shithub: mc

ref: 74d91a0021de012908cfdb35fb61a1473a376130
dir: /lib/math/atan-impl.myr/

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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, 0x3f7fffd559ac119a, 0x3fefff8001ffd883, 0xbf7fff0009e129ba, 0xbfd55355689e3241, 0x3f7ffdcf9a784a0c, 0x3fc995f4bed14247),
	(0x3f87ffffff37811c, 0x3f87ffb800bc4b32, 0x3feffee00a1fb7b6, 0xbf87fe5015f2c284, 0xbfd550d5b96bfd9c, 0x3f87fc076a4fd120, 0x3fc98fc8b892a7b2),
	(0x3f8fffffffc1eade, 0x3f8fff555b7d615a, 0x3feffe001ffe07ed, 0xbf8ffc005fa9b585, 0xbfd54d5693a075a5, 0x3f8ff657f61b9dd7, 0x3fc9869ee5cadbbd),
	(0x3f93fffffe0e9f70, 0x3f93ff595d2776fd, 0x3feffce04e18fad7, 0xbf93fc18907a7f60, 0xbfd548d85eda9a6c, 0x3f93f61bf2de1558, 0x3fc97abe5d2bf700),
	(0x3f97ffffffec7ecd, 0x3f97fee01836ddc0, 0x3feffb80a1e94266, 0xbf97f9416c2a7c14, 0xbfd5435ba3ccdd9c, 0x3f97eefc49b67b90, 0x3fc96b9f97155700),
	(0x3f9c000002043a2b, 0x3f9bfe36e12cee80, 0x3feff9e12be5b739, 0xbf9bf54b150fb215, 0xbfd53ce105d2bb59, 0x3f9be5104edd8ee0, 0x3fc9598af87d1c00),
	(0x3fa00000022e99a5, 0x3f9ffd55c0059267, 0x3feff801ff7df250, 0xbf9ff006025a63b6, 0xbfd5356946bb4ad9, 0x3f9fd7e22b0f4cc0, 0x3fc94484728b9380),
	(0x3fa200000058e037, 0x3fa1fe1a5c8788b8, 0x3feff5e3331c6ffa, 0xbfa1f4a166090962, 0xbfd52cf545c2c2d0, 0x3fa1e37e5b434560, 0x3fc92c909999c980),
	(0x3fa3fffffe4fe9f8, 0x3fa3fd65efba5c58, 0x3feff384e01a9106, 0xbfa3f0692146d108, 0xbfd5238601c8043d, 0x3fa3d95263db398f, 0x3fc9116fa0344708),
	(0x3fa5fffffd7d0899, 0x3fa5fc89a378731e, 0x3feff0e722c403ee, 0xbfa5eb42b2d510f3, 0xbfd5191c8f45f7ad, 0x3fa5cc83e91b1137, 0x3fc8f3b166e52899),
	(0x3fa7fffffff47d57, 0x3fa7fb8184255df7, 0x3fefee0a1a5143a8, 0xbfa7e516b6ddd4cd, 0xbfd50dba2684300b, 0x3fa7bd2cb777bd9a, 0x3fc8d317587dd676),
	(0x3fa9fffffc2c23a9, 0x3fa9fa4994982dbb, 0x3fefeaede8f76900, 0xbfa9ddcddc883e9e, 0xbfd501601a419c50, 0x3fa9ab141ce95d64, 0x3fc8afa883cd8ab3),
	(0x3fabfffffd0f0cbd, 0x3fabf8ddee4b0fc3, 0x3fefe792b3b38c6e, 0xbfabd5510839a77b, 0xbfd4f40fd8f0a91b, 0x3fab9602138cdbe9, 0x3fc8896c8f290450),
	(0x3fae000000ffdc11, 0x3fadf73aa09e1477, 0x3fefe3f8a27756be, 0xbfadcb8935b8415f, 0xbfd4e5cae94695f1, 0x3fad7d32cd68db68, 0x3fc860b09a86ff00),
	(0x3fb0000002ad31a4, 0x3faff55bbc820c23, 0x3fefe01fe0154058, 0xbfafc05f85dd78f3, 0xbfd4d692f8f482ea, 0x3faf617ebeb00bd8, 0x3fc834f3a9931b00),
	(0x3fb100000188824f, 0x3fb0f99ea8a41bcd, 0x3fefdc089a435d4f, 0xbfb0d9de9f1c8568, 0xbfd4c669ca505107, 0x3fb0a166428a466f, 0x3fc8063f36448e35),
	(0x3fb20000009bc757, 0x3fb1f86dbfa33080, 0x3fefd7b3016f7596, 0xbfb1d2c5eedc6080, 0xbfd4b55130ac22d8, 0x3fb18fd8c2bb0ee4, 0x3fc7d526db38f039),
	(0x3fb3000024c171ae, 0x3fb2f7195618302f, 0x3fefd31f48358ab4, 0xbfb2cadaa8a559c1, 0xbfd4a34b240cff5f, 0x3fb27c47c2ad49a6, 0x3fc7a1702e4d8d69),
	(0x3fb4000000068d16, 0x3fb3f59f0e82d887, 0x3fefce4da6ab73c7, 0xbfb3c2114d1e9dc2, 0xbfd49059bce708bf, 0x3fb36699156cde19, 0x3fc76b264479966d),
	(0x3fb4ffffffdaf503, 0x3fb4f3fd674dc761, 0x3fefc93e53a2ae2c, 0xbfb4b85f4a8c15c6, 0xbfd47c7f1d6673be, 0x3fb44eb3ca7ef180, 0x3fc7325487c15892),
	(0x3fb60000005453f7, 0x3fb5f23250268d66, 0x3fefc3f18b52cb50, 0xbfb5adb99d22e5c3, 0xbfd467bd8b4de796, 0x3fb5347ee36faada, 0x3fc6f706fed2d6e1),
	(0x3fb7000000057dfc, 0x3fb6f03bdcefbdc7, 0x3fefbe678c078564, 0xbfb6a2157e35172d, 0xbfd452176286cdb9, 0x3fb617e1fe0c1388, 0x3fc6b94a23d2539a),
	(0x3fb7fffffe7c548b, 0x3fb7ee182482a642, 0x3fefb8a096b5e8ba, 0xbfb7956844e7782f, 0xbfd43b8f17095f3f, 0x3fb6f8c53a2a94dc, 0x3fc6792ae7109ee3),
	(0x3fb9000000399ffa, 0x3fb8ebc544b20fc7, 0x3fefb29ceedb75bc, 0xbfb887a76b0a59ee, 0xbfd4242733f64aa6, 0x3fb7d7114183708e, 0x3fc636b6a9a36669),
	(0x3fba000000503ea3, 0x3fb9e941541f49d9, 0x3fefac5cdacc38c0, 0xbfb978c880930264, 0xbfd40be25c5cc9fc, 0x3fb8b2af3f76ad1f, 0x3fc5f1fb3caa1a83),
	(0x3fbafffffefd604e, 0x3fbae68a70c75b65, 0x3fefa5e0a3559ff9, 0xbfba68c13a88c726, 0xbfd3f2c349624f3b, 0x3fb98b88f1f263f7, 0x3fc5ab06d90ee046),
	(0x3fbc0000027ada67, 0x3fbbe39ec0e260e9, 0x3fef9f2893aa9d84, 0xbfbb578774c87524, 0xbfd3d8ccc2282c36, 0x3fba6104a32547f8, 0x3fc5622c6b5cdc00),
	(0x3fbcfffffe2aae0e, 0x3fbce07c5a6d6a81, 0x3fef9834fa03c879, 0xbfbc45111ba09bf8, 0xbfd3be01baf8bc7a, 0x3fbb3410872e3840, 0x3fc516f2437f8a00),
	(0x3fbe000001dab907, 0x3fbddd2171f3051f, 0x3fef910626538a70, 0xbfbd3154591dc7f9, 0xbfd3a2652c4d0b90, 0x3fbc04a65d78b257, 0x3fc4c967e0fd9461),
	(0x3fbeffffffa53a79, 0x3fbed98c21368e87, 0x3fef899c6bd1f677, 0xbfbe1c47601d7e8f, 0xbfd385fa159fbfca, 0x3fbcd19beca90b8c, 0x3fc47a256b36705a),
	(0x3fc000000f142d5a, 0x3fbfd5bab85dc3d0, 0x3fef81f81e981366, 0xbfbf05e0b86e7006, 0xbfd368c39b4690e4, 0x3fbd9b66d4a9b7f8, 0x3fc428f6923eacde),
	(0x3fc07ffffcdf1951, 0x3fc068d5810d5bd4, 0x3fef7a19994da3f1, 0xbfbfee169210a60e, 0xbfd34ac4fcd8967e, 0x3fbe61f4253945ff, 0x3fc3d5ebc0588a79),
	(0x3fc1000002aa39f4, 0x3fc0e6adcf926f2e, 0x3fef7201326bccc1, 0xbfc06a70037899d2, 0xbfd32c01735432c2, 0x3fbf25323b0051d6, 0x3fc38115615f521e),
	(0x3fc18000001fea3b, 0x3fc1646541255ca1, 0x3fef69af483d2f52, 0xbfc0dd19dadefc1f, 0xbfd30c7c5e62ba59, 0x3fbfe50f8f51460e, 0x3fc32a845cd3502a),
	(0x3fc1fffffe8823bc, 0x3fc1e1faf993a4c8, 0x3fef612438bab95f, 0xbfc14f04587e458a, 0xbfd2ec3924d36b33, 0x3fc050bdc309e406, 0x3fc2d249bb3c927d),
	(0x3fc27ffffc9337b0, 0x3fc25f6e13bf7bd7, 0x3fef586064c338f8, 0xbfc1c02b10b43292, 0xbfd2cb3b4055087c, 0x3fc0ad330fd21a61, 0x3fc27876c32ba8fe),
	(0x3fc2fffffdc6bf7b, 0x3fc2dcbdb0cea731, 0x3fef4f642f0fe16c, 0xbfc23089b1286391, 0xbfd2a98638fb0d98, 0x3fc107e008ee7ac7, 0x3fc21d1ce9880c12),
	(0x3fc3800000c2fb33, 0x3fc359e8eeaa28a7, 0x3fef462ffcd452fe, 0xbfc2a01bf88e9498, 0xbfd2871da71092b5, 0x3fc160bd5de46b03, 0x3fc1c04dd31f291d),
	(0x3fc4000000d8f876, 0x3fc3d6eee99a2f23, 0x3fef3cc435a04a95, 0xbfc30eddb88397f1, 0xbfd26405321912f4, 0x3fc1b7c415010900, 0x3fc1621b4e780103),
	(0x3fc48000000de728, 0x3fc453cec616b8c4, 0x3fef332142989e5d, 0xbfc37ccade8e0ca6, 0xbfd240408d609878, 0x3fc20ced92d9e381, 0x3fc1029747314f3d),
	(0x3fc4ffffff0e7ab0, 0x3fc4d087a8ef1e63, 0x3fef29478efdfd42, 0xbfc3e9df6d60cba2, 0xbfd21bd3796a9ce3, 0x3fc26033953388f3, 0x3fc0a1d3c688050b),
	(0x3fc57ffffc7377dc, 0x3fc54d18b69dbcdb, 0x3fef1f3788337a10, 0xbfc456177c933fea, 0xbfd1f6c1c37d2372, 0x3fc2b190331b73b1, 0x3fc03fe2eebeac95),
	(0x3fc5ffffff069c34, 0x3fc5c9811d4c8683, 0x3fef14f19ce2618f, 0xbfc4c16f418df8ce, 0xbfd1d10f41eb9d16, 0x3fc300fde3b8367f, 0x3fbfb9addc08f782),
	(0x3fc67ffffe3dace2, 0x3fc645bffdfed6b5, 0x3fef0a763ecf8d44, 0xbfc52be2fc09f17c, 0xbfd1aabfda46e407, 0x3fc34e776fde19e4, 0x3fbef18416e713f0),
	(0x3fc6fffffe88eb88, 0x3fc6c1d4881ddaae, 0x3feeffc5e08c9b3d, 0xbfc5956f0e16385c, 0xbfd183d77857c309, 0x3fc399f803dfed36, 0x3fbe276d18332ee6),
	(0x3fc77fffffbe804c, 0x3fc73dbde868750f, 0x3feef4e0f6d3f675, 0xbfc5fe0fed8db872, 0xbfd15c5a1291701e, 0x3fc3e37b24a92e8d, 0x3fbd5b8d76fa3afe),
	(0x3fc7fffffc35d1d7, 0x3fc7b97b48251fed, 0x3feee9c7f89a7009, 0xbfc665c223be9680, 0xbfd1344ba9bdc601, 0x3fc42afcaf9262f8, 0x3fbc8e09cf02b355),
	(0x3fc8800000a239dc, 0x3fc8350be43569e9, 0x3feede7b5cee6e27, 0xbfc6cc8260df5e45, 0xbfd10bb040c61a32, 0x3fc47078e771a1fe, 0x3fbbbf06908b1814),
	(0x3fc90000027df7fa, 0x3fc8b06ee4ee22e7, 0x3feed2fb9e8bb11b, 0xbfc7324d5aa33c81, 0xbfd0e28be90da705, 0x3fc4b3ec5d56fe5c, 0x3fbaeea83980b70d),
	(0x3fc9800002d31c2a, 0x3fc92ba37fbc8427, 0x3feec74938c8c904, 0xbfc7971fe886b4de, 0xbfd0b8e2b6d26476, 0x3fc4f55403930d95, 0x3fba1d131409b3ab),
	(0x3fc9fffffcddfcfd, 0x3fc9a6a8e66a4ab7, 0x3feebb64a91453c7, 0xbfc7faf6f60a4367, 0xbfd08eb8c5fc4160, 0x3fc534ad23ed705c, 0x3fb94a6b493e7c18),
	(0x3fca7fffff8b55ee, 0x3fca217e5fa0a315, 0x3feeaf4e6c86c253, 0xbfc85dcf96a2bb00, 0xbfd0641230fa59ed, 0x3fc571f56b9c5c8a, 0x3fb876d4adfc5ac3),
	(0x3fcafffffd414cf5, 0x3fca9c23189f6ea5, 0x3feea30704596a99, 0xbfc8bfa6e0659b60, 0xbfd038f3200791fd, 0x3fc5ad2ad32c8d7a, 0x3fb7a273097623dd),
	(0x3fcb7ffffe765c40, 0x3fcb169655d5327a, 0x3fee968ef1197db7, 0xbfc9207a13521ece, 0xbfd00d5fb7ad7b56, 0x3fc5e64bb64ff60a, 0x3fb6cd69b8f6e504),
	(0x3fcc000000abc4ff, 0x3fcb90d753364d8b, 0x3fee89e6b5bbd593, 0xbfc980467fefa357, 0xbfcfc2b8463315e8, 0x3fc61d56c362a69c, 0x3fb5f7dbddfa3d81),
	(0x3fcc7fffffda3e36, 0x3fcc0ae54d528b2b, 0x3fee7d0ed6b1420d, 0xbfc9df098ebe3358, 0xbfcf69d920a9cf8c, 0x3fc6524aff4b3332, 0x3fb521ec46597687),
	(0x3fccfffffff895a3, 0x3fcc84bf8a6d20c0, 0x3fee7007d8e2c735, 0xbfca3cc0c754ee99, 0xbfcf102a3a0ed514, 0x3fc6849fe6a47028, 0x3fb44c40e5060800),
	(0x3fcd800002e87b67, 0x3fccfe6550e0425b, 0x3fee62d242a27514, 0xbfca9969c88af14c, 0xbfceb5b42bd588bd, 0x3fc6b562a8259298, 0x3fb375f45f6e7800),
	(0x3fce0000007c3fa1, 0x3fcd77d5df961e94, 0x3fee556e9c79c2d5, 0xbfcaf502433f724e, 0xbfce5a7f5c4a951a, 0x3fc6e49a1c28bf4d, 0x3fb29f2902804d52),
	(0x3fce8000018c9844, 0x3fcdf11087c3e708, 0x3fee47dd6e4b27d1, 0xbfcb4f880dd31913, 0xbfcdfe93e4e6b22c, 0x3fc71130057e8a97, 0x3fb1c906570f0d02),
	(0x3fcf0000011971a7, 0x3fce6a148fa0c60b, 0x3fee3a1f427d6bd1, 0xbfcba8f90d9b0ad8, 0xbfcda1fa4468eaa4, 0x3fc73baf35ad6fbc, 0x3fb0f329b1d71c3b),
	(0x3fcf7fffffae07f0, 0x3fcee2e144d04e42, 0x3fee2c34a3e84e63, 0xbfcc015345086350, 0xbfcd44bad359c3db, 0x3fc76418a7b6f7b9, 0x3fb01db333c0e897),
	(0x3fd0000003c8cfe7, 0x3fcf5b76004c2654, 0x3fee1e1e1d47929d, 0xbfcc5894d623e3cd, 0xbfcce6ddde8c38ac, 0x3fc78a6da87d134e, 0x3fae9184c911f786),
	(0x3fd0400003e0f35b, 0x3fcfd3d2038a7c77, 0x3fee0fdc3d4407ce, 0xbfccaebbe9509c5a, 0xbfcc886bc113543a, 0x3fc7aeafca638093, 0x3face8ecd45e8a40),
	(0x3fd08000032761cb, 0x3fd025fa53fb759e, 0x3fee016f90dfb3ea, 0xbfcd03c6cfa46b13, 0xbfcc296cbe2c693a, 0x3fc7d0e0f31c6cc9, 0x3fab41db5e90e29b),
	(0x3fd0bffffd1bbae5, 0x3fd061ee9d889212, 0x3fedf2d8a7a1ddae, 0xbfcd57b3e8faad27, 0xbfcbc9e91c1a6c4e, 0x3fc7f1034f0598ee, 0x3fa99c8c39820b78),
	(0x3fd0fffffda7b6b5, 0x3fd09dc595a7aa29, 0x3fede4180e167b90, 0xbfcdaa81c348841c, 0xbfcb69e8ff46f764, 0x3fc80f195bae93a5, 0x3fa7f9398b9c2669),
	(0x3fd1400002df4b88, 0x3fd0d97ee7b72b9e, 0x3fedd52e54a32e74, 0xbfcdfc2ef2b6e464, 0xbfcb0974963c35fd, 0x3fc82b25d6d98068, 0x3fa6581c8d5f3f91),
	(0x3fd18000007fef6f, 0x3fd1151a369b3ad0, 0x3fedc61c0ed3a8b3, 0xbfce4cba17180e39, 0xbfcaa89413bba312, 0x3fc8452bc0669ca5, 0x3fa4b96d672b82b0),
	(0x3fd1bffffd750e56, 0x3fd1509738407215, 0x3fedb6e1cc5e56bc, 0xbfce9c22010b8ec2, 0xbfca474f8102a8b8, 0x3fc85d2e64dd2c0a, 0x3fa31d6268e4aace),
	(0x3fd2000003ca0de4, 0x3fd18bf5a68ee3b3, 0x3feda7801cbd34e3, 0xbfceea659ca4de3f, 0xbfc9e5aed4b250a4, 0x3fc87331536725ba, 0x3fa184306c2c9ae3),
	(0x3fd2400002036234, 0x3fd1c735230a790e, 0x3fed97f79692b2ec, 0xbfcf3783cc4f0a97, 0xbfc983ba20bb093a, 0x3fc887385275eb6c, 0x3f9fdc171f589263),
	(0x3fd27fffffdfc6c4, 0x3fd2025567c6bf7b, 0x3fed8848caf35b89, 0xbfcf837ba93cd7cf, 0xbfc9217940ea8bf9, 0x3fc89947704663ae, 0x3f9cb64bbd3e6b8a),
	(0x3fd2bffffed8628c, 0x3fd23d562a27d0d0, 0x3fed78744ce26842, 0xbfcfce4c5c6a3c83, 0xbfc8bef409ec6215, 0x3fc8a962f5b80e45, 0x3f999760297b6b81),
	(0x3fd2fffffd9ac75c, 0x3fd278371e2461ca, 0x3fed687ab0799710, 0xbfd00bfa9178bb91, 0xbfc85c32431fb914, 0x3fc8b78f661ebb08, 0x3f967fb3033009f6),
	(0x3fd33ffffe5a26b1, 0x3fd2b2f7fc188619, 0x3fed585c894c7f41, 0xbfd0303aaa9777f1, 0xbfc7f93b9bf3c072, 0x3fc8c3d17ee8c3ea, 0x3f936f9fb86fef86),
	(0x3fd37ffffec1b143, 0x3fd2ed98795ef86b, 0x3fed481a6c610cdd, 0xbfd053e62ea7a861, 0xbfc79617b7755508, 0x3fc8ce2e3400f439, 0x3f90677ee0a18b9f),
	(0x3fd3c000003531d6, 0x3fd328184fe61b16, 0x3fed37b4ee113238, 0xbfd076fce03b5660, 0xbfc732ce1eb44786, 0x3fc8d6aaaf852261, 0x3f8acf4b9ec59d46),
	(0x3fd40000014b4f1a, 0x3fd362773835c112, 0x3fed272ca3a66d1e, 0xbfd0997e8b99570c, 0xbfc6cf664a09a01d, 0x3fc8dd4c4eb42bdc, 0x3f84e0cdba8b378e),
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	(0x3fef7ffffdfbe7bb, 0x3fe8e17aa896a3fa, 0x3fe0407ffcf5ee0b, 0xbfd03ffbecf3660f, 0x3fb5513b2748a2d9, 0x3f6029586a500800, 0xbf9bdfc7884c0000),
	(0x3fef9fffff0fa180, 0x3fe8f1b30bcb577c, 0x3fe03047ff31c004, 0xbfd02ffe4a3a5b50, 0x3fb5530ac95cb722, 0x3f57c33cda611000, 0xbf9b59e0e3a28000),
	(0x3fefc00001a24bf4, 0x3fe901db3fc1bb6f, 0x3fe0201ffeecb132, 0xbfd01fff7e0f13c7, 0x3fb55452c9166f13, 0x3f4eba61be0c4000, 0xbf9ad57931778000),
	(0x3fefdfffffe38639, 0x3fe911f3518b3813, 0x3fe01008000a410c, 0xbfd00ffff01bf387, 0x3fb55515b9099fdd, 0x3f3c805a37250000, 0xbf9a52910b908000),
	(0x3feff0000008727f, 0x3fe919f953f31326, 0x3fe00801fffb8473, 0xbfd007fffdfb4766, 0x3fb555453ecbbd3f, 0x3f30045a21318845, 0xbf9a12c2520fd1de),
]

const atan32 = {x
	/*
	   Irritating special rounding cases that would require
	   more extra accuracy than they're worth to correctly round
	   back from 64 to 32 bits.
	 */
	var xb = std.flt32bits(x)
	if xb == 0x3d8d6b23
		-> std.flt32frombits(0x3d8d31c3)
	elif xb == 0xbd8d6b23
		-> std.flt32frombits(0xbd8d31c3)
	;;

	var r = atan264((x : flt64), 1.0)
	-> (r : flt32)
}

const atan64 = {x
	-> atan264(x, 1.0)
}

const atan232 = {y, x
	/* Handle the special cases of atan32 for consistency */
	if x == 1.0
		var yb = std.flt32bits(y)
		if yb == 0x3d8d6b23
			-> std.flt32frombits(0x3d8d31c3)
		elif yb == 0xbd8d6b23
			-> std.flt32frombits(0xbd8d31c3)
		;;
	;;

	var r = atan264((y : flt64), (x : flt64))
	-> (r : flt32)
}

const atan264 = {y, x
	if std.isnan(y) || std.isnan(x)
		-> std.flt64nan()
	;;

	var xb = std.flt64bits(x)
	var yb = std.flt64bits(y)
	var xpos, xe, ypos, ye
	(xpos, xe, _) = std.flt64explode(x)
	(ypos, ye, _) = std.flt64explode(y)
	xpos = !xpos
	ypos = !ypos

	/* All the special cases */
	if yb == 0x0000000000000000
		if xpos
			-> 0.0
		else
			-> std.flt64frombits(0x400921fb54442d18)
		;;
	elif yb == 0x8000000000000000
		if xpos
			-> std.flt64frombits(0x8000000000000000)
		else
			-> std.flt64frombits(0xc00921fb54442d18)
		;;
	elif xb == 0x0000000000000000 || xb == 0x8000000000000000
		if ypos
			-> std.flt64frombits(0x3ff921fb54442d18)
		else
			-> std.flt64frombits(0xbff921fb54442d18)
		;;
	;;

	var xinf = (xe == 1024)
	var yinf = (ye == 1024)

	match (yinf, ypos, xinf, xpos)
	| (true,  true,  true,  false): -> std.flt64frombits(0x4002d97c7f3321d2)
	| (true,  false, true,  false): -> std.flt64frombits(0xc002d97c7f3321d2)
	| (true,  true,  true,  true ): -> std.flt64frombits(0x3fe921fb54442d18)
	| (true,  false, true,  true ): -> std.flt64frombits(0xbfe921fb54442d18)
	| (true,  true,  false, _    ): -> std.flt64frombits(0x3ff921fb54442d18)
	| (true,  false, false, _    ): -> std.flt64frombits(0xbff921fb54442d18)
	| (false, true,  true,  false): -> std.flt64frombits(0x400921fb54442d18)
	| (false, false, true,  false): -> std.flt64frombits(0xc00921fb54442d18)
	| (false, true,  true,  true ): -> std.flt64frombits(0x0000000000000000)
	| (false, false, true,  true ): -> std.flt64frombits(0x8000000000000000)
	| _:
	;;

	/* Normal case. Here we just reduce y/x to [0, 1] */
	var xabs = std.flt64frombits(std.flt64bits(x) & (~(0 : uint64)) >> 1)
	var yabs = std.flt64frombits(std.flt64bits(y) & (~(0 : uint64)) >> 1)

	var then_negate = !ypos
	var then_subtract_from_pi = !xpos

	x = xabs
	y = yabs


	var then_reverse = false
	if y > x
		then_reverse = true
		std.swap(&x, &y)
	;;

	/* Compute y/x = u + du */
	var u = y / x
	var du
	if u * x == y
		du = 0.0
	else
		var t1, t2
		(t1, t2) = two_by_two64(u, x)
		du = ((y - t1) - t2)/x
	;;

	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
	;;
	if then_subtract_from_pi
		/* Compute pi - ret, or maybe -pi - ret */
		var pi_hi = scale2(std.flt64frombits(pi_over_2[0]), 1)
		var pi_lo = scale2(std.flt64frombits(pi_over_2[1]), 1)
		ret = (pi_hi - ret) + pi_lo
	;;

	if then_negate
		ret = -1.0 * ret
	;;

	-> ret
}

/* Handle arctan(z) with z in [0, 1] */
const atan_reduced = {u : flt64, du : flt64
	var ret : flt64

	/*
	   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[:])
		-> u*p + u*du
	;;

	/*
	   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)

	-> horner_polyu((u - xi) + du, c[:])
}