diff options
Diffstat (limited to 'sysdeps/aarch64/fpu/atanf_advsimd.c')
| -rw-r--r-- | sysdeps/aarch64/fpu/atanf_advsimd.c | 97 |
1 files changed, 59 insertions, 38 deletions
diff --git a/sysdeps/aarch64/fpu/atanf_advsimd.c b/sysdeps/aarch64/fpu/atanf_advsimd.c index 472865e..817a47e 100644 --- a/sysdeps/aarch64/fpu/atanf_advsimd.c +++ b/sysdeps/aarch64/fpu/atanf_advsimd.c @@ -22,26 +22,35 @@ static const struct data { + uint32x4_t sign_mask, pi_over_2; + float32x4_t neg_one; +#if WANT_SIMD_EXCEPT float32x4_t poly[8]; - float32x4_t pi_over_2; +} data = { + .poly = { V4 (-0x1.5554dcp-2), V4 (0x1.9978ecp-3), V4 (-0x1.230a94p-3), + V4 (0x1.b4debp-4), V4 (-0x1.3550dap-4), V4 (0x1.61eebp-5), + V4 (-0x1.0c17d4p-6), V4 (0x1.7ea694p-9) }, +#else + float32x4_t c0, c2, c4, c6; + float c1, c3, c5, c7; } data = { /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on [2**-128, 1.0]. Generated using fpminimax between FLT_MIN and 1. */ - .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f), - V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f), - V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) }, - .pi_over_2 = V4 (0x1.921fb6p+0f), + .c0 = V4 (-0x1.5554dcp-2), .c1 = 0x1.9978ecp-3, + .c2 = V4 (-0x1.230a94p-3), .c3 = 0x1.b4debp-4, + .c4 = V4 (-0x1.3550dap-4), .c5 = 0x1.61eebp-5, + .c6 = V4 (-0x1.0c17d4p-6), .c7 = 0x1.7ea694p-9, +#endif + .pi_over_2 = V4 (0x3fc90fdb), + .neg_one = V4 (-1.0f), + .sign_mask = V4 (0x80000000), }; -#define SignMask v_u32 (0x80000000) - -#define P(i) d->poly[i] - +#if WANT_SIMD_EXCEPT #define TinyBound 0x30800000 /* asuint(0x1p-30). */ #define BigBound 0x4e800000 /* asuint(0x1p30). */ -#if WANT_SIMD_EXCEPT static float32x4_t VPCS_ATTR NOINLINE special_case (float32x4_t x, float32x4_t y, uint32x4_t special) { @@ -51,19 +60,20 @@ special_case (float32x4_t x, float32x4_t y, uint32x4_t special) /* Fast implementation of vector atanf based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] - using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps: - _ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */ + using z=-1/x and shift = pi/2. Maximum observed error is 2.02 ulps: + _ZGVnN4v_atanf (0x1.03d4cep+0) got 0x1.95ed3ap-1 + want 0x1.95ed36p-1. */ float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x) { const struct data *d = ptr_barrier (&data); - /* Small cases, infs and nans are supported by our approximation technique, - but do not set fenv flags correctly. Only trigger special case if we need - fenv. */ uint32x4_t ix = vreinterpretq_u32_f32 (x); - uint32x4_t sign = vandq_u32 (ix, SignMask); + uint32x4_t sign = vandq_u32 (ix, d->sign_mask); #if WANT_SIMD_EXCEPT + /* Small cases, infs and nans are supported by our approximation technique, + but do not set fenv flags correctly. Only trigger special case if we need + fenv. */ uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000)); uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)), v_u32 (BigBound - TinyBound)); @@ -71,41 +81,52 @@ float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x) if (__glibc_unlikely (v_any_u32 (special))) return special_case (x, x, v_u32 (-1)); #endif - /* Argument reduction: - y := arctan(x) for x < 1 - y := pi/2 + arctan(-1/x) for x > 1 - Hence, use z=-1/a if x>=1, otherwise z=a. */ - uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0)); - /* Avoid dependency in abs(x) in division (and comparison). */ - float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x); + y := arctan(x) for |x| < 1 + y := arctan(-1/x) + pi/2 for x > +1 + y := arctan(-1/x) - pi/2 for x < -1 + Hence, use z=-1/a if x>=|-1|, otherwise z=a. */ + uint32x4_t red = vcagtq_f32 (x, d->neg_one); + + float32x4_t z = vbslq_f32 (red, vdivq_f32 (d->neg_one, x), x); + + /* Shift is calculated as +-pi/2 or 0, depending on the argument case. */ float32x4_t shift = vreinterpretq_f32_u32 ( - vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2))); - /* Use absolute value only when needed (odd powers of z). */ - float32x4_t az = vbslq_f32 ( - SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z); + vandq_u32 (red, veorq_u32 (d->pi_over_2, sign))); + + float32x4_t z2 = vmulq_f32 (z, z); + float32x4_t z3 = vmulq_f32 (z, z2); + float32x4_t z4 = vmulq_f32 (z2, z2); +#if WANT_SIMD_EXCEPT /* Calculate the polynomial approximation. Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However, a standard implementation using z8 creates spurious underflow in the very last fma (when z^8 is small enough). - Therefore, we split the last fma into a mul and an fma. - Horner and single-level Estrin have higher errors that exceed - threshold. */ - float32x4_t z2 = vmulq_f32 (z, z); - float32x4_t z4 = vmulq_f32 (z2, z2); - + Therefore, we split the last fma into a mul and an fma. */ float32x4_t y = vfmaq_f32 ( v_pairwise_poly_3_f32 (z2, z4, d->poly), z4, vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4))); - /* y = shift + z * P(z^2). */ - y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift); +#else + float32x4_t z8 = vmulq_f32 (z4, z4); + + /* Uses an Estrin scheme for polynomial approximation. */ + float32x4_t odd_coeffs = vld1q_f32 (&d->c1); + + float32x4_t p01 = vfmaq_laneq_f32 (d->c0, z2, odd_coeffs, 0); + float32x4_t p23 = vfmaq_laneq_f32 (d->c2, z2, odd_coeffs, 1); + float32x4_t p45 = vfmaq_laneq_f32 (d->c4, z2, odd_coeffs, 2); + float32x4_t p67 = vfmaq_laneq_f32 (d->c6, z2, odd_coeffs, 3); - /* y = atan(x) if x>0, -atan(-x) otherwise. */ - y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign)); + float32x4_t p03 = vfmaq_f32 (p01, z4, p23); + float32x4_t p47 = vfmaq_f32 (p45, z4, p67); - return y; + float32x4_t y = vfmaq_f32 (p03, z8, p47); +#endif + + /* y = shift + z * P(z^2). */ + return vfmaq_f32 (vaddq_f32 (shift, z), z3, y); } libmvec_hidden_def (V_NAME_F1 (atan)) HALF_WIDTH_ALIAS_F1 (atan) |