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Diffstat (limited to 'math/aarch64/experimental/erfc_1u8.c')
| -rw-r--r-- | math/aarch64/experimental/erfc_1u8.c | 153 |
1 files changed, 153 insertions, 0 deletions
diff --git a/math/aarch64/experimental/erfc_1u8.c b/math/aarch64/experimental/erfc_1u8.c new file mode 100644 index 000000000000..5357e9329433 --- /dev/null +++ b/math/aarch64/experimental/erfc_1u8.c @@ -0,0 +1,153 @@ +/* + * Double-precision erfc(x) function. + * + * Copyright (c) 2023-2024, Arm Limited. + * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception + */ + +#include "math_config.h" +#include "test_sig.h" +#include "test_defs.h" + +#define Shift 0x1p45 +#define P20 0x1.5555555555555p-2 /* 1/3. */ +#define P21 0x1.5555555555555p-1 /* 2/3. */ + +#define P40 0x1.999999999999ap-4 /* 1/10. */ +#define P41 0x1.999999999999ap-2 /* 2/5. */ +#define P42 0x1.11111111111111p-3 /* 2/15. */ + +#define P50 0x1.5555555555555p-3 /* 1/6. */ +#define P51 0x1.c71c71c71c71cp-3 /* 2/9. */ +#define P52 0x1.6c16c16c16c17p-5 /* 2/45. */ + +/* Qi = (i+1) / i. */ +#define Q5 0x1.3333333333333p0 +#define Q6 0x1.2aaaaaaaaaaabp0 +#define Q7 0x1.2492492492492p0 +#define Q8 0x1.2p0 +#define Q9 0x1.1c71c71c71c72p0 + +/* Ri = -2 * i / ((i+1)*(i+2)). */ +#define R5 -0x1.e79e79e79e79ep-3 +#define R6 -0x1.b6db6db6db6dbp-3 +#define R7 -0x1.8e38e38e38e39p-3 +#define R8 -0x1.6c16c16c16c17p-3 +#define R9 -0x1.4f2094f2094f2p-3 + +/* Fast erfc approximation based on series expansion near x rounded to + nearest multiple of 1/128. + Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r, + + erfc(x) ~ erfc(r) - scale * d * poly(r, d), with + + poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3 + + (2/15 r^4 - 2/5 r^2 + 1/10) d^4 + - r * (2/45 r^4 - 2/9 r^2 + 1/6) d^5 + + p6(r) d^6 + ... + p10(r) d^10 + + Polynomials p6(r) to p10(r) are computed using recurrence relation + + 2(i+1)p_i + 2r(i+2)p_{i+1} + (i+2)(i+3)p_{i+2} = 0, + with p0 = 1, and p1(r) = -r. + + Values of erfc(r) and scale(r) are read from lookup tables. Stored values + are scaled to avoid hitting the subnormal range. + + Note that for x < 0, erfc(x) = 2.0 - erfc(-x). + + Maximum measured error: 1.71 ULP + erfc(0x1.46cfe976733p+4) got 0x1.e15fcbea3e7afp-608 + want 0x1.e15fcbea3e7adp-608. */ +double +erfc (double x) +{ + /* Get top words and sign. */ + uint64_t ix = asuint64 (x); + uint64_t ia = ix & 0x7fffffffffffffff; + double a = asdouble (ia); + uint64_t sign = ix & ~0x7fffffffffffffff; + + /* erfc(nan)=nan, erfc(+inf)=0 and erfc(-inf)=2. */ + if (unlikely (ia >= 0x7ff0000000000000)) + return asdouble (sign >> 1) + 1.0 / x; /* Special cases. */ + + /* Return early for large enough negative values. */ + if (x < -6.0) + return 2.0; + + /* For |x| < 3487.0/128.0, the following approximation holds. */ + if (likely (ia < 0x403b3e0000000000)) + { + /* |x| < 0x1p-511 => accurate to 0.5 ULP. */ + if (unlikely (ia < asuint64 (0x1p-511))) + return 1.0 - x; + + /* Lookup erfc(r) and scale(r) in tables, e.g. set erfc(r) to 1 and scale + to 2/sqrt(pi), when x reduced to r = 0. */ + double z = a + Shift; + uint64_t i = asuint64 (z) - asuint64 (Shift); + double r = z - Shift; + /* These values are scaled by 2^128. */ + double erfcr = __v_erfc_data.tab[i].erfc; + double scale = __v_erfc_data.tab[i].scale; + + /* erfc(x) ~ erfc(r) - scale * d * poly (r, d). */ + double d = a - r; + double d2 = d * d; + double r2 = r * r; + /* Compute p_i as a regular (low-order) polynomial. */ + double p1 = -r; + double p2 = fma (P21, r2, -P20); + double p3 = -r * fma (P20, r2, -0.5); + double p4 = fma (fma (P42, r2, -P41), r2, P40); + double p5 = -r * fma (fma (P52, r2, -P51), r2, P50); + /* Compute p_i using recurrence relation: + p_{i+2} = (p_i + r * Q_{i+1} * p_{i+1}) * R_{i+1}. */ + double p6 = fma (Q5 * r, p5, p4) * R5; + double p7 = fma (Q6 * r, p6, p5) * R6; + double p8 = fma (Q7 * r, p7, p6) * R7; + double p9 = fma (Q8 * r, p8, p7) * R8; + double p10 = fma (Q9 * r, p9, p8) * R9; + /* Compute polynomial in d using pairwise Horner scheme. */ + double p90 = fma (p10, d, p9); + double p78 = fma (p8, d, p7); + double p56 = fma (p6, d, p5); + double p34 = fma (p4, d, p3); + double p12 = fma (p2, d, p1); + double y = fma (p90, d2, p78); + y = fma (y, d2, p56); + y = fma (y, d2, p34); + y = fma (y, d2, p12); + + y = fma (-fma (y, d2, d), scale, erfcr); + + /* Handle sign and scale back in a single fma. */ + double off = asdouble (sign >> 1); + double fac = asdouble (asuint64 (0x1p-128) | sign); + y = fma (y, fac, off); + + if (unlikely (x > 26.0)) + { + /* The underflow exception needs to be signaled explicitly when + result gets into the subnormal range. */ + if (unlikely (y < 0x1p-1022)) + force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); + /* Set errno to ERANGE if result rounds to 0. */ + return __math_check_uflow (y); + } + + return y; + } + /* Above the threshold (x > 3487.0/128.0) erfc is constant and needs to raise + underflow exception for positive x. */ + return __math_uflow (0); +} + +TEST_SIG (S, D, 1, erfc, -6.0, 28.0) +TEST_ULP (erfc, 1.21) +TEST_SYM_INTERVAL (erfc, 0, 0x1p-26, 40000) +TEST_INTERVAL (erfc, 0x1p-26, 28.0, 100000) +TEST_INTERVAL (erfc, -0x1p-26, -6.0, 100000) +TEST_INTERVAL (erfc, 28.0, inf, 40000) +TEST_INTERVAL (erfc, -6.0, -inf, 40000) |