ref: 16a7bb8f8cc86b22ce157e0944f66f60a556590f
dir: /lib/math/powr-impl.myr/
use std
use "fpmath"
use "log-impl"
use "util"
/*
This is an implementation of powr, not pow, so the special cases
are tailored more closely to the mathematical x^y = e^(y * log(x))
than to historical C implementations (pow was aligned to the C99
standard, which was aligned to codify existing practice).
Even then, some parts of the powr specification are unclear. For
example, IEEE 754-2008 does not specify what powr(infty, y) must
return when y is not 0.0 (an erratum was planned in 2010, but
does not appear to have been released as of 2018).
As a note: unlike many other functions in this library, there
has been no serious analysis of the accuracy and speed of this
particular implementation. Interested observers wishing to improve
this library will probably find this file goldmine of mistakes,
both theoretical and practical.
*/
pkg math =
pkglocal const powr32 : (x : flt32, y : flt32 -> flt32)
pkglocal const powr64 : (x : flt64, y : flt64 -> flt64)
;;
type fltdesc(@f, @u, @i) = struct
explode : (f : @f -> (bool, @i, @u))
assem : (n : bool, e : @i, s : @u -> @f)
tobits : (f : @f -> @u)
frombits : (u : @u -> @f)
nan : @u
inf : @u
expmask : @u
precision : @u
emax : @i
emin : @i
sgnmask : @u
sig8mask : @u
sig8last : @u
split_prec_mask : @u
split_prec_mask2 : @u
C : (@u, @u)[:]
eps_inf_border : @u
eps_zero_border : @u
exp_inf_border : @u
exp_zero_border : @u
exp_subnormal_border : @u
itercount : @u
;;
const desc32 : fltdesc(flt32, uint32, int32) = [
.explode = std.flt32explode,
.assem = std.flt32assem,
.tobits = std.flt32bits,
.frombits = std.flt32frombits,
.nan = 0x7fc00000,
.inf = 0x7f800000,
.expmask = 0x7f800000, /* mask to detect inf or NaN (inf, repeated for clarity) */
.precision = 24,
.emax = 127,
.emin = -126,
.sgnmask = 1 << 31,
.sig8mask = 0xffff0000, /* Mask to get 8 significant bits */
.sig8last = 16, /* Last bit kept when masking */
.split_prec_mask = 0xffff0000, /* 16 trailing zeros */
.split_prec_mask2 = 0xfffff000, /* 12 trailing zeros */
.C = accurate_logs32[0:130], /* See log-impl.myr */
.eps_inf_border = 0x4eb00f34, /* maximal y st. (1.00..1)^y < oo */
.eps_zero_border = 0x4ecff1b4, /* minimal y st. (0.99..9)^y > 0 */
.exp_inf_border = 0x42b17218, /* maximal y such that e^y < oo */
.exp_zero_border = 0xc2cff1b4, /* minimal y such that e^y > 0 */
.exp_subnormal_border = 0xc2aeac50, /* minimal y such that e^y is normal */
.itercount = 4, /* How many iterations of Taylor series for (1 + f)^y' */
]
const desc64 : fltdesc(flt64, uint64, int64) = [
.explode = std.flt64explode,
.assem = std.flt64assem,
.tobits = std.flt64bits,
.frombits = std.flt64frombits,
.nan = 0x7ff8000000000000,
.inf = 0x7ff0000000000000,
.expmask = 0x7ff0000000000000,
.precision = 53,
.emax = 1023,
.emin = -1022,
.sgnmask = 1 << 63,
.sig8mask = 0xffffe00000000000, /* Mask to get 8 significant bits */
.sig8last = 45, /* Last bit kept when masking */
.split_prec_mask = 0xffffff0000000000, /* 40 trailing zeroes */
.split_prec_mask2 = 0xfffffffffffc0000, /* 18 trailing zeroes */
.C = accurate_logs64[0:130], /* See log-impl.myr */
.eps_inf_border = 0x43d628b76e3a7b61, /* maximal y st. (1.00..1)^y < oo */
.eps_zero_border = 0x43d74e9c65eceee0, /* minimal y st. (0.99..9)^y > 0 */
.exp_inf_border = 0x40862e42fefa39ef, /* maximal y such that e^y < oo */
.exp_zero_border = 0xc0874910d52d3052, /* minimal y such that e^y > 0 */
.exp_subnormal_border = 0xc086232bdd7abcd2, /* minimal y such that e^y is normal */
.itercount = 8,
]
const powr32 = {x : flt32, y : flt32
-> powrgen(x, y, desc32)
}
const powr64 = {x : flt64, y : flt64
-> powrgen(x, y, desc64)
}
generic powrgen = {x : @f, y : @f, d : fltdesc(@f, @u, @i) :: numeric,floating,std.equatable @f, numeric,integral @u, numeric,integral @i
var xb, yb
xb = d.tobits(x)
yb = d.tobits(y)
var xn : bool, xe : @i, xs : @u
var yn : bool, ye : @i, ys : @u
(xn, xe, xs) = d.explode(x)
(yn, ye, ys) = d.explode(y)
/*
Special cases. Note we do not follow IEEE exceptions.
*/
if std.isnan(x) || std.isnan(y)
/* Propagate NaN */
-> d.frombits(d.nan)
elif (xb & ~d.sgnmask == 0)
if (yb & ~d.sgnmask == 0)
/* 0^0 is undefined. */
-> d.frombits(d.nan)
elif yn
/* 0^(< 0) is infinity */
-> d.frombits(d.inf)
else
/* otherwise, 0^y = 0. */
-> (0.0 : @f)
;;
elif xn
/*
(< 0)^(anything) is undefined. This comes from
thinking of floating-point numbers as representing
small ranges of real numbers. If you really want
to compute (-1.23)^5, use pown.
*/
-> d.frombits(d.nan)
elif (xb & ~d.sgnmask == d.inf)
if (yb & ~d.sgnmask == 0)
/* oo^0 is undefined */
-> d.frombits(d.nan)
elif yn
/* +/-oo^(< 0) is +/-0 */
-> d.assem(xn, 0, 0)
elif xn
/* (-oo)^(anything) is undefined */
-> d.frombits(d.nan)
else
/* oo^(> 0) is oo */
-> d.frombits(d.inf)
;;
elif std.eq(y, (1.0 : @f))
/* x^1 = x */
-> x
elif yb & ~d.sgnmask == 0
/* (finite, positive)^0 = 1 */
-> (1.0 : @f)
elif std.eq(x, (1.0 : @f))
if yb & ~d.sgnmask == d.inf
/* 1^oo is undefined */
-> d.frombits(d.nan)
else
/* 1^(finite, positive) = 1 */
-> (1.0 : @f)
;;
elif yb & ~d.sgnmask == d.inf
if xe < 0
/* (0 < x < 1)^oo = 0 */
-> (0.0 : @f)
else
/* (x > 1)^oo = oo */
-> d.frombits(d.inf)
;;
;;
/* Normalize x and y */
if xe < d.emin
var first_1 = find_first1_64((xs : uint64), (d.precision : int64))
var offset = (d.precision : @u) - 1 - (first_1 : @u)
xs = xs << offset
xe = d.emin - offset
;;
if ye < d.emin
var first_1 = find_first1_64((ys : uint64), (d.precision : int64))
var offset = (d.precision : @u) - 1 - (first_1 : @u)
ys = ys << offset
ye = d.emin - offset
;;
/*
Split x into 2^N * F * (1 + f), with F = 1 + j/128 (some
j) and f tiny. Compute F naively by truncation. Compute
f via f = (x' - 1 - F)/(1 + F), where 1/(1 + F) is
precomputed and x' is x/2^N. 128 is chosen so that we
can borrow some constants from log-impl.myr.
[Tan90] hints at a method of computing x^y which may be
comparable to this approach, but which is unfortunately
has not been elaborated on (as far as I can discover).
*/
var N = xe
var j, F, Fn, Fe, Fs
var xprime = d.assem(false, 0, xs)
if need_round_away(0, (xs : uint64), (d.sig8last : int64))
F = d.frombits((d.tobits(xprime) & d.sig8mask) + (1 << d.sig8last))
else
F = d.frombits(d.tobits(xprime) & d.sig8mask)
;;
(Fn, Fe, Fs) = d.explode(F)
if Fe != 0
j = 128
else
j = 0x7f & ((d.sig8mask & Fs) >> d.sig8last)
;;
var f = (xprime - F)/F
/*
y could actually be above integer infinity, in which
case x^y is most certainly infinity of 0. More importantly,
we can't safely compute M (below).
*/
if x > (1.0 : @f)
if y > d.frombits(d.eps_inf_border)
-> d.frombits(d.inf)
elif -y > d.frombits(d.eps_inf_border)
-> (0.0 : @f)
;;
elif x < (1.0 : @f)
if y > d.frombits(d.eps_zero_border) && x < (1.0 : @f)
-> (0.0 : @f)
elif -y > d.frombits(d.eps_zero_border) && x < (1.0 : @f)
-> d.frombits(d.inf)
;;
;;
/* Split y into M + y', with |y'| <= 0.5 and M an integer */
var M = floor(y)
var yprime = y - M
if yprime > (0.5 : @f)
M += (1.0 : @f)
yprime = y - M
elif yprime < (-0.5 : @f)
M -= (1.0: @f)
yprime = y - M
;;
/*
We'll multiply y' by log(2) and try to keep extra
precision, so we need to split y'. Since the high word
of C has 24 - 10 = 14 significant bits (53 - 16 = 37 in
flt64 case), we ensure 15 (39) trailing zeroes in
yprime_hi. (We also need this for y'*N, M, &c).
*/
var yprime_hi = d.frombits(d.tobits(yprime) & d.split_prec_mask)
var yprime_lo = yprime - yprime_hi
var yprimeN_hi = d.frombits(d.tobits((N : @f) * yprime) & d.split_prec_mask)
var yprimeN_lo = fma((N : @f), yprime, -yprimeN_hi)
var M_hi = d.frombits(d.tobits(M) & d.split_prec_mask)
var M_lo = M - M_hi
/*
At this point, we've built out
x^y = [ 2^N * F * (1 + f) ]^(M + y')
where N, M are integers, F is well-known, and f, y' are
tiny. So we can get to computing
/-1-\ /-------------------2--------------------------\ /-3--\
2^(N*M) * exp(log(F)*y' + log2*N*y' + log(F)*M + M*log(1+f)) * (1+f)^y'
where 1 can be handled by scale2, 2 we can mostly fake
by sticking high-precision values for log(F) and log(2)
through exp(), and 3 is composed of small numbers,
therefore can be reasonably approximated by a Taylor
expansion.
*/
/* t2 */
var log2_lo, log2_hi, Cu_hi, Cu_lo
(log2_hi, log2_lo) = d.C[128]
(Cu_hi, Cu_lo) = d.C[j]
var es : @f[20]
std.slfill(es[:], (0.0 : @f))
/* log(F) * y' */
es[0] = d.frombits(Cu_hi) * yprime_hi
es[1] = d.frombits(Cu_lo) * yprime_hi
es[2] = d.frombits(Cu_hi) * yprime_lo
es[3] = d.frombits(Cu_lo) * yprime_lo
/* log(2) * N * y' */
es[4] = d.frombits(log2_hi) * yprimeN_hi
es[5] = d.frombits(log2_lo) * yprimeN_hi
es[6] = d.frombits(log2_hi) * yprimeN_lo
es[7] = d.frombits(log2_lo) * yprimeN_lo
/* log(F) * M */
es[8] = d.frombits(Cu_hi) * M_hi
es[9] = d.frombits(Cu_lo) * M_hi
es[10] = d.frombits(Cu_hi) * M_lo
es[11] = d.frombits(Cu_lo) * M_lo
/* log(1 + f) * M */
var lf = log1p(f)
var lf_hi = d.frombits(d.tobits(lf) & d.split_prec_mask)
var lf_lo = lf - lf_hi
es[12] = lf_hi * M_hi
es[13] = lf_lo * M_hi
es[14] = lf_hi * M_lo
es[15] = lf_lo * M_lo
/*
The correct way to handle this would be to compare
magnitudes of eis and parenthesize the additions correctly.
We take the cheap way out.
*/
var exp_hi = priest_sum(es[0:16])
/*
We would like to just compute exp(exp_hi) * exp(exp_lo).
However, if that takes us into subnormal territory, yet
N * M is large, that will throw away a few bits of
information. We can correct for this by adding in a few
copies of P*log(2), then subtract off P when we compute
scale2() at the end.
We also have to be careful that P doesn't have too many
significant bits, otherwise we throw away some information
of log2_hi.
*/
var P = -rn(exp_hi / d.frombits(log2_hi))
var P_f = (P : @f)
P_f = d.frombits(d.tobits(P_f) & d.split_prec_mask2)
P = rn(P_f)
es[16] = P_f * d.frombits(log2_hi)
es[17] = P_f * d.frombits(log2_lo)
exp_hi = priest_sum(es[0:18])
es[18] = -exp_hi
var exp_lo = priest_sum(es[0:19])
var t2 = exp(exp_hi) * exp(exp_lo)
/*
t3: Abbreviated Taylor expansion for (1 + f)^y' - 1.
Since f is on the order of 2^-7 (and y' is on the order
of 2^-1), we need to go up to f^3 for single-precision,
and f^7 for double. We can then compute (1 + t3) * t2
The expansion is \Sum_{k=1}^{\infty} {y' \choose k} x^k
*/
var terms : @f[10] = [
(0.0 : @f), (0.0 : @f), (0.0 : @f), (0.0 : @f), (0.0 : @f),
(0.0 : @f), (0.0 : @f), (0.0 : @f), (0.0 : @f), (0.0 : @f),
]
var current = (1.0 : @f)
for var j = 0; j <= d.itercount; ++j
current = current * f * (yprime - (j : @f)) / ((j : @f) + (1.0 : @f))
terms[j] = current
;;
var t3 = priest_sum(terms[0:d.itercount + 1])
var total_exp_f = (N : @f) * M - (P : @f)
if total_exp_f > ((d.emax - d.emin + d.precision + 1) : @f)
-> d.frombits(d.inf)
elif total_exp_f < -((d.emax - d.emin + d.precision + 1) : @f)
-> (0.0 : @f)
;;
/*
Pull t2's exponent out so that we don't hit subnormal
calculation with the t3 multiplication
*/
var t2n, t2e, t2s
(t2n, t2e, t2s) = d.explode(t2)
if t2e < d.emin
var t2_first_1 = find_first1_64((t2s : uint64), (d.precision : int64))
var t2_offset = (d.precision : @u) - 1 - (t2_first_1 : @u)
t2s = t2s << t2_offset
t2e = d.emin - (t2_offset : @i)
;;
t2 = d.assem(t2n, 0, t2s)
P -= t2e
var base = fma(t2, t3, t2)
-> scale2(base, N * rn(M) - P)
}