1 /* 2 * Single-precision erf(x) function. 3 * 4 * Copyright (c) 2020, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include <stdint.h> 9 #include <math.h> 10 #include "math_config.h" 11 12 #define TwoOverSqrtPiMinusOne 0x1.06eba8p-3f 13 #define A __erff_data.erff_poly_A 14 #define B __erff_data.erff_poly_B 15 16 /* Top 12 bits of a float. */ 17 static inline uint32_t 18 top12 (float x) 19 { 20 return asuint (x) >> 20; 21 } 22 23 /* Efficient implementation of erff 24 using either a pure polynomial approximation or 25 the exponential of a polynomial. 26 Worst-case error is 1.09ulps at 0x1.c111acp-1. */ 27 float 28 erff (float x) 29 { 30 float r, x2, u; 31 32 /* Get top word. */ 33 uint32_t ix = asuint (x); 34 uint32_t sign = ix >> 31; 35 uint32_t ia12 = top12 (x) & 0x7ff; 36 37 /* Limit of both intervals is 0.875 for performance reasons but coefficients 38 computed on [0.0, 0.921875] and [0.921875, 4.0], which brought accuracy 39 from 0.94 to 1.1ulps. */ 40 if (ia12 < 0x3f6) 41 { /* a = |x| < 0.875. */ 42 43 /* Tiny and subnormal cases. */ 44 if (unlikely (ia12 < 0x318)) 45 { /* |x| < 2^(-28). */ 46 if (unlikely (ia12 < 0x040)) 47 { /* |x| < 2^(-119). */ 48 float y = fmaf (TwoOverSqrtPiMinusOne, x, x); 49 return check_uflowf (y); 50 } 51 return x + TwoOverSqrtPiMinusOne * x; 52 } 53 54 x2 = x * x; 55 56 /* Normalized cases (|x| < 0.921875). Use Horner scheme for x+x*P(x^2). */ 57 r = A[5]; 58 r = fmaf (r, x2, A[4]); 59 r = fmaf (r, x2, A[3]); 60 r = fmaf (r, x2, A[2]); 61 r = fmaf (r, x2, A[1]); 62 r = fmaf (r, x2, A[0]); 63 r = fmaf (r, x, x); 64 } 65 else if (ia12 < 0x408) 66 { /* |x| < 4.0 - Use a custom Estrin scheme. */ 67 68 float a = fabsf (x); 69 /* Start with Estrin scheme on high order (small magnitude) coefficients. */ 70 r = fmaf (B[6], a, B[5]); 71 u = fmaf (B[4], a, B[3]); 72 x2 = x * x; 73 r = fmaf (r, x2, u); 74 /* Then switch to pure Horner scheme. */ 75 r = fmaf (r, a, B[2]); 76 r = fmaf (r, a, B[1]); 77 r = fmaf (r, a, B[0]); 78 r = fmaf (r, a, a); 79 /* Single precision exponential with ~0.5ulps, 80 ensures erff has max. rel. error 81 < 1ulp on [0.921875, 4.0], 82 < 1.1ulps on [0.875, 4.0]. */ 83 r = expf (-r); 84 /* Explicit copysign (calling copysignf increases latency). */ 85 if (sign) 86 r = -1.0f + r; 87 else 88 r = 1.0f - r; 89 } 90 else 91 { /* |x| >= 4.0. */ 92 93 /* Special cases : erff(nan)=nan, erff(+inf)=+1 and erff(-inf)=-1. */ 94 if (unlikely (ia12 >= 0x7f8)) 95 return (1.f - (float) ((ix >> 31) << 1)) + 1.f / x; 96 97 /* Explicit copysign (calling copysignf increases latency). */ 98 if (sign) 99 r = -1.0f; 100 else 101 r = 1.0f; 102 } 103 return r; 104 } 105