160 lines
4.2 KiB
C
160 lines
4.2 KiB
C
/*
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* Double-precision scalar atan2(x) function.
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*
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* Copyright (c) 2021-2023, Arm Limited.
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* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
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*/
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#include <stdbool.h>
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#include "atan_common.h"
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#include "math_config.h"
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#include "pl_sig.h"
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#include "pl_test.h"
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#define Pi (0x1.921fb54442d18p+1)
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#define PiOver2 (0x1.921fb54442d18p+0)
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#define PiOver4 (0x1.921fb54442d18p-1)
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#define SignMask (0x8000000000000000)
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#define ExpMask (0x7ff0000000000000)
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/* We calculate atan2 by P(n/d), where n and d are similar to the input
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arguments, and P is a polynomial. Evaluating P(x) requires calculating x^8,
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which may underflow if n and d have very different magnitude.
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POW8_EXP_UFLOW_BOUND is the lower bound of the difference in exponents of n
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and d for which P underflows, and is used to special-case such inputs. */
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#define POW8_EXP_UFLOW_BOUND 62
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static inline int64_t
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biased_exponent (double f)
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{
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uint64_t fi = asuint64 (f);
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return (fi & ExpMask) >> 52;
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}
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/* Fast implementation of scalar atan2. Largest errors are when y and x are
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close together. The greatest observed error is 2.28 ULP:
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atan2(-0x1.5915b1498e82fp+732, 0x1.54d11ef838826p+732)
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got -0x1.954f42f1fa841p-1 want -0x1.954f42f1fa843p-1. */
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double
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atan2 (double y, double x)
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{
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uint64_t ix = asuint64 (x);
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uint64_t iy = asuint64 (y);
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uint64_t sign_x = ix & SignMask;
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uint64_t sign_y = iy & SignMask;
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uint64_t iax = ix & ~SignMask;
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uint64_t iay = iy & ~SignMask;
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bool xisnan = isnan (x);
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if (unlikely (isnan (y) && !xisnan))
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return __math_invalid (y);
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if (unlikely (xisnan))
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return __math_invalid (x);
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/* m = 2 * sign(x) + sign(y). */
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uint32_t m = ((iy >> 63) & 1) | ((ix >> 62) & 2);
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int64_t exp_diff = biased_exponent (x) - biased_exponent (y);
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/* y = 0. */
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if (iay == 0)
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{
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switch (m)
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{
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case 0:
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case 1:
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return y; /* atan(+-0,+anything)=+-0. */
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case 2:
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return Pi; /* atan(+0,-anything) = pi. */
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case 3:
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return -Pi; /* atan(-0,-anything) =-pi. */
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}
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}
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/* Special case for (x, y) either on or very close to the y axis. Either x =
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0, or y is much larger than x (difference in exponents >=
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POW8_EXP_UFLOW_BOUND). */
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if (unlikely (iax == 0 || exp_diff <= -POW8_EXP_UFLOW_BOUND))
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return sign_y ? -PiOver2 : PiOver2;
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/* Special case for either x is INF or (x, y) is very close to x axis and x is
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negative. */
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if (unlikely (iax == 0x7ff0000000000000
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|| (exp_diff >= POW8_EXP_UFLOW_BOUND && m >= 2)))
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{
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if (iay == 0x7ff0000000000000)
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{
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switch (m)
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{
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case 0:
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return PiOver4; /* atan(+INF,+INF). */
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case 1:
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return -PiOver4; /* atan(-INF,+INF). */
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case 2:
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return 3.0 * PiOver4; /* atan(+INF,-INF). */
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case 3:
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return -3.0 * PiOver4; /* atan(-INF,-INF). */
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}
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}
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else
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{
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switch (m)
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{
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case 0:
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return 0.0; /* atan(+...,+INF). */
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case 1:
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return -0.0; /* atan(-...,+INF). */
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case 2:
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return Pi; /* atan(+...,-INF). */
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case 3:
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return -Pi; /* atan(-...,-INF). */
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}
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}
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}
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/* y is INF. */
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if (iay == 0x7ff0000000000000)
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return sign_y ? -PiOver2 : PiOver2;
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uint64_t sign_xy = sign_x ^ sign_y;
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double ax = asdouble (iax);
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double ay = asdouble (iay);
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uint64_t pred_aygtax = (ay > ax);
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/* Set up z for call to atan. */
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double n = pred_aygtax ? -ax : ay;
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double d = pred_aygtax ? ay : ax;
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double z = n / d;
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double ret;
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if (unlikely (m < 2 && exp_diff >= POW8_EXP_UFLOW_BOUND))
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{
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/* If (x, y) is very close to x axis and x is positive, the polynomial
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will underflow and evaluate to z. */
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ret = z;
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}
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else
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{
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/* Work out the correct shift. */
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double shift = sign_x ? -2.0 : 0.0;
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shift = pred_aygtax ? shift + 1.0 : shift;
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shift *= PiOver2;
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ret = eval_poly (z, z, shift);
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}
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/* Account for the sign of x and y. */
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return asdouble (asuint64 (ret) ^ sign_xy);
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}
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/* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
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PL_SIG (S, D, 2, atan2)
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PL_TEST_ULP (atan2, 1.78)
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PL_TEST_INTERVAL (atan2, -10.0, 10.0, 50000)
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PL_TEST_INTERVAL (atan2, -1.0, 1.0, 40000)
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PL_TEST_INTERVAL (atan2, 0.0, 1.0, 40000)
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PL_TEST_INTERVAL (atan2, 1.0, 100.0, 40000)
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PL_TEST_INTERVAL (atan2, 1e6, 1e32, 40000)
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