INTERVAL (tanhf, 0x1.205966p+3, inf, 100) 89