1 /* 2 * Double-precision log(1+x) function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "poly_scalar_f64.h" 9 #include "math_config.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define Ln2Hi 0x1.62e42fefa3800p-1 14 #define Ln2Lo 0x1.ef35793c76730p-45 15 #define HfRt2Top 0x3fe6a09e /* top32(asuint64(sqrt(2)/2)). */ 16 #define OneMHfRt2Top \ 17 0x00095f62 /* top32(asuint64(1)) - top32(asuint64(sqrt(2)/2)). */ 18 #define OneTop12 0x3ff 19 #define BottomMask 0xffffffff 20 #define OneMHfRt2 0x3fd2bec333018866 21 #define Rt2MOne 0x3fda827999fcef32 22 #define AbsMask 0x7fffffffffffffff 23 #define ExpM63 0x3c00 24 25 static inline double 26 eval_poly (double f) 27 { 28 double f2 = f * f; 29 double f4 = f2 * f2; 30 double f8 = f4 * f4; 31 return estrin_18_f64 (f, f2, f4, f8, f8 * f8, __log1p_data.coeffs); 32 } 33 34 /* log1p approximation using polynomial on reduced interval. Largest 35 observed errors are near the lower boundary of the region where k 36 is 0. 37 Maximum measured error: 1.75ULP. 38 log1p(-0x1.2e1aea97b3e5cp-2) got -0x1.65fb8659a2f9p-2 39 want -0x1.65fb8659a2f92p-2. */ 40 double 41 log1p (double x) 42 { 43 uint64_t ix = asuint64 (x); 44 uint64_t ia = ix & AbsMask; 45 uint32_t ia16 = ia >> 48; 46 47 /* Handle special cases first. */ 48 if (unlikely (ia16 >= 0x7ff0 || ix >= 0xbff0000000000000 49 || ix == 0x8000000000000000)) 50 { 51 if (ix == 0x8000000000000000 || ix == 0x7ff0000000000000) 52 { 53 /* x == -0 => log1p(x) = -0. 54 x == Inf => log1p(x) = Inf. */ 55 return x; 56 } 57 if (ix == 0xbff0000000000000) 58 { 59 /* x == -1 => log1p(x) = -Inf. */ 60 return __math_divzero (-1); 61 ; 62 } 63 if (ia16 >= 0x7ff0) 64 { 65 /* x == +/-NaN => log1p(x) = NaN. */ 66 return __math_invalid (asdouble (ia)); 67 } 68 /* x < -1 => log1p(x) = NaN. 69 x == -Inf => log1p(x) = NaN. */ 70 return __math_invalid (x); 71 } 72 73 /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f 74 is in [sqrt(2)/2, sqrt(2)]): 75 log1p(x) = k*log(2) + log1p(f). 76 77 f may not be representable exactly, so we need a correction term: 78 let m = round(1 + x), c = (1 + x) - m. 79 c << m: at very small x, log1p(x) ~ x, hence: 80 log(1+x) - log(m) ~ c/m. 81 82 We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */ 83 84 uint64_t sign = ix & ~AbsMask; 85 if (ia <= OneMHfRt2 || (!sign && ia <= Rt2MOne)) 86 { 87 if (unlikely (ia16 <= ExpM63)) 88 { 89 /* If exponent of x <= -63 then shortcut the polynomial and avoid 90 underflow by just returning x, which is exactly rounded in this 91 region. */ 92 return x; 93 } 94 /* If x is in [sqrt(2)/2 - 1, sqrt(2) - 1] then we can shortcut all the 95 logic below, as k = 0 and f = x and therefore representable exactly. 96 All we need is to return the polynomial. */ 97 return fma (x, eval_poly (x) * x, x); 98 } 99 100 /* Obtain correctly scaled k by manipulation in the exponent. */ 101 double m = x + 1; 102 uint64_t mi = asuint64 (m); 103 uint32_t u = (mi >> 32) + OneMHfRt2Top; 104 int32_t k = (int32_t) (u >> 20) - OneTop12; 105 106 /* Correction term c/m. */ 107 double cm = (x - (m - 1)) / m; 108 109 /* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */ 110 uint32_t utop = (u & 0x000fffff) + HfRt2Top; 111 uint64_t u_red = ((uint64_t) utop << 32) | (mi & BottomMask); 112 double f = asdouble (u_red) - 1; 113 114 /* Approximate log1p(x) on the reduced input using a polynomial. Because 115 log1p(0)=0 we choose an approximation of the form: 116 x + C0*x^2 + C1*x^3 + C2x^4 + ... 117 Hence approximation has the form f + f^2 * P(f) 118 where P(x) = C0 + C1*x + C2x^2 + ... */ 119 double p = fma (f, eval_poly (f) * f, f); 120 121 double kd = k; 122 double y = fma (Ln2Lo, kd, cm); 123 return y + fma (Ln2Hi, kd, p); 124 } 125 126 PL_SIG (S, D, 1, log1p, -0.9, 10.0) 127 PL_TEST_ULP (log1p, 1.26) 128 PL_TEST_SYM_INTERVAL (log1p, 0.0, 0x1p-23, 50000) 129 PL_TEST_SYM_INTERVAL (log1p, 0x1p-23, 0.001, 50000) 130 PL_TEST_SYM_INTERVAL (log1p, 0.001, 1.0, 50000) 131 PL_TEST_SYM_INTERVAL (log1p, 1.0, inf, 5000) 132