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May 16, 2012, 11:11 AM

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cpython: Changes in mpd qexp():

http://hg.python.org/cpython/rev/db27a563fa9b
changeset: 77004:db27a563fa9b
parent: 77002:0f6a6f59b002
user: Stefan Krah <skrah [at] bytereef>
date: Wed May 16 20:10:21 2012 +0200
summary:
Changes in _mpd_qexp():
-----------------------

1) Reduce the number of iterations in the Horner scheme for operands with
a negative adjusted exponent. Previously the number was overestimated
quite generously.

2) The function _mpd_get_exp_iterations() now has an ACL2 proof and
is rewritten accordingly.

3) The proof relies on abs(op) > 9 * 10**(-prec-1), so operands without
that property are now handled by the new function _mpd_qexp_check_one().

4) The error analysis for the evaluation of the truncated Taylor series
in Hull&Abrham's paper relies on the fact that the reduced operand
'r' has fewer than context.prec digits.

Since the operands may have more than context.prec digits, a new ACL2
proof covers the case that r.digits > context.prec. To facilitate the
proof, the Horner step now uses fma instead of rounding twice in
multiply/add.


Changes in mpd_qexp():
----------------------

1) Fix a bound in the correct rounding loop that was too optimistic. In
practice results were always correctly rounded, because it is unlikely
that the error in _mpd_qexp() ever reaches the theoretical maximum.

files:
Modules/_decimal/libmpdec/mpdecimal.c | 173 +++++++++----
1 files changed, 122 insertions(+), 51 deletions(-)


diff --git a/Modules/_decimal/libmpdec/mpdecimal.c b/Modules/_decimal/libmpdec/mpdecimal.c
--- a/Modules/_decimal/libmpdec/mpdecimal.c
+++ b/Modules/_decimal/libmpdec/mpdecimal.c
@@ -3878,53 +3878,97 @@
}
#endif

-#if defined(_MSC_VER)
- /* conversion from 'double' to 'mpd_ssize_t', possible loss of data */
- #pragma warning(disable:4244)
-#endif
+/* Pad the result with trailing zeros if it has fewer digits than prec. */
+static void
+_mpd_zeropad(mpd_t *result, const mpd_context_t *ctx, uint32_t *status)
+{
+ if (!mpd_isspecial(result) && !mpd_iszero(result) &&
+ result->digits < ctx->prec) {
+ mpd_ssize_t shift = ctx->prec - result->digits;
+ mpd_qshiftl(result, result, shift, status);
+ result->exp -= shift;
+ }
+}
+
+/* Check if the result is guaranteed to be one. */
+static int
+_mpd_qexp_check_one(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
+ uint32_t *status)
+{
+ MPD_NEW_CONST(lim,0,-(ctx->prec+1),1,1,1,9);
+ MPD_NEW_SHARED(aa, a);
+
+ mpd_set_positive(&aa);
+
+ /* abs(a) <= 9 * 10**(-prec-1) */
+ if (_mpd_cmp(&aa, &lim) <= 0) {
+ _settriple(result, 0, 1, 0);
+ _mpd_zeropad(result, ctx, status);
+ *status = MPD_Rounded|MPD_Inexact;
+ return 1;
+ }
+
+ return 0;
+}
+
/*
* Get the number of iterations for the Horner scheme in _mpd_qexp().
*/
static inline mpd_ssize_t
-_mpd_get_exp_iterations(const mpd_t *a, mpd_ssize_t prec)
-{
- mpd_uint_t dummy;
- mpd_uint_t msdigits;
- double f;
-
- /* 9 is MPD_RDIGITS for 32 bit platforms */
- _mpd_get_msdigits(&dummy, &msdigits, a, 9);
- f = ((double)msdigits + 1) / mpd_pow10[mpd_word_digits(msdigits)];
+_mpd_get_exp_iterations(const mpd_t *r, mpd_ssize_t p)
+{
+ mpd_ssize_t log10pbyr; /* lower bound for log10(p / abs(r)) */
+ mpd_ssize_t n;
+
+ assert(p >= 10);
+ assert(!mpd_iszero(r));
+ assert(-p < mpd_adjexp(r) && mpd_adjexp(r) <= -1);

#ifdef CONFIG_64
- #ifdef USE_80BIT_LONG_DOUBLE
- return ceill((1.435*(long double)prec - 1.182)
- / log10l((long double)prec/f));
- #else
- /* prec > floor((1ULL<<53) / 1.435) */
- if (prec > 6276793905742851LL) {
+ if (p > (mpd_ssize_t)(1ULL<<52)) {
return MPD_SSIZE_MAX;
}
- return ceil((1.435*(double)prec - 1.182) / log10((double)prec/f));
- #endif
-#else /* CONFIG_32 */
- return ceil((1.435*(double)prec - 1.182) / log10((double)prec/f));
- #if defined(_MSC_VER)
- #pragma warning(default:4244)
- #endif
#endif
+
+ /*
+ * Lower bound for log10(p / abs(r)): adjexp(p) - (adjexp(r) + 1)
+ * At this point (for CONFIG_64, CONFIG_32 is not problematic):
+ * 1) 10 <= p <= 2**52
+ * 2) -p < adjexp(r) <= -1
+ * 3) 1 <= log10pbyr <= 2**52 + 14
+ */
+ log10pbyr = (mpd_word_digits(p)-1) - (mpd_adjexp(r)+1);
+
+ /*
+ * The numerator in the paper is 1.435 * p - 1.182, calculated
+ * exactly. We compensate for rounding errors by using 1.43503.
+ * ACL2 proofs:
+ * 1) exp-iter-approx-lower-bound: The term below evaluated
+ * in 53-bit floating point arithmetic is greater than or
+ * equal to the exact term used in the paper.
+ * 2) exp-iter-approx-upper-bound: The term below is less than
+ * or equal to 3/2 * p <= 3/2 * 2**52.
+ */
+ n = (mpd_ssize_t)ceil((1.43503*(double)p - 1.182) / (double)log10pbyr);
+ return n >= 3 ? n : 3;
}

/*
- * Internal function, specials have been dealt with.
+ * Internal function, specials have been dealt with. The result has a
+ * relative error of less than 0.5 * 10**(-ctx->prec).
*
* The algorithm is from Hull&Abrham, Variable Precision Exponential Function,
* ACM Transactions on Mathematical Software, Vol. 12, No. 2, June 1986.
*
* Main differences:
*
- * - The number of iterations for the Horner scheme is calculated using the
- * C log10() function.
+ * - The number of iterations for the Horner scheme is calculated using
+ * 53-bit floating point arithmetic.
+ *
+ * - In the error analysis for ER (relative error accumulated in the
+ * evaluation of the truncated series) the reduced operand r may
+ * have any number of digits.
+ * ACL2 proof: exponent-relative-error
*
* - The analysis for early abortion has been adapted for the mpd_t
* ranges.
@@ -3941,18 +3985,23 @@

assert(!mpd_isspecial(a));

+ if (mpd_iszerocoeff(a)) {
+ _settriple(result, MPD_POS, 1, 0);
+ return;
+ }
+
/*
- * We are calculating e^x = e^(r*10^t) = (e^r)^(10^t), where r < 1 and t >= 0.
+ * We are calculating e^x = e^(r*10^t) = (e^r)^(10^t), where abs(r) < 1 and t >= 0.
*
* If t > 0, we have:
*
- * (1) 0.1 <= r < 1, so e^r >= e^0.1. Overflow in the final power operation
- * will occur when (e^0.1)^(10^t) > 10^(emax+1). If we consider MAX_EMAX,
- * this will happen for t > 10 (32 bit) or (t > 19) (64 bit).
+ * (1) 0.1 <= r < 1, so e^0.1 <= e^r. If t > MAX_T, overflow occurs:
*
- * (2) -1 < r <= -0.1, so e^r > e^-1. Underflow in the final power operation
- * will occur when (e^-1)^(10^t) < 10^(etiny-1). If we consider MIN_ETINY,
- * this will also happen for t > 10 (32 bit) or (t > 19) (64 bit).
+ * MAX-EMAX+1 < log10(e^(0.1*10*t)) <= log10(e^(r*10^t)) < adjexp(e^(r*10^t))+1
+ *
+ * (2) -1 < r <= -0.1, so e^r <= e^-0.1. It t > MAX_T, underflow occurs:
+ *
+ * adjexp(e^(r*10^t)) <= log10(e^(r*10^t)) <= log10(e^(-0.1*10^t) < MIN-ETINY
*/
#if defined(CONFIG_64)
#define MPD_EXP_MAX_T 19
@@ -3974,20 +4023,33 @@
return;
}

+ /* abs(a) <= 9 * 10**(-prec-1) */
+ if (_mpd_qexp_check_one(result, a, ctx, status)) {
+ return;
+ }
+
mpd_maxcontext(&workctx);
workctx.prec = ctx->prec + t + 2;
- workctx.prec = (workctx.prec < 9) ? 9 : workctx.prec;
+ workctx.prec = (workctx.prec < 10) ? 10 : workctx.prec;
workctx.round = MPD_ROUND_HALF_EVEN;

- if ((n = _mpd_get_exp_iterations(a, workctx.prec)) == MPD_SSIZE_MAX) {
+ if (!mpd_qcopy(result, a, status)) {
+ return;
+ }
+ result->exp -= t;
+
+ /*
+ * At this point:
+ * 1) 9 * 10**(-prec-1) < abs(a)
+ * 2) 9 * 10**(-prec-t-1) < abs(r)
+ * 3) log10(9) - prec - t - 1 < log10(abs(r)) < adjexp(abs(r)) + 1
+ * 4) - prec - t - 2 < adjexp(abs(r)) <= -1
+ */
+ n = _mpd_get_exp_iterations(result, workctx.prec);
+ if (n == MPD_SSIZE_MAX) {
mpd_seterror(result, MPD_Invalid_operation, status); /* GCOV_UNLIKELY */
- goto finish; /* GCOV_UNLIKELY */
- }
-
- if (!mpd_qcopy(result, a, status)) {
- goto finish;
- }
- result->exp -= t;
+ return; /* GCOV_UNLIKELY */
+ }

_settriple(&sum, MPD_POS, 1, 0);

@@ -3995,8 +4057,7 @@
word.data[0] = j;
mpd_setdigits(&word);
mpd_qdiv(&tmp, result, &word, &workctx, &workctx.status);
- mpd_qmul(&sum, &sum, &tmp, &workctx, &workctx.status);
- mpd_qadd(&sum, &sum, &one, &workctx, &workctx.status);
+ mpd_qfma(&sum, &sum, &tmp, &one, &workctx, &workctx.status);
}

#ifdef CONFIG_64
@@ -4013,8 +4074,8 @@
}
#endif

-
-finish:
+ _mpd_zeropad(result, ctx, status);
+
mpd_del(&tmp);
mpd_del(&sum);
*status |= (workctx.status&MPD_Errors);
@@ -4069,8 +4130,18 @@
workctx.prec = prec;
_mpd_qexp(result, a, &workctx, status);
_ssettriple(&ulp, MPD_POS, 1,
- result->exp + result->digits-workctx.prec-1);
-
+ result->exp + result->digits-workctx.prec);
+
+ /*
+ * At this point:
+ * 1) abs(result - e**x) < 0.5 * 10**(-prec) * e**x
+ * 2) result - ulp < e**x < result + ulp
+ * 3) result - ulp < result < result + ulp
+ *
+ * If round(result-ulp)==round(result+ulp), then
+ * round(result)==round(e**x). Therefore the result
+ * is correctly rounded.
+ */
workctx.prec = ctx->prec;
mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status);
mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status);

--
Repository URL: http://hg.python.org/cpython

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