Quotient of products

Average Error: 11.3 → 2.5

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.2777989282357615 \cdot 10^{+289}:\\ \;\;\;\;a1 \cdot \left(\frac{1}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.2608332625522 \cdot 10^{-321} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.208734630485144 \cdot 10^{+290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

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))) <= -5.2777989282357615e+289)) {
		VAR = ((double) (a1 * ((double) ((1.0 / b1) * (a2 / b2)))));
	} else {
		double VAR_1;
		if ((((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -3.2608332625522e-321) || (!((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 0.0) && ((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 7.208734630485144e+290)))) {
			VAR_1 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
		} else {
			VAR_1 = ((double) ((a2 / b2) * (a1 / b1)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.6
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -5.2777989282357615e289

   1. Initial program 54.6

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

   2. Simplified 26.2

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

   4. Applied *-un-lft-identity 26.2

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

   5. Applied times-frac 17.8

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

   if -5.2777989282357615e289 < (/ (* a1 a2) (* b1 b2)) < -3.26083e-321 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 7.20873463048514406e290

   1. Initial program 0.9

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

   if -3.26083e-321 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 7.20873463048514406e290 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 22.5

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

   2. Simplified 14.8

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

   4. Applied *-un-lft-identity 14.8

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

   5. Applied times-frac 5.9

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

   6. Applied associate-*r* 3.5

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

   7. Simplified 3.5

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.2777989282357615 \cdot 10^{+289}:\\ \;\;\;\;a1 \cdot \left(\frac{1}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.2608332625522 \cdot 10^{-321} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.208734630485144 \cdot 10^{+290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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