diff options
Diffstat (limited to 'sysdeps/aarch64/fpu/expm1_sve.c')
| -rw-r--r-- | sysdeps/aarch64/fpu/expm1_sve.c | 202 |
1 files changed, 142 insertions, 60 deletions
diff --git a/sysdeps/aarch64/fpu/expm1_sve.c b/sysdeps/aarch64/fpu/expm1_sve.c index d4ba8cc..b1d940b 100644 --- a/sysdeps/aarch64/fpu/expm1_sve.c +++ b/sysdeps/aarch64/fpu/expm1_sve.c @@ -18,82 +18,164 @@ <https://www.gnu.org/licenses/>. */ #include "sv_math.h" -#include "poly_sve_f64.h" -#define SpecialBound 0x1.62b7d369a5aa9p+9 -#define ExponentBias 0x3ff0000000000000 +#define FexpaBound 0x1.4cb5ecef28adap-3 /* 15*ln2/64. */ +#define SpecialBound 0x1.628c2855bfaddp+9 /* ln(2^(1023 + 1/128)). */ static const struct data { - double poly[11]; - double shift, inv_ln2, special_bound; - /* To be loaded in one quad-word. */ + double c2, c4; + double inv_ln2; double ln2_hi, ln2_lo; + double c0, c1, c3; + double shift, thres; + uint64_t expm1_data[32]; } data = { - /* Generated using fpminimax. */ - .poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5, - 0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13, - 0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22, - 0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, }, - - .special_bound = SpecialBound, - .inv_ln2 = 0x1.71547652b82fep0, - .ln2_hi = 0x1.62e42fefa39efp-1, - .ln2_lo = 0x1.abc9e3b39803fp-56, - .shift = 0x1.8p52, + /* Table emulating FEXPA - 1, for values of FEXPA close to 1. + The table holds values of 2^(i/64) - 1, computed in arbitrary precision. + The first half of the table stores values associated to i from 0 to 15. + The second half of the table stores values associated to i from 0 to -15. */ + .expm1_data = { + 0x0000000000000000, 0x3f864d1f3bc03077, 0x3f966c34c5615d0f, 0x3fa0e8a30eb37901, + 0x3fa6ab0d9f3121ec, 0x3fac7d865a7a3440, 0x3fb1301d0125b50a, 0x3fb429aaea92ddfb, + 0x3fb72b83c7d517ae, 0x3fba35beb6fcb754, 0x3fbd4873168b9aa8, 0x3fc031dc431466b2, + 0x3fc1c3d373ab11c3, 0x3fc35a2b2f13e6e9, 0x3fc4f4efa8fef709, 0x3fc6942d3720185a, + 0x0000000000000000, 0xbfc331751ec3a814, 0xbfc20224341286e4, 0xbfc0cf85bed0f8b7, + 0xbfbf332113d56b1f, 0xbfbcc0768d4175a6, 0xbfba46f918837cb7, 0xbfb7c695afc3b424, + 0xbfb53f391822dbc7, 0xbfb2b0cfe1266bd4, 0xbfb01b466423250a, 0xbfaafd11874c009e, + 0xbfa5b505d5b6f268, 0xbfa05e4119ea5d89, 0xbf95f134923757f3, 0xbf860f9f985bc9f4, + }, + + /* Generated using Remez, in [-log(2)/128, log(2)/128]. */ + .c0 = 0x1p-1, + .c1 = 0x1.55555555548f9p-3, + .c2 = 0x1.5555555554c22p-5, + .c3 = 0x1.111123aaa2fb2p-7, + .c4 = 0x1.6c16d77d98e5bp-10, + .ln2_hi = 0x1.62e42fefa3800p-1, + .ln2_lo = 0x1.ef35793c76730p-45, + .inv_ln2 = 0x1.71547652b82fep+0, + .shift = 0x1.800000000ffc0p+46, /* 1.5*2^46+1023. */ + .thres = SpecialBound, }; -static svfloat64_t NOINLINE -special_case (svfloat64_t x, svfloat64_t y, svbool_t pg) +#define SpecialOffset 0x6000000000000000 /* 0x1p513. */ +/* SpecialBias1 + SpecialBias1 = asuint(1.0). */ +#define SpecialBias1 0x7000000000000000 /* 0x1p769. */ +#define SpecialBias2 0x3010000000000000 /* 0x1p-254. */ + +static NOINLINE svfloat64_t +special_case (svbool_t pg, svfloat64_t y, svfloat64_t s, svfloat64_t p, + svfloat64_t n) { - return sv_call_f64 (expm1, x, y, pg); + /* s=2^n may overflow, break it up into s=s1*s2, + such that exp = s + s*y can be computed as s1*(s2+s2*y) + and s1*s1 overflows only if n>0. */ + + /* If n<=0 then set b to 0x6, 0 otherwise. */ + svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */ + svuint64_t b + = svdup_u64_z (p_sign, SpecialOffset); /* Inactive lanes set to 0. */ + + /* Set s1 to generate overflow depending on sign of exponent n, + ie. s1 = 0x70...0 - b. */ + svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1)); + /* Offset s to avoid overflow in final result if n is below threshold. + ie. s2 = as_u64 (s) - 0x3010...0 + b. */ + svfloat64_t s2 = svreinterpret_f64 ( + svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b)); + + /* |n| > 1280 => 2^(n) overflows. */ + svbool_t p_cmp = svacgt (pg, n, 1280.0); + + svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1); + svfloat64_t r2 = svmla_x (pg, s2, s2, p); + svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1); + + svbool_t is_safe = svacle (pg, n, 1023); /* Only correct special lanes. */ + return svsel (is_safe, y, svsub_x (pg, svsel (p_cmp, r1, r0), 1.0)); } -/* Double-precision vector exp(x) - 1 function. - The maximum error observed error is 2.18 ULP: - _ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2 - want 0x1.a8b9ea8d66e2p-2. */ +/* FEXPA based SVE expm1 algorithm. + Maximum measured error is 2.81 + 0.5 ULP: + _ZGVsMxv_expm1 (0x1.974060e619bfp-3) got 0x1.c290e5858bb53p-3 + want 0x1.c290e5858bb5p-3. */ svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg) { const struct data *d = ptr_barrier (&data); - /* Large, Nan/Inf. */ - svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound)); - - /* Reduce argument to smaller range: - Let i = round(x / ln2) - and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. - exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 - where 2^i is exact because i is an integer. */ - svfloat64_t shift = sv_f64 (d->shift); - svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift); - svint64_t i = svcvt_s64_x (pg, n); - svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi); - svfloat64_t f = svmls_lane (x, n, ln2, 0); - f = svmls_lane (f, n, ln2, 1); - - /* Approximate expm1(f) using polynomial. - Taylor expansion for expm1(x) has the form: - x + ax^2 + bx^3 + cx^4 .... - So we calculate the polynomial P(f) = a + bf + cf^2 + ... - and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ - svfloat64_t f2 = svmul_x (pg, f, f); - svfloat64_t f4 = svmul_x (pg, f2, f2); - svfloat64_t f8 = svmul_x (pg, f4, f4); - svfloat64_t p - = svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, f, f2, f4, f8, d->poly)); - - /* Assemble the result. - expm1(x) ~= 2^i * (p + 1) - 1 - Let t = 2^i. */ - svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias); - svfloat64_t t = svreinterpret_f64 (u); - - /* expm1(x) ~= p * t + (t - 1). */ - svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t); + svbool_t special = svacgt (pg, x, d->thres); - if (__glibc_unlikely (svptest_any (pg, special))) - return special_case (x, y, special); + svfloat64_t z = svmla_x (pg, sv_f64 (d->shift), x, d->inv_ln2); + svuint64_t u = svreinterpret_u64 (z); + svfloat64_t n = svsub_x (pg, z, d->shift); + /* r = x - n * ln2, r is in [-ln2/128, ln2/128]. */ + svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi); + svfloat64_t r = x; + r = svmls_lane (r, n, ln2, 0); + r = svmls_lane (r, n, ln2, 1); + + /* y = exp(r) - 1 ~= r + C0 r^2 + C1 r^3 + C2 r^4 + C3 r^5 + C4 r^6. */ + svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r); + svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2); + + svfloat64_t p; + svfloat64_t c12 = svmla_lane (sv_f64 (d->c1), r, c24, 0); + svfloat64_t c34 = svmla_lane (sv_f64 (d->c3), r, c24, 1); + p = svmad_x (pg, c34, r2, c12); + p = svmad_x (pg, p, r, sv_f64 (d->c0)); + p = svmad_x (pg, p, r2, r); + + svfloat64_t scale = svexpa (u); + svfloat64_t scalem1 = svsub_x (pg, scale, sv_f64 (1.0)); + + /* We want to construct expm1(x) = (scale - 1) + scale * poly. + However, for values of scale close to 1, scale-1 causes large ULP errors + due to cancellation. + + This can be circumvented by using a small lookup for scale-1 + when our input is below a certain bound, otherwise we can use FEXPA. + + This bound is based upon the table size: + Bound = (TableSize-1/64) * ln2. + The current bound is based upon a table size of 16. */ + svbool_t is_small = svaclt (pg, x, FexpaBound); + + if (svptest_any (pg, is_small)) + { + /* Index via the input of FEXPA, but we only care about the lower 4 bits. + */ + svuint64_t base_idx = svand_x (pg, u, 0xf); + + /* We can use the sign of x as a fifth bit to account for the asymmetry + of e^x around 0. */ + svuint64_t signBit + = svlsl_x (pg, svlsr_x (pg, svreinterpret_u64 (x), 63), 4); + svuint64_t idx = svorr_x (pg, base_idx, signBit); + + /* Lookup values for scale - 1 for small x. */ + svfloat64_t lookup = svreinterpret_f64 ( + svld1_gather_index (is_small, d->expm1_data, idx)); + + /* Select the appropriate scale - 1 value based on x. */ + scalem1 = svsel (is_small, lookup, scalem1); + } + + svfloat64_t y = svmla_x (pg, scalem1, scale, p); + + /* FEXPA returns nan for large inputs so we special case those. */ + if (__glibc_unlikely (svptest_any (pg, special))) + { + /* FEXPA zeroes the sign bit, however the sign is meaningful to the + special case function so needs to be copied. + e = sign bit of u << 46. */ + svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000); + /* Copy sign to s. */ + scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale))); + return special_case (pg, y, scale, p, n); + } + + /* return expm1 = (scale - 1) + (scale * poly). */ return y; } |