ref: 2002d2fa999d7efeb71aa64a32a1a87c2f4b8399
parent: 64c5e8f1227c1a98ab996e95425ab43791c0ee2e
author: S. Gilles <sgilles@math.umd.edu>
date: Sat Jun 8 11:30:55 EDT 2019
Apply changes of pown to rootn. Faster, better edge handling.
--- a/lib/math/ancillary/ulp.gp
+++ b/lib/math/ancillary/ulp.gp
@@ -31,7 +31,7 @@
e = bitand(a, 0x7f800000) >> 23;
s = bitand(a, 0x007fffff);
- if(e != 0, s = bitor(s, 0x00800000),);
+ if(e != 0, s = bitor(s, 0x00800000), s = 2.0 * s);
s = s * 2.0^(-23);
e = e - 127;
return((-1)^n * s * 2^(e));
@@ -42,7 +42,7 @@
e = bitand(a, 0x7ff0000000000000) >> 52;
s = bitand(a, 0x000fffffffffffff);
- if(e != 0, s = bitor(s, 0x0010000000000000),);
+ if(e != 0, s = bitor(s, 0x0010000000000000), s = 2.0 * s);
s = s * 2.0^(-52);
e = e - 1023;
return((-1)^n * 2^(e) * s);
--- a/lib/math/log-overkill.myr
+++ b/lib/math/log-overkill.myr
@@ -40,8 +40,6 @@
pkglocal const logoverkill64 : (x : flt64 -> (flt64, flt64))
;;
-
-
/*
Ci is a table such that, for Ci[j] = (L1, L2, I1, I2),
L1, L2 are log(1 + j·2^-(5i))
--- a/lib/math/pown-impl.myr
+++ b/lib/math/pown-impl.myr
@@ -38,6 +38,8 @@
floor : (x : @f -> @f)
log_overkill : (x : @f -> (@f, @f))
precision : @i
+ nosgn_mask : @u
+ implicit_bit : @u
emin : @i
emax : @i
imax : @i
@@ -62,6 +64,8 @@
.floor = floor32,
.log_overkill = logoverkill32,
.precision = 24,
+ .nosgn_mask = 0x7fffffff,
+ .implicit_bit = 23,
.emin = -126,
.emax = 127,
.imax = 2147483647, /* For detecting overflow in final exponent */
@@ -86,6 +90,8 @@
.floor = floor64,
.log_overkill = logoverkill64,
.precision = 53,
+ .nosgn_mask = 0x7fffffffffffffff,
+ .implicit_bit = 52,
.emin = -1022,
.emax = 1023,
.imax = 9223372036854775807,
@@ -181,6 +187,9 @@
But first: do some rough calculations: if we can show n*log(xs) has the
same sign as n*e, and n*e would cause overflow, then we might as well
return right now.
+
+ This also takes care of subnormals very nicely, so we don't have to do
+ any special handling to reconstitute xs "right", as we do in rootn.
*/
var exp_rough_estimate = n * xe
if n > 0 && (exp_rough_estimate > d.emax + 1 || (exp_rough_estimate / n != xe))
@@ -286,6 +295,19 @@
elif q == 1
/* Anything^1/1 is itself */
-> x
+ elif xe < d.emin
+ /*
+ Subnormals are actually a problem. If we naively reconstitute xs, it
+ will be wildly wrong and won't match up with the exponent. So let's
+ pretend we have unbounded exponent range. We know the loop terminates
+ because we covered the +/-0.0 case above.
+ */
+ xe++
+ var check = 1 << d.implicit_bit
+ while xs & check == 0
+ xs <<= 1
+ xe--
+ ;;
;;
/* As in pown */
@@ -294,6 +316,33 @@
ult_sgn = -1.0
;;
+ /*
+ If we're looking at (1 + 2^-h)^1/q, and the answer will be 1 + e, with
+ (1 + e)^q = 1 + 2^-h, then for q and h large enough, e might be below
+ the representable range. Specifically,
+
+ (1 + e)^q ≅ 1 + qe + (q choose 2)e^2 + ...
+
+ So (using single-precision as the example)
+
+ (1 + 2^-23)^q ≅ 1 + q 2^-23 + (absolutely tiny terms)
+
+ And anything in [1, 1 + q 2^-24) will just truncate to 1.0 when
+ calculated.
+ */
+ if xe == 0
+ var cutoff = scale2(qf, -1 * d.precision - 1) + 1.0
+ if (xb & d.nosgn_mask) < d.tobits(cutoff)
+ -> 1.0
+ ;;
+ elif xe == -1
+ /* Something similar for (1 - e)^q */
+ var cutoff = 1.0 - scale2(qf, -1 * d.precision - 1)
+ if (xb & d.nosgn_mask) > d.tobits(cutoff)
+ -> 1.0
+ ;;
+ ;;
+
/* Similar to pown. Let e/q = E + psi, with E an integer.
x^(1/q) = e^(log(xs)/q) * 2^(e/q)
@@ -307,15 +356,14 @@
*/
/* Calculate 1/q in very high precision */
- var r1 = 1.0 / qf
- var r2 = -math.fma(r1, qf, -1.0) / qf
+ var qinv_hi = 1.0 / qf
+ var qinv_lo = -math.fma(qinv_hi, qf, -1.0) / qf
var ln_xs_hi, ln_xs_lo
(ln_xs_hi, ln_xs_lo) = d.log_overkill(d.assem(false, 0, xs))
- var ls1 : @f[12]
- (ls1[0], ls1[1]) = d.two_by_two(ln_xs_hi, r1)
- (ls1[2], ls1[3]) = d.two_by_two(ln_xs_hi, r2)
- (ls1[4], ls1[5]) = d.two_by_two(ln_xs_lo, r1)
+ var G1, G2
+ (G1, G2) = d.split_mul(ln_xs_hi, ln_xs_lo, qinv_hi, qinv_lo)
+
var E : @i
if q > std.abs(xe)
/* Don't cast q to @i unless we're sure it's in small range */
@@ -328,15 +376,20 @@
var psi_lo = -math.fma(psi_hi, qf, -(qpsi : @f)) / qf
var log2_hi, log2_lo
(log2_hi, log2_lo) = d.C[128]
- (ls1[ 6], ls1[ 7]) = d.two_by_two(psi_hi, d.frombits(log2_hi))
- (ls1[ 8], ls1[ 9]) = d.two_by_two(psi_hi, d.frombits(log2_lo))
- (ls1[10], ls1[11]) = d.two_by_two(psi_lo, d.frombits(log2_hi))
+ var H1, H2
+ (H1, H2) = d.split_mul(psi_hi, psi_lo, d.frombits(log2_hi), d.frombits(log2_lo))
- var G1, G2
- (G1, G2) = double_compensated_sum(ls1[0:12])
- /* G1 + G2 approximates log(xs)/q + log(2)*psi */
+ var J1, J2, t1, t2
+ /*
+ We can't use split_add; we don't kow the relative magitudes of G and H
+ */
+ (t1, t2) = slow2sum(G2, H2)
+ (J2, t1) = slow2sum(H1, t1)
+ (J1, J2) = slow2sum(G1, J2)
+ J2 = J2 + (t1 + t2)
- var base = exp(G1) + G2
+ /* J1 + J2 approximates log(xs)/q + log(2)*psi */
+ var base = exp(J1) + J2
-> ult_sgn * scale2(base, E)
}
--- a/lib/math/test/pown-impl.myr
+++ b/lib/math/test/pown-impl.myr
@@ -457,6 +457,12 @@
(0xe0bc4cbf6bd74d8f, 0x000000000000bd8b, 0xbff01ed2c4e821fc),
(0x31d4f2baa91a9e8e, 0x000000000000244d, 0x3fef774c954b40bf),
(0x01283d1c679f5652, 0x0000000000008647, 0x3fef5bc18f5e292f),
+ (0x80003d8a341ee060, 0x0000000000009c71, 0xbfef6f873f76b7cd),
+ (0xbfecf0fc4dc97b93, 0x0000000000005f4b, 0xbfeffff75d0a25fe),
+ (0xbfe2fb84944a35ee, 0x000000000000e947, 0xbfefffeda94c6d07),
+ (0xbfe0ef0c05bd84ab, 0x0000000000007165, 0xbfefffd205e2a1c8),
+ (0xbfe7354e962bdcb3, 0x000000000000076b, 0xbfeffe9d4844aad6),
+ (0xbfea556dd1eb1e58, 0x00000000000095cb, 0xbfeffff557890356),
][:]
for (x, y, z) : inputs