Quotient of products

Average Error: 10.9 → 2.7

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.64742855356953027 \cdot 10^{-304} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 7.5253061419253405 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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)))) <= -inf.0)) {
		VAR = ((double) (a1 * ((double) (a2 / ((double) (b1 * b2))))));
	} else {
		double VAR_1;
		if (((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.6474285535695303e-304) || (!(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -0.0) && (((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 7.5253061419253405e+301)))) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			VAR_1 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.2
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 32.2

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.64742855356953027e-304 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 7.5253061419253405e301

   1. Initial program 0.8

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

   if -1.64742855356953027e-304 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 7.5253061419253405e301 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 21.9

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

   2. Simplified 14.0

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 14.0

      \[\leadsto a1 \cdot \frac{\color{blue}{1 \cdot a2}}{b1 \cdot b2}\]

   5. Applied times-frac 5.6

      \[\leadsto a1 \cdot \color{blue}{\left(\frac{1}{b1} \cdot \frac{a2}{b2}\right)}\]

   6. Applied associate-*r* 3.1

      \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1}\right) \cdot \frac{a2}{b2}}\]

   7. Simplified 3.0

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.64742855356953027 \cdot 10^{-304} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 7.5253061419253405 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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