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authorS. Gilles <sgilles@math.umd.edu>2018-07-31 12:03:40 -0400
committerS. Gilles <sgilles@math.umd.edu>2018-07-31 12:03:40 -0400
commit56a659f634f2a591cf641eb875a4e7f457c40321 (patch)
treea3d7f0330b1d3cf4b6f4ddfa48d379231f5592c5 /lib/math
parent527b74f79517336b3ef11c963e86658527aa96cd (diff)
downloadmc-56a659f634f2a591cf641eb875a4e7f457c40321.tar.gz
First compiling atan.
Diffstat (limited to 'lib/math')
-rw-r--r--lib/math/ancillary/generate-arctan-tuples-for-GB91.c3
-rw-r--r--lib/math/ancillary/generate-minimax-by-Remez.gp11
-rw-r--r--lib/math/atan-impl.myr422
-rw-r--r--lib/math/bld.sub9
-rw-r--r--lib/math/fpmath.myr16
-rw-r--r--lib/math/sin-impl.myr4
6 files changed, 460 insertions, 5 deletions
diff --git a/lib/math/ancillary/generate-arctan-tuples-for-GB91.c b/lib/math/ancillary/generate-arctan-tuples-for-GB91.c
index 5bd0e0e..251877d 100644
--- a/lib/math/ancillary/generate-arctan-tuples-for-GB91.c
+++ b/lib/math/ancillary/generate-arctan-tuples-for-GB91.c
@@ -1,4 +1,5 @@
/* cc -o generate-arctan-tuples-for-GB91 generate-arctan-tuples-for-GB91.c -lmpfr # -fno-strict-aliasing */
+/* cc -static -std=c99 -D_POSIX_C_SOURCE=999999999 -fno-strict-aliasing -O2 -o generate-arctan-tuples-for-GB91 generate-arctan-tuples-for-GB91.c -lmpfr -lgmp */
#include <errno.h>
#include <stdint.h>
#include <stdio.h>
@@ -244,7 +245,7 @@ static int find_tuple(int ii, int min_leeway)
/* Compute (xij)^(-1) */
invert_poorly(xij, xijinv);
- while (r < (1 << 26)) {
+ while (r < (1 << 28)) {
xi = xi_orig + r;
xi_d = UINT64_TO_FLT64(xi);
mpfr_set_d(xi_m, xi_d, MPFR_RNDN);
diff --git a/lib/math/ancillary/generate-minimax-by-Remez.gp b/lib/math/ancillary/generate-minimax-by-Remez.gp
index 7ac32ff..a388f4e 100644
--- a/lib/math/ancillary/generate-minimax-by-Remez.gp
+++ b/lib/math/ancillary/generate-minimax-by-Remez.gp
@@ -127,6 +127,10 @@
return(if(x == 0, 1, tan(x)/x));
}
+{ atanxoverx(x) =
+ return(if(x == 0, 1, atan(x)/x));
+}
+
{ cotx(x) =
return(1/tanoverx(x));
}
@@ -164,4 +168,11 @@ print("\n");
print("(You'll need to add a 0x0 at the beginning to make a degree 11...\n");
print("\n");
print("---\n");
+print("\n");
+print("Minmimaxing atan(x) / x, degree 12, on [0, 15.5/256]:");
+find_minimax(atanxoverx, 12, 0, 1/16)
+print("\n");
+print("(You'll need to add a 0x0 at the beginning to make a degree 13...\n");
+print("\n");
+print("---\n");
print("Remember that there's that extra, ugly E term at the end of the vector that you want to lop off.\n");
diff --git a/lib/math/atan-impl.myr b/lib/math/atan-impl.myr
new file mode 100644
index 0000000..911c99c
--- /dev/null
+++ b/lib/math/atan-impl.myr
@@ -0,0 +1,422 @@
+use std
+
+use "fpmath"
+
+use "sin-impl"
+use "util"
+
+/*
+ As with other trig functions, we follow [GB91] here. This is a
+ relatively straight-forward "tabulate and approximate by local
+ polynomials" approach. The (relative) lack of sophistication can
+ be seen by the huge number of constants required. The only fancy
+ trick is that
+
+ atan(x) + atan(1/x) = pi/2
+
+ which allows quick reduction to the range [-1, 1].
+
+ The Highly Accurate Table approach of [GB91] then appears in the
+ constants for the polynomial approximations. This acts as a few
+ bits of insurance against subtractive cancellation of the first
+ few terms of the polynomial approximations.
+
+ See files generate-atan-tuples-for-GB91.c, and
+ generate-minimax-by-Remez.gp for where the constants come from.
+ */
+pkg math =
+ pkglocal const atan32 : (x : flt32 -> flt32)
+ pkglocal const atan64 : (x : flt64 -> flt64)
+
+ pkglocal const atan232 : (y : flt32, x : flt32 -> flt32)
+ pkglocal const atan264 : (y : flt64, x : flt64 -> flt64)
+;;
+
+/*
+ Coefficients for p(x), where arctan(x) is approximated in degree
+ 13 by x*p(x^2).
+ */
+const atan_coeffs : uint64[7] = [
+ 0x3ff0000000000000,
+ 0xbfd5555555555555,
+ 0x3fc9999999999128,
+ 0xbfc2492493302077,
+ 0x3fbc71bd4ec2b9ae,
+ 0xbfb750eefa902315,
+ 0x3fb226324f0122e0,
+]
+
+/*
+ Coefficients for degree 5 polynomial approximations of arctan(x)
+ on intervals of width 1/256, roughly filling the [0, 1] space.
+ Because of the slight shifting of the pivot xi variables by up
+ to 2^26 ulps, there are tiny gaps, of up to size 2*2^(-53 + 26)
+ = 2^-26, on which the minimax guarantees do not apply. This is
+ not expected to be a significant issue.
+
+ Note that the first few entries aren't actually used.
+ */
+const C : (uint64, uint64, uint64, uint64, uint64, uint64, uint64)[257] = [
+ (0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000),
+ (0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000),
+ (0x3f80000001f83362, 0xbf7fffd559ac119a, 0x3fefff8001ffd883, 0x3f7fff0009e129ba, 0xbfd55355689e3241, 0xbf7ffdcf9a784a0c, 0x3fc995f4bed14247),
+ (0x3f87ffffff37811c, 0xbf87ffb800bc4b32, 0x3feffee00a1fb7b6, 0x3f87fe5015f2c284, 0xbfd550d5b96bfd9c, 0xbf87fc076a4fd120, 0x3fc98fc8b892a7b2),
+ (0x3f8fffffffc1eade, 0xbf8fff555b7d615a, 0x3feffe001ffe07ed, 0x3f8ffc005fa9b585, 0xbfd54d5693a075a5, 0xbf8ff657f61b9dd7, 0x3fc9869ee5cadbbd),
+ (0x3f93fffffe0e9f70, 0xbf93ff595d2776fd, 0x3feffce04e18fad7, 0x3f93fc18907a7f60, 0xbfd548d85eda9a6c, 0xbf93f61bf2de1558, 0x3fc97abe5d2bf700),
+ (0x3f97ffffffec7ecd, 0xbf97fee01836ddc0, 0x3feffb80a1e94266, 0x3f97f9416c2a7c14, 0xbfd5435ba3ccdd9c, 0xbf97eefc49b67b90, 0x3fc96b9f97155700),
+ (0x3f9c000002043a2b, 0xbf9bfe36e12cee80, 0x3feff9e12be5b739, 0x3f9bf54b150fb215, 0xbfd53ce105d2bb59, 0xbf9be5104edd8ee0, 0x3fc9598af87d1c00),
+ (0x3fa00000022e99a5, 0xbf9ffd55c0059267, 0x3feff801ff7df250, 0x3f9ff006025a63b6, 0xbfd5356946bb4ad9, 0xbf9fd7e22b0f4cc0, 0x3fc94484728b9380),
+ (0x3fa200000058e037, 0xbfa1fe1a5c8788b8, 0x3feff5e3331c6ffa, 0x3fa1f4a166090962, 0xbfd52cf545c2c2d0, 0xbfa1e37e5b434560, 0x3fc92c909999c980),
+ (0x3fa3fffffe4fe9f8, 0xbfa3fd65efba5c58, 0x3feff384e01a9106, 0x3fa3f0692146d108, 0xbfd5238601c8043d, 0xbfa3d95263db398f, 0x3fc9116fa0344708),
+ (0x3fa5fffffd7d0899, 0xbfa5fc89a378731e, 0x3feff0e722c403ee, 0x3fa5eb42b2d510f3, 0xbfd5191c8f45f7ad, 0xbfa5cc83e91b1137, 0x3fc8f3b166e52899),
+ (0x3fa7fffffff47d57, 0xbfa7fb8184255df7, 0x3fefee0a1a5143a8, 0x3fa7e516b6ddd4cd, 0xbfd50dba2684300b, 0xbfa7bd2cb777bd9a, 0x3fc8d317587dd676),
+ (0x3fa9fffffc2c23a9, 0xbfa9fa4994982dbb, 0x3fefeaede8f76900, 0x3fa9ddcddc883e9e, 0xbfd501601a419c50, 0xbfa9ab141ce95d64, 0x3fc8afa883cd8ab3),
+ (0x3fabfffffd0f0cbd, 0xbfabf8ddee4b0fc3, 0x3fefe792b3b38c6e, 0x3fabd5510839a77b, 0xbfd4f40fd8f0a91b, 0xbfab9602138cdbe9, 0x3fc8896c8f290450),
+ (0x3fae000000ffdc11, 0xbfadf73aa09e1477, 0x3fefe3f8a27756be, 0x3fadcb8935b8415f, 0xbfd4e5cae94695f1, 0xbfad7d32cd68db68, 0x3fc860b09a86ff00),
+ (0x3fb0000002ad31a4, 0xbfaff55bbc820c23, 0x3fefe01fe0154058, 0x3fafc05f85dd78f3, 0xbfd4d692f8f482ea, 0xbfaf617ebeb00bd8, 0x3fc834f3a9931b00),
+ (0x3fb100000188824f, 0xbfb0f99ea8a41bcd, 0x3fefdc089a435d4f, 0x3fb0d9de9f1c8568, 0xbfd4c669ca505107, 0xbfb0a166428a466f, 0x3fc8063f36448e35),
+ (0x3fb20000009bc757, 0xbfb1f86dbfa33080, 0x3fefd7b3016f7596, 0x3fb1d2c5eedc6080, 0xbfd4b55130ac22d8, 0xbfb18fd8c2bb0ee4, 0x3fc7d526db38f039),
+ (0x3fb3000024c171ae, 0xbfb2f7195618302f, 0x3fefd31f48358ab4, 0x3fb2cadaa8a559c1, 0xbfd4a34b240cff5f, 0xbfb27c47c2ad49a6, 0x3fc7a1702e4d8d69),
+ (0x3fb4000000068d16, 0xbfb3f59f0e82d887, 0x3fefce4da6ab73c7, 0x3fb3c2114d1e9dc2, 0xbfd49059bce708bf, 0xbfb36699156cde19, 0x3fc76b264479966d),
+ (0x3fb4ffffffdaf503, 0xbfb4f3fd674dc761, 0x3fefc93e53a2ae2c, 0x3fb4b85f4a8c15c6, 0xbfd47c7f1d6673be, 0xbfb44eb3ca7ef180, 0x3fc7325487c15892),
+ (0x3fb60000005453f7, 0xbfb5f23250268d66, 0x3fefc3f18b52cb50, 0x3fb5adb99d22e5c3, 0xbfd467bd8b4de796, 0xbfb5347ee36faada, 0x3fc6f706fed2d6e1),
+ (0x3fb7000000057dfc, 0xbfb6f03bdcefbdc7, 0x3fefbe678c078564, 0x3fb6a2157e35172d, 0xbfd452176286cdb9, 0xbfb617e1fe0c1388, 0x3fc6b94a23d2539a),
+ (0x3fb7fffffe7c548b, 0xbfb7ee182482a642, 0x3fefb8a096b5e8ba, 0x3fb7956844e7782f, 0xbfd43b8f17095f3f, 0xbfb6f8c53a2a94dc, 0x3fc6792ae7109ee3),
+ (0x3fb9000000399ffa, 0xbfb8ebc544b20fc7, 0x3fefb29ceedb75bc, 0x3fb887a76b0a59ee, 0xbfd4242733f64aa6, 0xbfb7d7114183708e, 0x3fc636b6a9a36669),
+ (0x3fba000000503ea3, 0xbfb9e941541f49d9, 0x3fefac5cdacc38c0, 0x3fb978c880930264, 0xbfd40be25c5cc9fc, 0xbfb8b2af3f76ad1f, 0x3fc5f1fb3caa1a83),
+ (0x3fbafffffefd604e, 0xbfbae68a70c75b65, 0x3fefa5e0a3559ff9, 0x3fba68c13a88c726, 0xbfd3f2c349624f3b, 0xbfb98b88f1f263f7, 0x3fc5ab06d90ee046),
+ (0x3fbc0000027ada67, 0xbfbbe39ec0e260e9, 0x3fef9f2893aa9d84, 0x3fbb578774c87524, 0xbfd3d8ccc2282c36, 0xbfba6104a32547f8, 0x3fc5622c6b5cdc00),
+ (0x3fbcfffffe2aae0e, 0xbfbce07c5a6d6a81, 0x3fef9834fa03c879, 0x3fbc45111ba09bf8, 0xbfd3be01baf8bc7a, 0xbfbb3410872e3840, 0x3fc516f2437f8a00),
+ (0x3fbe000001dab907, 0xbfbddd2171f3051f, 0x3fef910626538a70, 0x3fbd3154591dc7f9, 0xbfd3a2652c4d0b90, 0xbfbc04a65d78b257, 0x3fc4c967e0fd9461),
+ (0x3fbeffffffa53a79, 0xbfbed98c21368e87, 0x3fef899c6bd1f677, 0x3fbe1c47601d7e8f, 0xbfd385fa159fbfca, 0xbfbcd19beca90b8c, 0x3fc47a256b36705a),
+ (0x3fc000000f142d5a, 0xbfbfd5bab85dc3d0, 0x3fef81f81e981366, 0x3fbf05e0b86e7006, 0xbfd368c39b4690e4, 0xbfbd9b66d4a9b7f8, 0x3fc428f6923eacde),
+ (0x3fc07ffffcdf1951, 0xbfc068d5810d5bd4, 0x3fef7a19994da3f1, 0x3fbfee169210a60e, 0xbfd34ac4fcd8967e, 0xbfbe61f4253945ff, 0x3fc3d5ebc0588a79),
+ (0x3fc1000002aa39f4, 0xbfc0e6adcf926f2e, 0x3fef7201326bccc1, 0x3fc06a70037899d2, 0xbfd32c01735432c2, 0xbfbf25323b0051d6, 0x3fc38115615f521e),
+ (0x3fc18000001fea3b, 0xbfc1646541255ca1, 0x3fef69af483d2f52, 0x3fc0dd19dadefc1f, 0xbfd30c7c5e62ba59, 0xbfbfe50f8f51460e, 0x3fc32a845cd3502a),
+ (0x3fc1fffffe8823bc, 0xbfc1e1faf993a4c8, 0x3fef612438bab95f, 0x3fc14f04587e458a, 0xbfd2ec3924d36b33, 0xbfc050bdc309e406, 0x3fc2d249bb3c927d),
+ (0x3fc27ffffc9337b0, 0xbfc25f6e13bf7bd7, 0x3fef586064c338f8, 0x3fc1c02b10b43292, 0xbfd2cb3b4055087c, 0xbfc0ad330fd21a61, 0x3fc27876c32ba8fe),
+ (0x3fc2fffffdc6bf7b, 0xbfc2dcbdb0cea731, 0x3fef4f642f0fe16c, 0x3fc23089b1286391, 0xbfd2a98638fb0d98, 0xbfc107e008ee7ac7, 0x3fc21d1ce9880c12),
+ (0x3fc3800000c2fb33, 0xbfc359e8eeaa28a7, 0x3fef462ffcd452fe, 0x3fc2a01bf88e9498, 0xbfd2871da71092b5, 0xbfc160bd5de46b03, 0x3fc1c04dd31f291d),
+ (0x3fc4000000d8f876, 0xbfc3d6eee99a2f23, 0x3fef3cc435a04a95, 0x3fc30eddb88397f1, 0xbfd26405321912f4, 0xbfc1b7c415010900, 0x3fc1621b4e780103),
+ (0x3fc48000000de728, 0xbfc453cec616b8c4, 0x3fef332142989e5d, 0x3fc37ccade8e0ca6, 0xbfd240408d609878, 0xbfc20ced92d9e381, 0x3fc1029747314f3d),
+ (0x3fc4ffffff0e7ab0, 0xbfc4d087a8ef1e63, 0x3fef29478efdfd42, 0x3fc3e9df6d60cba2, 0xbfd21bd3796a9ce3, 0xbfc26033953388f3, 0x3fc0a1d3c688050b),
+ (0x3fc57ffffc7377dc, 0xbfc54d18b69dbcdb, 0x3fef1f3788337a10, 0x3fc456177c933fea, 0xbfd1f6c1c37d2372, 0xbfc2b190331b73b1, 0x3fc03fe2eebeac95),
+ (0x3fc5ffffff069c34, 0xbfc5c9811d4c8683, 0x3fef14f19ce2618f, 0x3fc4c16f418df8ce, 0xbfd1d10f41eb9d16, 0xbfc300fde3b8367f, 0x3fbfb9addc08f782),
+ (0x3fc67ffffe3dace2, 0xbfc645bffdfed6b5, 0x3fef0a763ecf8d44, 0x3fc52be2fc09f17c, 0xbfd1aabfda46e407, 0xbfc34e776fde19e4, 0x3fbef18416e713f0),
+ (0x3fc6fffffe88eb88, 0xbfc6c1d4881ddaae, 0x3feeffc5e08c9b3d, 0x3fc5956f0e16385c, 0xbfd183d77857c309, 0xbfc399f803dfed36, 0x3fbe276d18332ee6),
+ (0x3fc77fffffbe804c, 0xbfc73dbde868750f, 0x3feef4e0f6d3f675, 0x3fc5fe0fed8db872, 0xbfd15c5a1291701e, 0xbfc3e37b24a92e8d, 0x3fbd5b8d76fa3afe),
+ (0x3fc7fffffc35d1d7, 0xbfc7b97b48251fed, 0x3feee9c7f89a7009, 0x3fc665c223be9680, 0xbfd1344ba9bdc601, 0xbfc42afcaf9262f8, 0x3fbc8e09cf02b355),
+ (0x3fc8800000a239dc, 0xbfc8350be43569e9, 0x3feede7b5cee6e27, 0x3fc6cc8260df5e45, 0xbfd10bb040c61a32, 0xbfc47078e771a1fe, 0x3fbbbf06908b1814),
+ (0x3fc90000027df7fa, 0xbfc8b06ee4ee22e7, 0x3feed2fb9e8bb11b, 0x3fc7324d5aa33c81, 0xbfd0e28be90da705, 0xbfc4b3ec5d56fe5c, 0x3fbaeea83980b70d),
+ (0x3fc9800002d31c2a, 0xbfc92ba37fbc8427, 0x3feec74938c8c904, 0x3fc7971fe886b4de, 0xbfd0b8e2b6d26476, 0xbfc4f55403930d95, 0x3fba1d131409b3ab),
+ (0x3fc9fffffcddfcfd, 0xbfc9a6a8e66a4ab7, 0x3feebb64a91453c7, 0x3fc7faf6f60a4367, 0xbfd08eb8c5fc4160, 0xbfc534ad23ed705c, 0x3fb94a6b493e7c18),
+ (0x3fca7fffff8b55ee, 0xbfca217e5fa0a315, 0x3feeaf4e6c86c253, 0x3fc85dcf96a2bb00, 0xbfd0641230fa59ed, 0xbfc571f56b9c5c8a, 0x3fb876d4adfc5ac3),
+ (0x3fcafffffd414cf5, 0xbfca9c23189f6ea5, 0x3feea30704596a99, 0x3fc8bfa6e0659b60, 0xbfd038f3200791fd, 0xbfc5ad2ad32c8d7a, 0x3fb7a273097623dd),
+ (0x3fcb7ffffe765c40, 0xbfcb169655d5327a, 0x3fee968ef1197db7, 0x3fc9207a13521ece, 0xbfd00d5fb7ad7b56, 0xbfc5e64bb64ff60a, 0x3fb6cd69b8f6e504),
+ (0x3fcc000000abc4ff, 0xbfcb90d753364d8b, 0x3fee89e6b5bbd593, 0x3fc980467fefa357, 0xbfcfc2b8463315e8, 0xbfc61d56c362a69c, 0x3fb5f7dbddfa3d81),
+ (0x3fcc7fffffda3e36, 0xbfcc0ae54d528b2b, 0x3fee7d0ed6b1420d, 0x3fc9df098ebe3358, 0xbfcf69d920a9cf8c, 0xbfc6524aff4b3332, 0x3fb521ec46597687),
+ (0x3fccfffffff895a3, 0xbfcc84bf8a6d20c0, 0x3fee7007d8e2c735, 0x3fca3cc0c754ee99, 0xbfcf102a3a0ed514, 0xbfc6849fe6a47028, 0x3fb44c40e5060800),
+ (0x3fcd800002e87b67, 0xbfccfe6550e0425b, 0x3fee62d242a27514, 0x3fca9969c88af14c, 0xbfceb5b42bd588bd, 0xbfc6b562a8259298, 0x3fb375f45f6e7800),
+ (0x3fce0000007c3fa1, 0xbfcd77d5df961e94, 0x3fee556e9c79c2d5, 0x3fcaf502433f724e, 0xbfce5a7f5c4a951a, 0xbfc6e49a1c28bf4d, 0x3fb29f2902804d52),
+ (0x3fce8000018c9844, 0xbfcdf11087c3e708, 0x3fee47dd6e4b27d1, 0x3fcb4f880dd31913, 0xbfcdfe93e4e6b22c, 0xbfc71130057e8a97, 0x3fb1c906570f0d02),
+ (0x3fcf0000011971a7, 0xbfce6a148fa0c60b, 0x3fee3a1f427d6bd1, 0x3fcba8f90d9b0ad8, 0xbfcda1fa4468eaa4, 0xbfc73baf35ad6fbc, 0x3fb0f329b1d71c3b),
+ (0x3fcf7fffffae07f0, 0xbfcee2e144d04e42, 0x3fee2c34a3e84e63, 0x3fcc015345086350, 0xbfcd44bad359c3db, 0xbfc76418a7b6f7b9, 0x3fb01db333c0e897),
+ (0x3fd0000003c8cfe7, 0xbfcf5b76004c2654, 0x3fee1e1e1d47929d, 0x3fcc5894d623e3cd, 0xbfcce6ddde8c38ac, 0xbfc78a6da87d134e, 0x3fae9184c911f786),
+ (0x3fd0400003e0f35b, 0xbfcfd3d2038a7c77, 0x3fee0fdc3d4407ce, 0x3fccaebbe9509c5a, 0xbfcc886bc113543a, 0xbfc7aeafca638093, 0x3face8ecd45e8a40),
+ (0x3fd08000032761cb, 0xbfd025fa53fb759e, 0x3fee016f90dfb3ea, 0x3fcd03c6cfa46b13, 0xbfcc296cbe2c693a, 0xbfc7d0e0f31c6cc9, 0x3fab41db5e90e29b),
+ (0x3fd0bffffd1bbae5, 0xbfd061ee9d889212, 0x3fedf2d8a7a1ddae, 0x3fcd57b3e8faad27, 0xbfcbc9e91c1a6c4e, 0xbfc7f1034f0598ee, 0x3fa99c8c39820b78),
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+ (0x3fee00000039743a, 0xbfe819d0b7341e61, 0x3fe107fbbde28c9e, 0x3fd0feeb406dd82e, 0x3fb50e5af0d5593e, 0xbf82a72ef1e0c167, 0xbfa14d527b168f8a),
+ (0x3fee1ffffe5cc00a, 0xbfe82ad03521bbb6, 0x3fe0f704ba259f97, 0x3fd0ef1d13273fc2, 0x3fb5175a44a65f21, 0xbf8135cae0939400, 0xbfa1010aa6568000),
+ (0x3fee4000019b115b, 0xbfe83bbec6a702d7, 0x3fe0e61d852e5196, 0x3fd0df4863d03473, 0x3fb51fb00570e1c7, 0xbf7fce36d18fd400, 0xbfa0b60a74c20000),
+ (0x3fee5ffffdf4f7e1, 0xbfe84c9c7540d5f2, 0x3fe0d5462b922e3d, 0x3fd0cf6db5f37b20, 0x3fb5275fea93680c, 0xbf7d3c75fad99000, 0xbfa06bc035858000),
+ (0x3fee80000323ee64, 0xbfe85d6959120279, 0x3fe0c47eaad03167, 0x3fd0bf8d7d13d57e, 0x3fb52e6cdb8a684c, 0xbf7ab635583b0400, 0xbfa0222d3764c000),
+ (0x3feea000027f9467, 0xbfe86e257a464e49, 0x3fe0b3c70fe40a0e, 0x3fd0afa8399b9a17, 0x3fb534d9b198efb2, 0xbf783b593b7d2000, 0xbf9fb2a5de1a8000),
+ (0x3feec000016d28f0, 0xbfe87ed0eb9a3410, 0x3fe0a31f5cc9486a, 0x3fd09fbe5fe0dff2, 0x3fb53aa9435bdcf3, 0xbf75cbc42e130c00, 0xbf9f226507280000),
+ (0x3feee0000186b339, 0xbfe88f6bbdcc87be, 0x3fe092879506eb54, 0x3fd08fd063fd06b3, 0x3fb53fde5f2950c0, 0xbf736758dd6aac00, 0xbf9e939a0cb08000),
+ (0x3feeffffffce4723, 0xbfe89ff5ff3e4b8a, 0x3fe081ffbe119e2e, 0x3fd07fdeba3b0de8, 0x3fb5447bcb488540, 0xbf710dfa27bda400, 0xbf9e0646ebfc0000),
+ (0x3fef20000017d916, 0xbfe8b06fc1db7f25, 0x3fe07187d965874a, 0x3fd06fe9d16dfdc9, 0x3fb5488447a9ed9c, 0xbf6d7f1490008000, 0xbf9d7a6d4a5b0000),
+ (0x3fef3ffffe33911c, 0xbfe8c0d9137147b0, 0x3fe0611fec30af7f, 0x3fd05ff21a3a42d4, 0x3fb54bf9b699774b, 0xbf696e589b06ec6f, 0xbf9cf12a32a34060),
+ (0x3fef5ffffe084599, 0xbfe8d13205f7eff9, 0x3fe050c7f70b73f8, 0x3fd04ff7ffb69502, 0x3fb54ee148abcb5e, 0xbf648602474d4800, 0xbf9c672c6e7a8000),
+ (0x3fef7ffffdfbe7bb, 0xbfe8e17aa896a3fa, 0x3fe0407ffcf5ee0b, 0x3fd03ffbecf3660f, 0x3fb5513b2748a2d9, 0xbf6029586a500800, 0xbf9bdfc7884c0000),
+ (0x3fef9fffff0fa180, 0xbfe8f1b30bcb577c, 0x3fe03047ff31c004, 0x3fd02ffe4a3a5b50, 0x3fb5530ac95cb722, 0xbf57c33cda611000, 0xbf9b59e0e3a28000),
+ (0x3fefc00001a24bf4, 0xbfe901db3fc1bb6f, 0x3fe0201ffeecb132, 0x3fd01fff7e0f13c7, 0x3fb55452c9166f13, 0xbf4eba61be0c4000, 0xbf9ad57931778000),
+ (0x3fefdfffffe38639, 0xbfe911f3518b3813, 0x3fe01008000a410c, 0x3fd00ffff01bf387, 0x3fb55515b9099fdd, 0xbf3c805a37250000, 0xbf9a52910b908000),
+ (0x3feff0000008727f, 0xbfe919f953f31326, 0x3fe00801fffb8473, 0x3fd007fffdfb4766, 0x3fb555453ecbbd3f, 0xbf30045a21318845, 0xbf9a12c2520fd1de),
+]
+
+const atan32 = {x
+ var r = atan264((x : flt64), 1.0)
+ -> (r : flt32)
+}
+
+const atan64 = {x
+ -> atan264(x, 1.0)
+}
+
+const atan232 = {y, x
+ var r = atan264((y : flt64), (x : flt64))
+ -> (r : flt32)
+}
+
+const atan264 = {y, x
+ if std.isnan(y) || std.isnan(x)
+ -> std.flt64nan()
+ ;;
+
+ var xabs = std.flt64frombits(std.flt64bits(x) & (~(0 : uint64)) >> 1)
+ var yabs = std.flt64frombits(std.flt64bits(y) & (~(0 : uint64)) >> 1)
+
+ var xb = std.flt64bits(x)
+ if xb == 0x0 || xb == 0x8000000000000000
+ if (y >= 0.0) == (x >= 0.0)
+ -> std.flt64frombits(0x3ff921fb54442d18)
+ else
+ -> std.flt64frombits(0xbff921fb54442d18)
+ ;;
+ ;;
+
+ var then_reverse = false
+ if yabs > xabs
+ then_reverse = true
+ std.swap(&x, &y)
+ ;;
+
+ /* Compute y/x = u + du; du is one Newton's iteration of division */
+ var u = y / x
+ var t1, t2
+ (t1, t2) = two_by_two(u, x)
+ var du = u * ((1.0 - t1) - t2)
+
+ var ret = atan_reduced(u, du)
+
+ if then_reverse
+ /* Compute pi/2 - ret; pi_over_2 is in sin-impl.myr */
+ var po2_hi = std.flt64frombits(pi_over_2[0])
+ var po2_lo = std.flt64frombits(pi_over_2[1])
+ ret = (po2_hi - ret) + po2_lo
+ ;;
+
+ -> ret
+}
+
+/* Handle arctan(z) with z in (-1, 1) */
+const atan_reduced = {u : flt64, du : flt64
+ var then_negate : bool = false
+ var ret : flt64
+
+ if u < 0.0
+ then_negate = true
+ u = -1.0 * u
+ du = -1.0 * du
+ ;;
+
+ /*
+ If u is less than 1/16, [GB91] uses a single polynomial
+ approximation. I am not quite sure why they don't employ
+ tables in this case: the speed appears to be about the
+ same. Perhaps to save a few bits for the first entries
+ of the table?
+ */
+ if u < 0.0625
+ var s = u * u
+ var p = horner_polyu(s, atan_coeffs[:])
+ ret = u*p + du
+ goto have_result
+ ;;
+
+ /*
+ Extract a polynomial from the table.
+ */
+ var j = rn(scale2(u, 8))
+ if j < 0
+ j = 0
+ elif j > 256
+ j = 256
+ ;;
+
+ var xi_u : uint64, c : uint64[6]
+ (xi_u, c[0], c[1], c[2], c[3], c[4], c[5]) = C[j]
+ var xi : flt64 = std.flt64frombits(xi_u)
+
+ ret = horner_polyu((u - xi) + du, c[:])
+
+:have_result
+
+ if then_negate
+ ret = -1.0 * ret
+ ;;
+
+ -> ret
+}
diff --git a/lib/math/bld.sub b/lib/math/bld.sub
index 0a6af63..c2e9da4 100644
--- a/lib/math/bld.sub
+++ b/lib/math/bld.sub
@@ -1,13 +1,16 @@
lib math =
fpmath.myr
- # exp and expm1
- exp-impl.myr
-
# rounding (to actual integers)
round-impl+posixy-x64-sse4.s
round-impl.myr
+ # atan and atan2
+ atan-impl.myr
+
+ # exp and expm1
+ exp-impl.myr
+
# fused-multiply-add
fma-impl+posixy-x64-fma.s
fma-impl.myr
diff --git a/lib/math/fpmath.myr b/lib/math/fpmath.myr
index 1bc26ef..e470ec1 100644
--- a/lib/math/fpmath.myr
+++ b/lib/math/fpmath.myr
@@ -3,6 +3,10 @@ use std
pkg math =
trait fpmath @f =
+ /* atan-impl */
+ atan : (x : @f -> @f)
+ atan2 : (y : @f, x : @f -> @f)
+
/* exp-impl */
exp : (x : @f -> @f)
expm1 : (x : @f -> @f)
@@ -84,6 +88,9 @@ impl roundable flt64 -> int64 =
;;
impl fpmath flt32 =
+ atan = {x; -> atan32(x)}
+ atan2 = {y, x; -> atan232(y, x)}
+
fma = {x, y, z; -> fma32(x, y, z)}
exp = {x; -> exp32(x)}
@@ -118,6 +125,9 @@ impl fpmath flt32 =
;;
impl fpmath flt64 =
+ atan = {x; -> atan64(x)}
+ atan2 = {y, x; -> atan264(y, x)}
+
fma = {x, y, z; -> fma64(x, y, z)}
exp = {x; -> exp64(x)}
@@ -153,6 +163,12 @@ impl fpmath flt64 =
extern const rn32 : (x : flt32 -> int32)
extern const rn64 : (x : flt64 -> int64)
+extern const atan32 : (x : flt32 -> flt32)
+extern const atan64 : (x : flt64 -> flt64)
+
+extern const atan232 : (y : flt32, x : flt32 -> flt32)
+extern const atan264 : (y : flt64, x : flt64 -> flt64)
+
extern const fma32 : (x : flt32, y : flt32, z : flt32 -> flt32)
extern const fma64 : (x : flt64, y : flt64, z : flt64 -> flt64)
diff --git a/lib/math/sin-impl.myr b/lib/math/sin-impl.myr
index 7bb44e2..4a9b0bc 100644
--- a/lib/math/sin-impl.myr
+++ b/lib/math/sin-impl.myr
@@ -58,10 +58,12 @@ pkg math =
pkglocal const trig_reduce : (x : flt64 -> (int64, flt64, flt64))
pkglocal const trig_table_approx : (x1 : flt64, C : (uint64, uint64, uint64)[257] -> (uint64, uint64, uint64))
+
+ pkglocal const pi_over_2 : uint64[4]
;;
/* Pi/2 in lots of detail, for range reducing sin(2^18) or so */
-const pi_over_2 : uint64[4] = [
+const pi_over_2 = [
0x3ff921fb54442d18,
0x3c91a62633145c07,
0xb91f1976b7ed8fbc,