diff options
Diffstat (limited to 'sysdeps/aarch64/fpu/tanh_sve.c')
| -rw-r--r-- | sysdeps/aarch64/fpu/tanh_sve.c | 154 |
1 files changed, 94 insertions, 60 deletions
diff --git a/sysdeps/aarch64/fpu/tanh_sve.c b/sysdeps/aarch64/fpu/tanh_sve.c index 789cc68..5869419 100644 --- a/sysdeps/aarch64/fpu/tanh_sve.c +++ b/sysdeps/aarch64/fpu/tanh_sve.c @@ -18,83 +18,117 @@ <https://www.gnu.org/licenses/>. */ #include "sv_math.h" -#include "poly_sve_f64.h" static const struct data { - float64_t poly[11]; - float64_t inv_ln2, ln2_hi, ln2_lo, shift; - uint64_t thresh, tiny_bound; + double ln2_hi, ln2_lo; + double c2, c4; + double c0, c1, c3; + double two_over_ln2, shift; + uint64_t tiny_bound; + double large_bound, fexpa_bound; + uint64_t e2xm1_data[20]; } data = { - /* Generated using Remez, deg=12 in [-log(2)/2, log(2)/2]. */ - .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, }, - - .inv_ln2 = 0x1.71547652b82fep0, - .ln2_hi = -0x1.62e42fefa39efp-1, - .ln2_lo = -0x1.abc9e3b39803fp-56, - .shift = 0x1.8p52, - + /* 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, + .two_over_ln2 = 0x1.71547652b82fep+1, + .shift = 0x1.800000000ffc0p+46, /* 1.5*2^46+1023. */ .tiny_bound = 0x3e40000000000000, /* asuint64 (0x1p-27). */ - /* asuint64(0x1.241bf835f9d5fp+4) - asuint64(tiny_bound). */ - .thresh = 0x01f241bf835f9d5f, + .large_bound = 0x1.30fc1931f09cap+4, /* arctanh(1 - 2^-54). */ + .fexpa_bound = 0x1.a56ef8ec924ccp-4, /* 19/64 * ln2/2. */ + /* Table lookup of 2^(i/64) - 1, for values of i from 0..19. */ + .e2xm1_data = { + 0x0000000000000000, 0x3f864d1f3bc03077, 0x3f966c34c5615d0f, 0x3fa0e8a30eb37901, + 0x3fa6ab0d9f3121ec, 0x3fac7d865a7a3440, 0x3fb1301d0125b50a, 0x3fb429aaea92ddfb, + 0x3fb72b83c7d517ae, 0x3fba35beb6fcb754, 0x3fbd4873168b9aa8, 0x3fc031dc431466b2, + 0x3fc1c3d373ab11c3, 0x3fc35a2b2f13e6e9, 0x3fc4f4efa8fef709, 0x3fc6942d3720185a, + 0x3fc837f0518db8a9, 0x3fc9e0459320b7fa, 0x3fcb8d39b9d54e55, 0x3fcd3ed9a72cffb7, + }, }; +/* An expm1 inspired, FEXPA based helper function that returns an + accurate estimate for e^2x - 1. With no special case or support for + negative inputs of x. */ static inline svfloat64_t -expm1_inline (svfloat64_t x, const svbool_t pg, const struct data *d) -{ - /* Helper routine for calculating exp(x) - 1. Vector port of the helper from - the scalar variant of tanh. */ - - /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */ - svfloat64_t j - = svsub_x (pg, svmla_x (pg, sv_f64 (d->shift), x, d->inv_ln2), d->shift); - svint64_t i = svcvt_s64_x (pg, j); - svfloat64_t f = svmla_x (pg, x, j, d->ln2_hi); - f = svmla_x (pg, f, j, d->ln2_lo); - - /* Approximate expm1(f) using polynomial. */ - svfloat64_t f2 = svmul_x (pg, f, f); - svfloat64_t f4 = svmul_x (pg, f2, f2); - svfloat64_t p = svmla_x ( - pg, f, f2, - sv_estrin_10_f64_x (pg, f, f2, f4, svmul_x (pg, f4, f4), d->poly)); - - /* t = 2 ^ i. */ - svfloat64_t t = svscale_x (pg, sv_f64 (1), i); - /* expm1(x) = p * t + (t - 1). */ - return svmla_x (pg, svsub_x (pg, t, 1), p, t); -} - -static svfloat64_t NOINLINE -special_case (svfloat64_t x, svfloat64_t y, svbool_t special) +e2xm1_inline (const svbool_t pg, svfloat64_t x, const struct data *d) { - return sv_call_f64 (tanh, x, y, special); + svfloat64_t z = svmla_x (pg, sv_f64 (d->shift), x, d->two_over_ln2); + svuint64_t u = svreinterpret_u64 (z); + svfloat64_t n = svsub_x (pg, z, d->shift); + + /* r = x - n * ln2/2, r is in [-ln2/(2N), ln2/(2N)]. */ + svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi); + svfloat64_t r = svadd_x (pg, x, 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); + + /* We want to construct e2xm1(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. */ + svbool_t is_small = svaclt (pg, x, d->fexpa_bound); + + /* Index via the input of FEXPA, but we only care about the lower 5 bits. */ + svuint64_t base_idx = svand_x (pg, u, 0x1f); + + /* Compute scale - 1 from FEXPA, and lookup values where this fails. */ + svfloat64_t scalem1_estimate = svsub_x (pg, scale, sv_f64 (1.0)); + svuint64_t scalem1_lookup + = svld1_gather_index (is_small, d->e2xm1_data, base_idx); + + /* Select the appropriate scale - 1 value based on x. */ + svfloat64_t scalem1 + = svsel (is_small, svreinterpret_f64 (scalem1_lookup), scalem1_estimate); + return svmla_x (pg, scalem1, scale, p); } -/* SVE approximation for double-precision tanh(x), using a simplified - version of expm1. The greatest observed error is 2.77 ULP: - _ZGVsMxv_tanh(-0x1.c4a4ca0f9f3b7p-3) got -0x1.bd6a21a163627p-3 - want -0x1.bd6a21a163624p-3. */ +/* SVE approximation for double-precision tanh(x), using a modified version of + FEXPA expm1 to calculate e^2x - 1. + The greatest observed error is 2.79 + 0.5 ULP: + _ZGVsMxv_tanh (0x1.fff868eb3c223p-9) got 0x1.fff7be486cae6p-9 + want 0x1.fff7be486cae9p-9. */ svfloat64_t SV_NAME_D1 (tanh) (svfloat64_t x, svbool_t pg) { const struct data *d = ptr_barrier (&data); - svuint64_t ia = svreinterpret_u64 (svabs_x (pg, x)); + svbool_t large = svacge (pg, x, d->large_bound); - /* Trigger special-cases for tiny, boring and infinity/NaN. */ - svbool_t special = svcmpgt (pg, svsub_x (pg, ia, d->tiny_bound), d->thresh); + /* We can use tanh(x) = (e^2x - 1) / (e^2x + 1) to approximate tanh. + As an additional optimisation, we can ensure more accurate values of e^x + by only using positive inputs. So we calculate tanh(|x|), and restore the + sign of the input before returning. */ + svfloat64_t ax = svabs_x (pg, x); + svuint64_t sign_bit + = sveor_x (pg, svreinterpret_u64 (x), svreinterpret_u64 (ax)); - svfloat64_t u = svadd_x (pg, x, x); + svfloat64_t p = e2xm1_inline (pg, ax, d); + svfloat64_t q = svadd_x (pg, p, 2); - /* tanh(x) = (e^2x - 1) / (e^2x + 1). */ - svfloat64_t q = expm1_inline (u, pg, d); - svfloat64_t qp2 = svadd_x (pg, q, 2); + /* For sufficiently high inputs, the result of tanh(|x|) is 1 when correctly + rounded, at this point we can return 1 directly, with sign correction. + This will also act as a guard against our approximation overflowing. */ + svfloat64_t y = svsel (large, sv_f64 (1.0), svdiv_x (pg, p, q)); - if (__glibc_unlikely (svptest_any (pg, special))) - return special_case (x, svdiv_x (pg, q, qp2), special); - return svdiv_x (pg, q, qp2); + return svreinterpret_f64 (svorr_x (pg, sign_bit, svreinterpret_u64 (y))); } |