Quotient of products

Average Error: 11.8 → 2.7

Time: 3.2s

Precision: binary64

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.28691639061212305 \cdot 10^{272} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.19361 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.3939702079453936 \cdot 10^{269}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if (((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -3.286916390612123e+272) || !((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.1936131937879e-319) || !((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -0.0) || !(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 3.3939702079453936e+269))))) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		VAR = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.8
Target	11.2
Herbie	2.7

\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -3.28691639061212305e272 or -1.19361e-319 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 3.3939702079453936e269 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.1

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied times-frac 4.9

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]

   if -3.28691639061212305e272 < (/ (* a1 a2) (* b1 b2)) < -1.19361e-319 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 3.3939702079453936e269

   1. Initial program 0.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.28691639061212305 \cdot 10^{272} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.19361 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.3939702079453936 \cdot 10^{269}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))