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Diffstat (limited to 'pl/math/tanf_3u3.c')
| -rw-r--r-- | pl/math/tanf_3u3.c | 202 |
1 files changed, 202 insertions, 0 deletions
diff --git a/pl/math/tanf_3u3.c b/pl/math/tanf_3u3.c new file mode 100644 index 0000000..ec006dc --- /dev/null +++ b/pl/math/tanf_3u3.c @@ -0,0 +1,202 @@ +/* + * Single-precision scalar tan(x) function. + * + * Copyright (c) 2021-2023, Arm Limited. + * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception + */ +#include "math_config.h" +#include "pl_sig.h" +#include "pl_test.h" +#include "pairwise_hornerf.h" + +/* Useful constants. */ +#define NegPio2_1 (-0x1.921fb6p+0f) +#define NegPio2_2 (0x1.777a5cp-25f) +#define NegPio2_3 (0x1.ee59dap-50f) +/* Reduced from 0x1p20 to 0x1p17 to ensure 3.5ulps. */ +#define RangeVal (0x1p17f) +#define InvPio2 ((0x1.45f306p-1f)) +#define Shift (0x1.8p+23f) +#define AbsMask (0x7fffffff) +#define Pio4 (0x1.921fb6p-1) +/* 2PI * 2^-64. */ +#define Pio2p63 (0x1.921FB54442D18p-62) + +#define P(i) __tanf_poly_data.poly_tan[i] +#define Q(i) __tanf_poly_data.poly_cotan[i] + +static inline float +eval_P (float z) +{ + return PAIRWISE_HORNER_5 (z, z * z, P); +} + +static inline float +eval_Q (float z) +{ + return PAIRWISE_HORNER_3 (z, z * z, Q); +} + +/* Reduction of the input argument x using Cody-Waite approach, such that x = r + + n * pi/2 with r lives in [-pi/4, pi/4] and n is a signed integer. */ +static inline float +reduce (float x, int32_t *in) +{ + /* n = rint(x/(pi/2)). */ + float r = x; + float q = fmaf (InvPio2, r, Shift); + float n = q - Shift; + /* There is no rounding here, n is representable by a signed integer. */ + *in = (int32_t) n; + /* r = x - n * (pi/2) (range reduction into -pi/4 .. pi/4). */ + r = fmaf (NegPio2_1, n, r); + r = fmaf (NegPio2_2, n, r); + r = fmaf (NegPio2_3, n, r); + return r; +} + +/* Table with 4/PI to 192 bit precision. To avoid unaligned accesses + only 8 new bits are added per entry, making the table 4 times larger. */ +static const uint32_t __inv_pio4[24] + = {0x000000a2, 0x0000a2f9, 0x00a2f983, 0xa2f9836e, 0xf9836e4e, 0x836e4e44, + 0x6e4e4415, 0x4e441529, 0x441529fc, 0x1529fc27, 0x29fc2757, 0xfc2757d1, + 0x2757d1f5, 0x57d1f534, 0xd1f534dd, 0xf534ddc0, 0x34ddc0db, 0xddc0db62, + 0xc0db6295, 0xdb629599, 0x6295993c, 0x95993c43, 0x993c4390, 0x3c439041}; + +/* Reduce the range of XI to a multiple of PI/2 using fast integer arithmetic. + XI is a reinterpreted float and must be >= 2.0f (the sign bit is ignored). + Return the modulo between -PI/4 and PI/4 and store the quadrant in NP. + Reduction uses a table of 4/PI with 192 bits of precision. A 32x96->128 bit + multiply computes the exact 2.62-bit fixed-point modulo. Since the result + can have at most 29 leading zeros after the binary point, the double + precision result is accurate to 33 bits. */ +static inline double +reduce_large (uint32_t xi, int *np) +{ + const uint32_t *arr = &__inv_pio4[(xi >> 26) & 15]; + int shift = (xi >> 23) & 7; + uint64_t n, res0, res1, res2; + + xi = (xi & 0xffffff) | 0x800000; + xi <<= shift; + + res0 = xi * arr[0]; + res1 = (uint64_t) xi * arr[4]; + res2 = (uint64_t) xi * arr[8]; + res0 = (res2 >> 32) | (res0 << 32); + res0 += res1; + + n = (res0 + (1ULL << 61)) >> 62; + res0 -= n << 62; + double x = (int64_t) res0; + *np = n; + return x * Pio2p63; +} + +/* Top 12 bits of the float representation with the sign bit cleared. */ +static inline uint32_t +top12 (float x) +{ + return (asuint (x) >> 20); +} + +/* Fast single-precision tan implementation. + Maximum ULP error: 3.293ulps. + tanf(0x1.c849eap+16) got -0x1.fe8d98p-1 want -0x1.fe8d9ep-1. */ +float +tanf (float x) +{ + /* Get top words. */ + uint32_t ix = asuint (x); + uint32_t ia = ix & AbsMask; + uint32_t ia12 = ia >> 20; + + /* Dispatch between no reduction (small numbers), fast reduction and + slow large numbers reduction. The reduction step determines r float + (|r| < pi/4) and n signed integer such that x = r + n * pi/2. */ + int32_t n; + float r; + if (ia12 < top12 (Pio4)) + { + /* Optimize small values. */ + if (unlikely (ia12 < top12 (0x1p-12f))) + { + if (unlikely (ia12 < top12 (0x1p-126f))) + /* Force underflow for tiny x. */ + force_eval_float (x * x); + return x; + } + + /* tan (x) ~= x + x^3 * P(x^2). */ + float x2 = x * x; + float y = eval_P (x2); + return fmaf (x2, x * y, x); + } + /* Similar to other trigonometric routines, fast inaccurate reduction is + performed for values of x from pi/4 up to RangeVal. In order to keep errors + below 3.5ulps, we set the value of RangeVal to 2^17. This might differ for + other trigonometric routines. Above this value more advanced but slower + reduction techniques need to be implemented to reach a similar accuracy. + */ + else if (ia12 < top12 (RangeVal)) + { + /* Fast inaccurate reduction. */ + r = reduce (x, &n); + } + else if (ia12 < 0x7f8) + { + /* Slow accurate reduction. */ + uint32_t sign = ix & ~AbsMask; + double dar = reduce_large (ia, &n); + float ar = (float) dar; + r = asfloat (asuint (ar) ^ sign); + } + else + { + /* tan(Inf or NaN) is NaN. */ + return __math_invalidf (x); + } + + /* If x lives in an interval where |tan(x)| + - is finite then use an approximation of tangent in the form + tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2). + - grows to infinity then use an approximation of cotangent in the form + cotan(z) ~ 1/z + z * Q(z^2), where the reciprocal can be computed early. + Using symmetries of tangent and the identity tan(r) = cotan(pi/2 - r), + we only need to change the sign of r to obtain tan(x) from cotan(r). + This 2-interval approach requires 2 different sets of coefficients P and + Q, where Q is a lower order polynomial than P. */ + + /* Determine if x lives in an interval where |tan(x)| grows to infinity. */ + uint32_t alt = (uint32_t) n & 1; + + /* Perform additional reduction if required. */ + float z = alt ? -r : r; + + /* Prepare backward transformation. */ + float z2 = r * r; + float offset = alt ? 1.0f / z : z; + float scale = alt ? z : z * z2; + + /* Evaluate polynomial approximation of tan or cotan. */ + float p = alt ? eval_Q (z2) : eval_P (z2); + + /* A unified way of assembling the result on both interval types. */ + return fmaf (scale, p, offset); +} + +PL_SIG (S, F, 1, tan, -3.1, 3.1) +PL_TEST_ULP (tanf, 2.80) +PL_TEST_INTERVAL (tanf, 0, 0xffff0000, 10000) +PL_TEST_INTERVAL (tanf, 0x1p-127, 0x1p-14, 50000) +PL_TEST_INTERVAL (tanf, -0x1p-127, -0x1p-14, 50000) +PL_TEST_INTERVAL (tanf, 0x1p-14, 0.7, 50000) +PL_TEST_INTERVAL (tanf, -0x1p-14, -0.7, 50000) +PL_TEST_INTERVAL (tanf, 0.7, 1.5, 50000) +PL_TEST_INTERVAL (tanf, -0.7, -1.5, 50000) +PL_TEST_INTERVAL (tanf, 1.5, 0x1p17, 50000) +PL_TEST_INTERVAL (tanf, -1.5, -0x1p17, 50000) +PL_TEST_INTERVAL (tanf, 0x1p17, 0x1p54, 50000) +PL_TEST_INTERVAL (tanf, -0x1p17, -0x1p54, 50000) +PL_TEST_INTERVAL (tanf, 0x1p54, inf, 50000) +PL_TEST_INTERVAL (tanf, -0x1p54, -inf, 50000) |