Quotient of products

Average Error: 11.6 → 2.0

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\left(\frac{a1}{b1} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\right)\right) \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.4278497164812 \cdot 10^{-321} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq -0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.914732966838874 \cdot 10^{+288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return (((double) (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) (((double) ((a1 / b1) * ((double) (((double) cbrt(a2)) * (((double) cbrt(a2)) / ((double) (((double) cbrt(b2)) * ((double) cbrt(b2))))))))) * (((double) cbrt(a2)) / ((double) cbrt(b2)))));
	} else {
		double VAR_1;
		if ((((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -1.4278497164812e-321) || (!((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -0.0) && ((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 7.914732966838874e+288)))) {
			VAR_1 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
		} else {
			VAR_1 = ((double) ((a1 / b1) * (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	11.6
Target	11.2
Herbie	2.0

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

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 28.4

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

   4. Applied *-un-lft-identity 28.4

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

   5. Applied times-frac 17.3

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

   6. Applied associate-*r* 10.5

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

   7. Simplified 10.5

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

   9. Applied add-cube-cbrt 11.3

      \[\leadsto \frac{a1}{b1} \cdot \frac{a2}{\color{blue}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}}\]

   10. Applied add-cube-cbrt 11.6

       \[\leadsto \frac{a1}{b1} \cdot \frac{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}\]

   11. Applied times-frac 11.6

       \[\leadsto \frac{a1}{b1} \cdot \color{blue}{\left(\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\right)}\]

   12. Applied associate-*r* 7.5

       \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\right) \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}\]

   13. Simplified 7.5

       \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\right)\right)} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\]

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.42785e-321 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 7.9147329668388741e288

   1. Initial program 0.8

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

   if -1.42785e-321 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 7.9147329668388741e288 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 23.2

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

   2. Simplified 15.3

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

   4. Applied *-un-lft-identity 15.3

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

   5. Applied times-frac 5.9

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

   6. Applied associate-*r* 3.3

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

   7. Simplified 3.3

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\left(\frac{a1}{b1} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\right)\right) \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.4278497164812 \cdot 10^{-321} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq -0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.914732966838874 \cdot 10^{+288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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)))