ref: f2e8e641bf8ee06fa5128c522b151a7f4d9d003b
dir: /lib/math/exp-impl.myr/
use std
use "fpmath"
use "util"
/*
See [Mul16] (6.2.1), [Tan89], and [Tan92]. While the exp and
expm1 functions are similar enough to be in the same file (Tang's
algorithms use the same S constants), they are not quite similar
enough to be in the same function.
*/
pkg math =
pkglocal const exp32 : (x : flt32 -> flt32)
pkglocal const exp64 : (x : flt64 -> flt64)
pkglocal const expm132 : (x : flt32 -> flt32)
pkglocal const expm164 : (x : flt64 -> flt64)
;;
extern const horner_polyu32 : (x : flt32, a : uint32[:] -> flt32)
extern const horner_polyu64 : (x : flt64, a : uint64[:] -> flt64)
type fltdesc(@f, @u, @i) = struct
explode : (f : @f -> (bool, @i, @u))
assem : (n : bool, e : @i, s : @u -> @f)
horner : (f : @f, a : @u[:] -> @f)
sgnmask : @u
tobits : (f : @f -> @u)
frombits : (u : @u -> @f)
inf : @u
nan : @u
/* For exp */
thresh_1_min : @u
thresh_1_max : @u
thresh_2 : @u
Ai : @u[:]
/* For expm1 */
thresh_tiny : @u
thresh_huge_neg : @u
Log3_4 : @u
Log5_4 : @u
Bi : @u[:]
precision : @u
/* For both */
nabs : @u
inv_L : @u
L1 : @u
L2 : @u
S : (@u, @u)[32]
;;
const desc32 : fltdesc(flt32, uint32, int32) = [
.explode = std.flt32explode,
.assem = std.flt32assem,
.horner = horner_polyu32,
.sgnmask = (1 << 31),
.tobits = std.flt32bits,
.frombits = std.flt32frombits,
.inf = 0x7f800000,
.nan = 0x7fc00000,
.thresh_1_min = 0xc2cff1b4, /* ln(2^(-127 - 23)) ~= -103.9720770839917 */
.thresh_1_max = 0x42b17218, /* ln([2 - 2^(-24)] * 2^127) ~= 88.72283905206 */
.inv_L = 0x4238aa3b, /* L = log(2) / 32, this is 1/L in working precision */
.L1 = 0x3cb17200, /* L1 and L2 sum to give L in high precision, */
.L2 = 0x333fbe8e, /* and L1 has some trailing zeroes. */
.nabs = 9, /* L1 has 9 trailing zeroes in binary */
.Ai = [ /* Coefficients for approximating expm1 in a tiny range */
0x3f000044,
0x3e2aaaec,
][:],
.Log3_4 = 0xbe934b11, /* log(1 - 1/4) < x < log(1 + 1/4) */
.Log5_4 = 0x3e647fbf, /* triggers special expm1 case */
.thresh_tiny = 0x33000000, /* similar to thresh_1_{min,max}, but for expm1 */
.thresh_huge_neg = 0xc18aa122,
.Bi = [ /* Coefficients for approximating expm1 between log(3/4) and log(5/4) */
0x3e2aaaaa,
0x3d2aaaa0,
0x3c0889ff,
0x3ab64de5,
0x394ab327,
][:],
.S = [ /* Double-precise expansions of 2^(J/32) */
(0x3f800000, 0x00000000),
(0x3f82cd80, 0x35531585),
(0x3f85aac0, 0x34d9f312),
(0x3f889800, 0x35e8092e),
(0x3f8b95c0, 0x3471f546),
(0x3f8ea400, 0x36e62d17),
(0x3f91c3c0, 0x361b9d59),
(0x3f94f4c0, 0x36bea3fc),
(0x3f9837c0, 0x36c14637),
(0x3f9b8d00, 0x36e6e755),
(0x3f9ef500, 0x36c98247),
(0x3fa27040, 0x34c0c312),
(0x3fa5fec0, 0x36354d8b),
(0x3fa9a140, 0x3655a754),
(0x3fad5800, 0x36fba90b),
(0x3fb123c0, 0x36d6074b),
(0x3fb504c0, 0x36cccfe7),
(0x3fb8fb80, 0x36bd1d8c),
(0x3fbd0880, 0x368e7d60),
(0x3fc12c40, 0x35cca667),
(0x3fc56700, 0x36a84554),
(0x3fc9b980, 0x36f619b9),
(0x3fce2480, 0x35c151f8),
(0x3fd2a800, 0x366c8f89),
(0x3fd744c0, 0x36f32b5a),
(0x3fdbfb80, 0x36de5f6c),
(0x3fe0ccc0, 0x36776155),
(0x3fe5b900, 0x355cef90),
(0x3feac0c0, 0x355cfba5),
(0x3fefe480, 0x36e66f73),
(0x3ff52540, 0x36f45492),
(0x3ffa8380, 0x36cb6dc9),
],
.precision = 24, /* threshold to prevent underflow in a S[high] + 2^-m */
]
const desc64 : fltdesc(flt64, uint64, int64) = [
.explode = std.flt64explode,
.assem = std.flt64assem,
.horner = horner_polyu64,
.sgnmask = (1 << 63),
.tobits = std.flt64bits,
.frombits = std.flt64frombits,
.inf = 0x7ff0000000000000,
.nan = 0x7ff8000000000000,
.thresh_1_min = 0xc0874910d52d3052, /* ln(2^(-1023 - 52)) ~= -745.1332191019 */
.thresh_1_max = 0x40862e42fefa39ef, /* ln([2 - 2^(-53)] * 2^1023) ~= 709.78271289 */
.inv_L = 0x40471547652b82fe, /* below as in exp32 */
.L1 = 0x3f962e42fef00000,
.L2 = 0x3d8473de6af278ed,
.nabs = 20,
.Ai = [
0x3fe0000000000000,
0x3fc5555555548f7c,
0x3fa5555555545d4e,
0x3f811115b7aa905e,
0x3f56c1728d739765,
][:],
.Log3_4 = 0xbfd269621134db93,
.Log5_4 = 0x3fcc8ff7c79a9a22,
.thresh_tiny = 0x3c90000000000000,
.thresh_huge_neg = 0xc042b708872320e1,
.Bi = [
0x3fc5555555555549,
0x3fa55555555554b6,
0x3f8111111111a9f3,
0x3f56c16c16ce14c6,
0x3f2a01a01159dd2d,
0x3efa019f635825c4,
0x3ec71e14bfe3db59,
0x3e928295484734ea,
0x3e5a2836aa646b96,
][:],
.S = [
(0x3ff0000000000000, 0x0000000000000000),
(0x3ff059b0d3158540, 0x3d0a1d73e2a475b4),
(0x3ff0b5586cf98900, 0x3ceec5317256e308),
(0x3ff11301d0125b40, 0x3cf0a4ebbf1aed93),
(0x3ff172b83c7d5140, 0x3d0d6e6fbe462876),
(0x3ff1d4873168b980, 0x3d053c02dc0144c8),
(0x3ff2387a6e756200, 0x3d0c3360fd6d8e0b),
(0x3ff29e9df51fdec0, 0x3d009612e8afad12),
(0x3ff306fe0a31b700, 0x3cf52de8d5a46306),
(0x3ff371a7373aa9c0, 0x3ce54e28aa05e8a9),
(0x3ff3dea64c123400, 0x3d011ada0911f09f),
(0x3ff44e0860618900, 0x3d068189b7a04ef8),
(0x3ff4bfdad5362a00, 0x3d038ea1cbd7f621),
(0x3ff5342b569d4f80, 0x3cbdf0a83c49d86a),
(0x3ff5ab07dd485400, 0x3d04ac64980a8c8f),
(0x3ff6247eb03a5580, 0x3cd2c7c3e81bf4b7),
(0x3ff6a09e667f3bc0, 0x3ce921165f626cdd),
(0x3ff71f75e8ec5f40, 0x3d09ee91b8797785),
(0x3ff7a11473eb0180, 0x3cdb5f54408fdb37),
(0x3ff82589994cce00, 0x3cf28acf88afab35),
(0x3ff8ace5422aa0c0, 0x3cfb5ba7c55a192d),
(0x3ff93737b0cdc5c0, 0x3d027a280e1f92a0),
(0x3ff9c49182a3f080, 0x3cf01c7c46b071f3),
(0x3ffa5503b23e2540, 0x3cfc8b424491caf8),
(0x3ffae89f995ad380, 0x3d06af439a68bb99),
(0x3ffb7f76f2fb5e40, 0x3cdbaa9ec206ad4f),
(0x3ffc199bdd855280, 0x3cfc2220cb12a092),
(0x3ffcb720dcef9040, 0x3d048a81e5e8f4a5),
(0x3ffd5818dcfba480, 0x3cdc976816bad9b8),
(0x3ffdfc97337b9b40, 0x3cfeb968cac39ed3),
(0x3ffea4afa2a490c0, 0x3cf9858f73a18f5e),
(0x3fff50765b6e4540, 0x3c99d3e12dd8a18b),
],
.precision = 53,
]
const exp32 = {x : flt32
-> expgen(x, desc32)
}
const exp64 = {x : flt64
-> expgen(x, desc64)
}
generic expgen = {x : @f, d : fltdesc(@f, @u, @i) :: numeric,floating,std.equatable @f, numeric,integral @u, numeric,integral @i, roundable @f -> @i
var b = d.tobits(x)
var n, e, s
(n, e, s) = d.explode(x)
/*
Detect if exp(f) would round to outside representability.
We don't do bias adjustment, so Tang's thresholds can
be tightened.
*/
if !n && b >= d.thresh_1_max
-> d.frombits(d.inf)
elif n && b > d.thresh_1_min
-> (0.0 : @f)
;;
/* Detect if exp(f) is close to 1. */
if (b & ~d.sgnmask) <= d.thresh_2
-> (1.0 : @f)
;;
/* Argument reduction to [ -ln(2)/64, ln(2)/64 ] */
var inv_L = d.frombits(d.inv_L)
var N = rn(x * inv_L)
var N2 = N % (32 : @i)
if N2 < 0
N2 += (32 : @i)
;;
var N1 = N - N2
var R1 : @f, R2 : @f
var Nf = (N : @f)
/*
There are few enough significant bits that these are all
exact, without need for an FMA. R1 + R2 will approximate
(very well) f reduced into [ -ln(2)/64, ln(2)/64 ]
*/
if std.abs(N) >= (1 << d.nabs)
R1 = (x - (N1 : @f) * d.frombits(d.L1)) - ((N2 : @f) * d.frombits(d.L1))
else
R1 = x - (N : @f) * d.frombits(d.L1)
;;
R2 = -1.0 * (N : @f) * d.frombits(d.L2)
var M = (N1 : @i) / 32
var J = (N2 : std.size)
/* Compute a polynomial approximation of exp1m */
var R = R1 + R2
var Q = R * R * d.horner(R, d.Ai)
var P = R1 + (R2 + Q)
/* Expand out from reduced range */
var Su_hi, Su_lo
(Su_hi, Su_lo) = d.S[J]
var S_hi = d.frombits(Su_hi)
var S_lo = d.frombits(Su_lo)
var S = S_hi + S_lo
var unscaled = S_hi + (S_lo + (P * S))
var nu, eu, su
(nu, eu, su) = d.explode(unscaled)
var exp = d.assem(nu, eu + M, su)
if (d.tobits(exp) == 0)
/* Make sure we don't quite return 0 */
-> d.frombits(1)
;;
-> exp
}
const expm132 = {x : flt32
-> expm1gen(x, desc32)
}
const expm164 = {x : flt64
-> expm1gen(x, desc64)
}
generic expm1gen = {x : @f, d : fltdesc(@f, @u, @i) :: \
numeric,floating,std.equatable @f,
numeric,integral @u,
numeric,integral @i,
roundable @f -> @i
var b = d.tobits(x)
var n, e, s
(n, e, s) = d.explode(x)
/* Special cases: +/- 0, inf, NaN, tiny, and huge */
if (b & ~d.sgnmask == 0)
-> x
elif n && (b & ~d.sgnmask == d.inf)
-> (-1.0 : @f)
elif (b & ~d.sgnmask == d.inf)
-> x
elif std.isnan(x)
-> d.frombits(d.nan)
elif (b & ~d.sgnmask) <= d.thresh_tiny
var two_to_large = d.assem(false, 100, 0)
var two_to_small = d.assem(false, -100, 0)
var abs_x = d.assem(false, e, s)
-> (two_to_large * x + abs_x) * two_to_small
elif !n && b >= d.thresh_1_max /* exp(x) = oo <=> expm1(x) = oo, as it turns out */
-> d.frombits(d.inf)
elif n && b >= d.thresh_huge_neg
-> (-1.0 : @f)
;;
if ((n && b < d.Log3_4) || (!n && b < d.Log5_4))
/* Procedure 2 */
/* compute x^2 / 2 with extra precision */
var u = round(x, d)
var v = x - u
var y = u * u * (0.5 : @f)
var z = v * (x + u) * (0.5 : @f)
var q = x * x * x * d.horner(x, d.Bi)
var yn, ye, ys
(yn, ye, ys) = d.explode(y)
if (ye >= -7)
-> (u + y) + (q + (v + z))
else
-> x + (y + (q + z))
;;
;;
/* Procedure 1 */
var inv_L = d.frombits(d.inv_L)
var N = rn(x * inv_L)
var N2 = N % (32 : @i)
if N2 < 0
N2 += (32 : @i)
;;
var N1 = N - N2
var R1 : @f, R2 : @f
/*
As in the exp case, R1 + R2 very well approximates f
reduced into [ -ln(2)/64, ln(2)/64 ]
*/
if std.abs(N) >= (1 << d.nabs)
R1 = (x - (N1 : @f) * d.frombits(d.L1)) - ((N2 : @f) * d.frombits(d.L1))
else
R1 = x - (N : @f) * d.frombits(d.L1)
;;
R2 = -1.0 * (N : @f) * d.frombits(d.L2)
var M = (N1 : @i) / 32
var J = (N2 : std.size)
/* Compute a polynomial approximation of exp1m */
var R = R1 + R2
var Q = R * R * d.horner(R, d.Ai)
var P = R1 + (R2 + Q)
/* Expand out */
var Su_hi, Su_lo
(Su_hi, Su_lo) = d.S[J]
var S_hi = d.frombits(Su_hi)
var S_lo = d.frombits(Su_lo)
var S = S_hi + S_lo
/*
Note that [Tan92] includes cases to avoid tripping the
underflow flag when not technically required. We currently
do not care about such flags, so that logic is omitted.
*/
if M >= d.precision
-> scale2(S_hi + (S * P + (S_lo - scale2(1.0, -M))), M)
elif M <= -8
-> scale2(S_hi + (S * P + S_lo), M) - (1.0 : @f)
;;
-> scale2((S_hi - scale2(1.0, -M)) + (S_hi * P + S_lo * (P + (1.0 : @f))), M)
}
generic round = {f : @f, d : fltdesc(@f, @u, @i) :: numeric,floating,std.equatable @f, numeric,integral @u, numeric,integral @i, roundable @f -> @i
var b = d.tobits(f)
var n, e, s
(n, e, s) = d.explode(f)
var mask = ~((1 << d.nabs) - 1)
if need_round_away(0, ((s & mask) : uint64), (d.nabs : int64) + 1)
-> d.assem(n, e, 1 + s & mask)
;;
-> d.assem(n, e, s & mask)
}