286 lines
11 KiB
C++
Executable File
286 lines
11 KiB
C++
Executable File
/*============================================================================
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This source file is an extension to the SoftFloat IEC/IEEE Floating-point
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Arithmetic Package, Release 2b, written for Bochs (x86 achitecture simulator)
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floating point emulation.
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THIS SOFTWARE IS DISTRIBUTED AS IS, FOR FREE. Although reasonable effort has
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been made to avoid it, THIS SOFTWARE MAY CONTAIN FAULTS THAT WILL AT TIMES
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RESULT IN INCORRECT BEHAVIOR. USE OF THIS SOFTWARE IS RESTRICTED TO PERSONS
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AND ORGANIZATIONS WHO CAN AND WILL TAKE FULL RESPONSIBILITY FOR ALL LOSSES,
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COSTS, OR OTHER PROBLEMS THEY INCUR DUE TO THE SOFTWARE, AND WHO FURTHERMORE
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EFFECTIVELY INDEMNIFY JOHN HAUSER AND THE INTERNATIONAL COMPUTER SCIENCE
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INSTITUTE (possibly via similar legal warning) AGAINST ALL LOSSES, COSTS, OR
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OTHER PROBLEMS INCURRED BY THEIR CUSTOMERS AND CLIENTS DUE TO THE SOFTWARE.
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Derivative works are acceptable, even for commercial purposes, so long as
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(1) the source code for the derivative work includes prominent notice that
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the work is derivative, and (2) the source code includes prominent notice with
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these four paragraphs for those parts of this code that are retained.
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=============================================================================*/
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/*============================================================================
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* Written for Bochs (x86 achitecture simulator) by
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* Stanislav Shwartsman (gate at fidonet.org.il)
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* ==========================================================================*/
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#define FLOAT128
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#include "softfloatx80.h"
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#include "softfloat-round-pack.h"
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#include "fpu_constant.h"
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#define FPATAN_ARR_SIZE 11
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static const float128 float128_one =
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packFloat128(BX_CONST64(0x3fff000000000000), BX_CONST64(0x0000000000000000));
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static const float128 float128_sqrt3 =
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packFloat128(BX_CONST64(0x3fffbb67ae8584ca), BX_CONST64(0xa73b25742d7078b8));
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static const floatx80 floatx80_pi =
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packFloatx80(0, 0x4000, BX_CONST64(0xc90fdaa22168c235));
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static const floatx80 floatx80_pi2 =
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packFloatx80(0, 0x3fff, BX_CONST64(0xc90fdaa22168c235));
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static const floatx80 floatx80_pi4 =
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packFloatx80(0, 0x3ffe, BX_CONST64(0xc90fdaa22168c235));
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static const floatx80 floatx80_pi6 =
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packFloatx80(0, 0x3ffe, BX_CONST64(0x860a91c16b9b2c23));
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static float128 atan_arr[FPATAN_ARR_SIZE] =
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{
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packFloat128(BX_CONST64(0x3fff000000000000), BX_CONST64(0x0000000000000000)), /* 1 */
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packFloat128(BX_CONST64(0xbffd555555555555), BX_CONST64(0x5555555555555555)), /* 3 */
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packFloat128(BX_CONST64(0x3ffc999999999999), BX_CONST64(0x999999999999999a)), /* 5 */
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packFloat128(BX_CONST64(0xbffc249249249249), BX_CONST64(0x2492492492492492)), /* 7 */
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packFloat128(BX_CONST64(0x3ffbc71c71c71c71), BX_CONST64(0xc71c71c71c71c71c)), /* 9 */
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packFloat128(BX_CONST64(0xbffb745d1745d174), BX_CONST64(0x5d1745d1745d1746)), /* 11 */
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packFloat128(BX_CONST64(0x3ffb3b13b13b13b1), BX_CONST64(0x3b13b13b13b13b14)), /* 13 */
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packFloat128(BX_CONST64(0xbffb111111111111), BX_CONST64(0x1111111111111111)), /* 15 */
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packFloat128(BX_CONST64(0x3ffae1e1e1e1e1e1), BX_CONST64(0xe1e1e1e1e1e1e1e2)), /* 17 */
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packFloat128(BX_CONST64(0xbffaaf286bca1af2), BX_CONST64(0x86bca1af286bca1b)), /* 19 */
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packFloat128(BX_CONST64(0x3ffa861861861861), BX_CONST64(0x8618618618618618)) /* 21 */
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};
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extern float128 OddPoly(float128 x, float128 *arr, unsigned n, float_status_t &status);
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/* |x| < 1/4 */
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static float128 poly_atan(float128 x1, float_status_t &status)
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{
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/*
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// 3 5 7 9 11 13 15 17
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// x x x x x x x x
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// atan(x) ~ x - --- + --- - --- + --- - ---- + ---- - ---- + ----
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// 3 5 7 9 11 13 15 17
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//
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// 2 4 6 8 10 12 14 16
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// x x x x x x x x
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// = x * [ 1 - --- + --- - --- + --- - ---- + ---- - ---- + ---- ]
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// 3 5 7 9 11 13 15 17
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//
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// 5 5
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// -- 4k -- 4k+2
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// p(x) = > C * x q(x) = > C * x
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// -- 2k -- 2k+1
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// k=0 k=0
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//
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// 2
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// atan(x) ~ x * [ p(x) + x * q(x) ]
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//
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*/
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return OddPoly(x1, atan_arr, FPATAN_ARR_SIZE, status);
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}
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// =================================================
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// FPATAN Compute y * log (x)
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// 2
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// =================================================
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//
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// Uses the following identities:
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//
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// 1. ----------------------------------------------------------
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//
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// atan(-x) = -atan(x)
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//
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// 2. ----------------------------------------------------------
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//
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// x + y
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// atan(x) + atan(y) = atan -------, xy < 1
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// 1-xy
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//
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// x + y
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// atan(x) + atan(y) = atan ------- + PI, x > 0, xy > 1
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// 1-xy
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//
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// x + y
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// atan(x) + atan(y) = atan ------- - PI, x < 0, xy > 1
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// 1-xy
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//
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// 3. ----------------------------------------------------------
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//
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// atan(x) = atan(INF) + atan(- 1/x)
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//
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// x-1
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// atan(x) = PI/4 + atan( ----- )
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// x+1
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//
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// x * sqrt(3) - 1
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// atan(x) = PI/6 + atan( ----------------- )
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// x + sqrt(3)
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//
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// 4. ----------------------------------------------------------
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// 3 5 7 9 2n+1
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// x x x x n x
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// atan(x) = x - --- + --- - --- + --- - ... + (-1) ------ + ...
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// 3 5 7 9 2n+1
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//
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floatx80 fpatan(floatx80 a, floatx80 b, float_status_t &status)
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{
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// handle unsupported extended double-precision floating encodings
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if (floatx80_is_unsupported(a)) {
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float_raise(status, float_flag_invalid);
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return floatx80_default_nan;
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}
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Bit64u aSig = extractFloatx80Frac(a);
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Bit32s aExp = extractFloatx80Exp(a);
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int aSign = extractFloatx80Sign(a);
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Bit64u bSig = extractFloatx80Frac(b);
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Bit32s bExp = extractFloatx80Exp(b);
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int bSign = extractFloatx80Sign(b);
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int zSign = aSign ^ bSign;
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if (bExp == 0x7FFF)
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{
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if ((Bit64u) (bSig<<1))
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return propagateFloatx80NaN(a, b, status);
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if (aExp == 0x7FFF) {
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if ((Bit64u) (aSig<<1))
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return propagateFloatx80NaN(a, b, status);
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if (aSign) { /* return 3PI/4 */
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return roundAndPackFloatx80(80, bSign,
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FLOATX80_3PI4_EXP, FLOAT_3PI4_HI, FLOAT_3PI4_LO, status);
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}
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else { /* return PI/4 */
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return roundAndPackFloatx80(80, bSign,
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FLOATX80_PI4_EXP, FLOAT_PI_HI, FLOAT_PI_LO, status);
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}
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}
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if (aSig && (aExp == 0))
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float_raise(status, float_flag_denormal);
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/* return PI/2 */
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return roundAndPackFloatx80(80, bSign, FLOATX80_PI2_EXP, FLOAT_PI_HI, FLOAT_PI_LO, status);
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}
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if (aExp == 0x7FFF)
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{
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if ((Bit64u) (aSig<<1))
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return propagateFloatx80NaN(a, b, status);
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if (bSig && (bExp == 0))
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float_raise(status, float_flag_denormal);
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return_PI_or_ZERO:
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if (aSign) { /* return PI */
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return roundAndPackFloatx80(80, bSign, FLOATX80_PI_EXP, FLOAT_PI_HI, FLOAT_PI_LO, status);
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} else { /* return 0 */
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return packFloatx80(bSign, 0, 0);
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}
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}
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if (bExp == 0)
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{
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if (bSig == 0) {
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if (aSig && (aExp == 0)) float_raise(status, float_flag_denormal);
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goto return_PI_or_ZERO;
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}
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float_raise(status, float_flag_denormal);
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normalizeFloatx80Subnormal(bSig, &bExp, &bSig);
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}
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if (aExp == 0)
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{
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if (aSig == 0) /* return PI/2 */
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return roundAndPackFloatx80(80, bSign, FLOATX80_PI2_EXP, FLOAT_PI_HI, FLOAT_PI_LO, status);
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float_raise(status, float_flag_denormal);
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normalizeFloatx80Subnormal(aSig, &aExp, &aSig);
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}
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float_raise(status, float_flag_inexact);
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/* |a| = |b| ==> return PI/4 */
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if (aSig == bSig && aExp == bExp)
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return roundAndPackFloatx80(80, bSign, FLOATX80_PI4_EXP, FLOAT_PI_HI, FLOAT_PI_LO, status);
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/* ******************************** */
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/* using float128 for approximation */
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/* ******************************** */
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float128 a128 = normalizeRoundAndPackFloat128(0, aExp-0x10, aSig, 0, status);
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float128 b128 = normalizeRoundAndPackFloat128(0, bExp-0x10, bSig, 0, status);
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float128 x;
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int swap = 0, add_pi6 = 0, add_pi4 = 0;
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if (aExp > bExp || (aExp == bExp && aSig > bSig))
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{
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x = float128_div(b128, a128, status);
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}
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else {
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x = float128_div(a128, b128, status);
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swap = 1;
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}
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Bit32s xExp = extractFloat128Exp(x);
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floatx80 result;
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if (xExp <= EXP_BIAS-40) {
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result = float128_to_floatx80(x, status);
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goto approximation_completed;
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}
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if (x.hi >= BX_CONST64(0x3ffe800000000000)) // 3/4 < x < 1
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{
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/*
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arctan(x) = arctan((x-1)/(x+1)) + pi/4
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*/
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float128 t1 = float128_sub(x, float128_one, status);
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float128 t2 = float128_add(x, float128_one, status);
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x = float128_div(t1, t2, status);
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add_pi4 = 1;
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}
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else
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{
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/* argument correction */
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if (xExp >= 0x3FFD) // 1/4 < x < 3/4
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{
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/*
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arctan(x) = arctan((x*sqrt(3)-1)/(x+sqrt(3))) + pi/6
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*/
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float128 t1 = float128_mul(x, float128_sqrt3, status);
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float128 t2 = float128_add(x, float128_sqrt3, status);
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x = float128_sub(t1, float128_one, status);
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x = float128_div(x, t2, status);
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add_pi6 = 1;
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}
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}
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x = poly_atan(x, status);
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result = float128_to_floatx80(x, status);
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if (add_pi6) result = floatx80_add(result, floatx80_pi6, status);
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if (add_pi4) result = floatx80_add(result, floatx80_pi4, status);
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approximation_completed:
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if (swap) result = floatx80_sub(floatx80_pi2, result, status);
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if (zSign) floatx80_chs(result);
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int rSign = extractFloatx80Sign(result);
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if (!bSign && rSign)
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return floatx80_add(result, floatx80_pi, status);
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if (bSign && !rSign)
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return floatx80_sub(result, floatx80_pi, status);
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return result;
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}
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