* Copyright (c) 1998-2008, PostgreSQL Global Development Group
*
* IDENTIFICATION
- * $PostgreSQL: pgsql/src/backend/utils/adt/numeric.c,v 1.108 2008/01/01 19:45:52 momjian Exp $
+ * $PostgreSQL: pgsql/src/backend/utils/adt/numeric.c,v 1.109 2008/04/04 18:45:36 tgl Exp $
*
*-------------------------------------------------------------------------
*/
* NBASE that's less than sqrt(INT_MAX), in practice we are only interested
* in NBASE a power of ten, so that I/O conversions and decimal rounding
* are easy. Also, it's actually more efficient if NBASE is rather less than
- * sqrt(INT_MAX), so that there is "headroom" for mul_var and div_var to
+ * sqrt(INT_MAX), so that there is "headroom" for mul_var and div_var_fast to
* postpone processing carries.
* ----------
*/
/* ----------
+ * NumericVar is the format we use for arithmetic. The digit-array part
+ * is the same as the NumericData storage format, but the header is more
+ * complex.
+ *
* The value represented by a NumericVar is determined by the sign, weight,
* ndigits, and digits[] array.
* Note: the first digit of a NumericVar's value is assumed to be multiplied
* NumericVar. digits points at the first digit in actual use (the one
* with the specified weight). We normally leave an unused digit or two
* (preset to zeroes) between buf and digits, so that there is room to store
- * a carry out of the top digit without special pushups. We just need to
+ * a carry out of the top digit without reallocating space. We just need to
* decrement digits (and increment weight) to make room for the carry digit.
* (There is no such extra space in a numeric value stored in the database,
* only in a NumericVar in memory.)
int rscale);
static void div_var(NumericVar *var1, NumericVar *var2, NumericVar *result,
int rscale, bool round);
+static void div_var_fast(NumericVar *var1, NumericVar *var2, NumericVar *result,
+ int rscale, bool round);
static int select_div_scale(NumericVar *var1, NumericVar *var2);
static void mod_var(NumericVar *var1, NumericVar *var2, NumericVar *result);
static void ceil_var(NumericVar *var, NumericVar *result);
}
+/*
+ * numeric_div_trunc() -
+ *
+ * Divide one numeric into another, truncating the result to an integer
+ */
+Datum
+numeric_div_trunc(PG_FUNCTION_ARGS)
+{
+ Numeric num1 = PG_GETARG_NUMERIC(0);
+ Numeric num2 = PG_GETARG_NUMERIC(1);
+ NumericVar arg1;
+ NumericVar arg2;
+ NumericVar result;
+ Numeric res;
+
+ /*
+ * Handle NaN
+ */
+ if (NUMERIC_IS_NAN(num1) || NUMERIC_IS_NAN(num2))
+ PG_RETURN_NUMERIC(make_result(&const_nan));
+
+ /*
+ * Unpack the arguments
+ */
+ init_var(&arg1);
+ init_var(&arg2);
+ init_var(&result);
+
+ set_var_from_num(num1, &arg1);
+ set_var_from_num(num2, &arg2);
+
+ /*
+ * Do the divide and return the result
+ */
+ div_var(&arg1, &arg2, &result, 0, false);
+
+ res = make_result(&result);
+
+ free_var(&arg1);
+ free_var(&arg2);
+ free_var(&result);
+
+ PG_RETURN_NUMERIC(res);
+}
+
+
/*
* numeric_mod() -
*
/*
* div_var() -
*
- * Division on variable level. Quotient of var1 / var2 is stored
- * in result. Result is rounded to no more than rscale fractional digits.
+ * Division on variable level. Quotient of var1 / var2 is stored in result.
+ * The quotient is figured to exactly rscale fractional digits.
+ * If round is true, it is rounded at the rscale'th digit; if false, it
+ * is truncated (towards zero) at that digit.
*/
static void
div_var(NumericVar *var1, NumericVar *var2, NumericVar *result,
int rscale, bool round)
+{
+ int div_ndigits;
+ int res_ndigits;
+ int res_sign;
+ int res_weight;
+ int carry;
+ int borrow;
+ int divisor1;
+ int divisor2;
+ NumericDigit *dividend;
+ NumericDigit *divisor;
+ NumericDigit *res_digits;
+ int i;
+ int j;
+
+ /* copy these values into local vars for speed in inner loop */
+ int var1ndigits = var1->ndigits;
+ int var2ndigits = var2->ndigits;
+
+ /*
+ * First of all division by zero check; we must not be handed an
+ * unnormalized divisor.
+ */
+ if (var2ndigits == 0 || var2->digits[0] == 0)
+ ereport(ERROR,
+ (errcode(ERRCODE_DIVISION_BY_ZERO),
+ errmsg("division by zero")));
+
+ /*
+ * Now result zero check
+ */
+ if (var1ndigits == 0)
+ {
+ zero_var(result);
+ result->dscale = rscale;
+ return;
+ }
+
+ /*
+ * Determine the result sign, weight and number of digits to calculate.
+ * The weight figured here is correct if the emitted quotient has no
+ * leading zero digits; otherwise strip_var() will fix things up.
+ */
+ if (var1->sign == var2->sign)
+ res_sign = NUMERIC_POS;
+ else
+ res_sign = NUMERIC_NEG;
+ res_weight = var1->weight - var2->weight;
+ /* The number of accurate result digits we need to produce: */
+ res_ndigits = res_weight + 1 + (rscale + DEC_DIGITS - 1) / DEC_DIGITS;
+ /* ... but always at least 1 */
+ res_ndigits = Max(res_ndigits, 1);
+ /* If rounding needed, figure one more digit to ensure correct result */
+ if (round)
+ res_ndigits++;
+ /*
+ * The working dividend normally requires res_ndigits + var2ndigits
+ * digits, but make it at least var1ndigits so we can load all of var1
+ * into it. (There will be an additional digit dividend[0] in the
+ * dividend space, but for consistency with Knuth's notation we don't
+ * count that in div_ndigits.)
+ */
+ div_ndigits = res_ndigits + var2ndigits;
+ div_ndigits = Max(div_ndigits, var1ndigits);
+
+ /*
+ * We need a workspace with room for the working dividend (div_ndigits+1
+ * digits) plus room for the possibly-normalized divisor (var2ndigits
+ * digits). It is convenient also to have a zero at divisor[0] with
+ * the actual divisor data in divisor[1 .. var2ndigits]. Transferring the
+ * digits into the workspace also allows us to realloc the result (which
+ * might be the same as either input var) before we begin the main loop.
+ * Note that we use palloc0 to ensure that divisor[0], dividend[0], and
+ * any additional dividend positions beyond var1ndigits, start out 0.
+ */
+ dividend = (NumericDigit *)
+ palloc0((div_ndigits + var2ndigits + 2) * sizeof(NumericDigit));
+ divisor = dividend + (div_ndigits + 1);
+ memcpy(dividend + 1, var1->digits, var1ndigits * sizeof(NumericDigit));
+ memcpy(divisor + 1, var2->digits, var2ndigits * sizeof(NumericDigit));
+
+ /*
+ * Now we can realloc the result to hold the generated quotient digits.
+ */
+ alloc_var(result, res_ndigits);
+ res_digits = result->digits;
+
+ if (var2ndigits == 1)
+ {
+ /*
+ * If there's only a single divisor digit, we can use a fast path
+ * (cf. Knuth section 4.3.1 exercise 16).
+ */
+ divisor1 = divisor[1];
+ carry = 0;
+ for (i = 0; i < res_ndigits; i++)
+ {
+ carry = carry * NBASE + dividend[i + 1];
+ res_digits[i] = carry / divisor1;
+ carry = carry % divisor1;
+ }
+ }
+ else
+ {
+ /*
+ * The full multiple-place algorithm is taken from Knuth volume 2,
+ * Algorithm 4.3.1D.
+ *
+ * We need the first divisor digit to be >= NBASE/2. If it isn't,
+ * make it so by scaling up both the divisor and dividend by the
+ * factor "d". (The reason for allocating dividend[0] above is to
+ * leave room for possible carry here.)
+ */
+ if (divisor[1] < HALF_NBASE)
+ {
+ int d = NBASE / (divisor[1] + 1);
+
+ carry = 0;
+ for (i = var2ndigits; i > 0; i--)
+ {
+ carry += divisor[i] * d;
+ divisor[i] = carry % NBASE;
+ carry = carry / NBASE;
+ }
+ Assert(carry == 0);
+ carry = 0;
+ /* at this point only var1ndigits of dividend can be nonzero */
+ for (i = var1ndigits; i >= 0; i--)
+ {
+ carry += dividend[i] * d;
+ dividend[i] = carry % NBASE;
+ carry = carry / NBASE;
+ }
+ Assert(carry == 0);
+ Assert(divisor[1] >= HALF_NBASE);
+ }
+ /* First 2 divisor digits are used repeatedly in main loop */
+ divisor1 = divisor[1];
+ divisor2 = divisor[2];
+
+ /*
+ * Begin the main loop. Each iteration of this loop produces the
+ * j'th quotient digit by dividing dividend[j .. j + var2ndigits]
+ * by the divisor; this is essentially the same as the common manual
+ * procedure for long division.
+ */
+ for (j = 0; j < res_ndigits; j++)
+ {
+ /* Estimate quotient digit from the first two dividend digits */
+ int next2digits = dividend[j] * NBASE + dividend[j+1];
+ int qhat;
+
+ /*
+ * If next2digits are 0, then quotient digit must be 0 and there's
+ * no need to adjust the working dividend. It's worth testing
+ * here to fall out ASAP when processing trailing zeroes in
+ * a dividend.
+ */
+ if (next2digits == 0)
+ {
+ res_digits[j] = 0;
+ continue;
+ }
+
+ if (dividend[j] == divisor1)
+ qhat = NBASE - 1;
+ else
+ qhat = next2digits / divisor1;
+ /*
+ * Adjust quotient digit if it's too large. Knuth proves that
+ * after this step, the quotient digit will be either correct
+ * or just one too large. (Note: it's OK to use dividend[j+2]
+ * here because we know the divisor length is at least 2.)
+ */
+ while (divisor2 * qhat >
+ (next2digits - qhat * divisor1) * NBASE + dividend[j+2])
+ qhat--;
+
+ /* As above, need do nothing more when quotient digit is 0 */
+ if (qhat > 0)
+ {
+ /*
+ * Multiply the divisor by qhat, and subtract that from the
+ * working dividend. "carry" tracks the multiplication,
+ * "borrow" the subtraction (could we fold these together?)
+ */
+ carry = 0;
+ borrow = 0;
+ for (i = var2ndigits; i >= 0; i--)
+ {
+ carry += divisor[i] * qhat;
+ borrow -= carry % NBASE;
+ carry = carry / NBASE;
+ borrow += dividend[j + i];
+ if (borrow < 0)
+ {
+ dividend[j + i] = borrow + NBASE;
+ borrow = -1;
+ }
+ else
+ {
+ dividend[j + i] = borrow;
+ borrow = 0;
+ }
+ }
+ Assert(carry == 0);
+
+ /*
+ * If we got a borrow out of the top dividend digit, then
+ * indeed qhat was one too large. Fix it, and add back the
+ * divisor to correct the working dividend. (Knuth proves
+ * that this will occur only about 3/NBASE of the time; hence,
+ * it's a good idea to test this code with small NBASE to be
+ * sure this section gets exercised.)
+ */
+ if (borrow)
+ {
+ qhat--;
+ carry = 0;
+ for (i = var2ndigits; i >= 0; i--)
+ {
+ carry += dividend[j + i] + divisor[i];
+ if (carry >= NBASE)
+ {
+ dividend[j + i] = carry - NBASE;
+ carry = 1;
+ }
+ else
+ {
+ dividend[j + i] = carry;
+ carry = 0;
+ }
+ }
+ /* A carry should occur here to cancel the borrow above */
+ Assert(carry == 1);
+ }
+ }
+
+ /* And we're done with this quotient digit */
+ res_digits[j] = qhat;
+ }
+ }
+
+ pfree(dividend);
+
+ /*
+ * Finally, round or truncate the result to the requested precision.
+ */
+ result->weight = res_weight;
+ result->sign = res_sign;
+
+ /* Round or truncate to target rscale (and set result->dscale) */
+ if (round)
+ round_var(result, rscale);
+ else
+ trunc_var(result, rscale);
+
+ /* Strip leading and trailing zeroes */
+ strip_var(result);
+}
+
+
+/*
+ * div_var_fast() -
+ *
+ * This has the same API as div_var, but is implemented using the division
+ * algorithm from the "FM" library, rather than Knuth's schoolbook-division
+ * approach. This is significantly faster but can produce inaccurate
+ * results, because it sometimes has to propagate rounding to the left,
+ * and so we can never be entirely sure that we know the requested digits
+ * exactly. We compute DIV_GUARD_DIGITS extra digits, but there is
+ * no certainty that that's enough. We use this only in the transcendental
+ * function calculation routines, where everything is approximate anyway.
+ */
+static void
+div_var_fast(NumericVar *var1, NumericVar *var2, NumericVar *result,
+ int rscale, bool round)
{
int div_ndigits;
int res_sign;
mod_var(NumericVar *var1, NumericVar *var2, NumericVar *result)
{
NumericVar tmp;
- int rscale;
init_var(&tmp);
/* ---------
* We do this using the equation
* mod(x,y) = x - trunc(x/y)*y
- * We set rscale the same way numeric_div and numeric_mul do
- * to get the right answer from the equation. The final result,
- * however, need not be displayed to more precision than the inputs.
+ * div_var can be persuaded to give us trunc(x/y) directly.
* ----------
*/
- rscale = select_div_scale(var1, var2);
+ div_var(var1, var2, &tmp, 0, false);
- div_var(var1, var2, &tmp, rscale, false);
-
- trunc_var(&tmp, 0);
-
- mul_var(var2, &tmp, &tmp, var2->dscale + tmp.dscale);
+ mul_var(var2, &tmp, &tmp, var2->dscale);
sub_var(var1, &tmp, result);
- round_var(result, Max(var1->dscale, var2->dscale));
-
free_var(&tmp);
}
for (;;)
{
- div_var(&tmp_arg, result, &tmp_val, local_rscale, true);
+ div_var_fast(&tmp_arg, result, &tmp_val, local_rscale, true);
add_var(result, &tmp_val, result);
mul_var(result, &const_zero_point_five, result, local_rscale);
/* Compensate for input sign, and round to requested rscale */
if (xneg)
- div_var(&const_one, result, result, rscale, true);
+ div_var_fast(&const_one, result, result, rscale, true);
else
round_var(result, rscale);
add_var(&ni, &const_one, &ni);
mul_var(&xpow, &x, &xpow, local_rscale);
mul_var(&ifac, &ni, &ifac, 0);
- div_var(&xpow, &ifac, &elem, local_rscale, true);
+ div_var_fast(&xpow, &ifac, &elem, local_rscale, true);
if (elem.ndigits == 0)
break;
*/
sub_var(&x, &const_one, result);
add_var(&x, &const_one, &elem);
- div_var(result, &elem, result, local_rscale, true);
+ div_var_fast(result, &elem, result, local_rscale, true);
set_var_from_var(result, &xx);
mul_var(result, result, &x, local_rscale);
{
add_var(&ni, &const_two, &ni);
mul_var(&xx, &x, &xx, local_rscale);
- div_var(&xx, &ni, &elem, local_rscale, true);
+ div_var_fast(&xx, &ni, &elem, local_rscale, true);
if (elem.ndigits == 0)
break;
/* Select scale for division result */
rscale = select_div_scale(&ln_num, &ln_base);
- div_var(&ln_num, &ln_base, result, rscale, true);
+ div_var_fast(&ln_num, &ln_base, result, rscale, true);
free_var(&ln_num);
free_var(&ln_base);
/* Compensate for input sign, and round to requested rscale */
if (neg)
- div_var(&const_one, result, result, rscale, true);
+ div_var_fast(&const_one, result, result, rscale, true);
else
round_var(result, rscale);
}
/*
* trunc_var
*
- * Truncate the value of a variable at rscale decimal digits after the
- * decimal point. NOTE: we allow rscale < 0 here, implying
+ * Truncate (towards zero) the value of a variable at rscale decimal digits
+ * after the decimal point. NOTE: we allow rscale < 0 here, implying
* truncation before the decimal point.
*/
static void
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