Quotient of products

Average Error: 11.4 → 2.9

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.8300642559644884 \cdot 10^{-306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.3534308712433477 \cdot 10^{+306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array}\]

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= ((double) -(((double) INFINITY))))) {
		VAR = (((double) (a2 * (a1 / b1))) / b2);
	} else {
		double VAR_1;
		if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -4.8300642559644884e-306)) {
			VAR_1 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
		} else {
			double VAR_2;
			if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -0.0)) {
				VAR_2 = ((double) ((a1 / b1) * (a2 / b2)));
			} else {
				double VAR_3;
				if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 1.3534308712433477e+306)) {
					VAR_3 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
				} else {
					VAR_3 = (((double) (a2 * (a1 / b1))) / b2);
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.4
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or 1.35343087124334768e306 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 64.0

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

   3. Applied associate-/r* 47.1

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

   4. Simplified 16.0

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.8300642559644884e-306 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.35343087124334768e306

   1. Initial program 0.6

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

   if -4.8300642559644884e-306 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 13.8

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

   3. Applied times-frac 2.6

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.8300642559644884 \cdot 10^{-306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.3534308712433477 \cdot 10^{+306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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