Quotient of products

Average Error: 11.6 → 5.4

Time: 27.2min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.95890435295476721 \cdot 10^{211}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.2498994217787959 \cdot 10^{-213} \lor \neg \left(b1 \cdot b2 \le -0.0\right) \land b1 \cdot b2 \le 1.023298139213398 \cdot 10^{209}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

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) (b1 * b2)) <= -2.958904352954767e+211)) {
		VAR = ((double) ((a2 / b1) * (a1 / b2)));
	} else {
		double VAR_1;
		if (((((double) (b1 * b2)) <= -9.249899421778796e-213) || (!(((double) (b1 * b2)) <= -0.0) && (((double) (b1 * b2)) <= 1.023298139213398e+209)))) {
			VAR_1 = ((a1 / ((double) (b1 * b2))) / (1.0 / a2));
		} else {
			VAR_1 = ((double) ((a2 / b1) * (a1 / 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	11.4
Herbie	5.4

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

Derivation

1. Split input into 2 regimes

2. if (* b1 b2) < -2.95890435295476721e211 or -9.2498994217787959e-213 < (* b1 b2) < -0.0 or 1.023298139213398e209 < (* b1 b2)

   1. Initial program 23.1

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

   3. Applied associate-/l* 23.0

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

   5. Applied add-cube-cbrt 23.2

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

   6. Applied times-frac 10.9

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

   7. Applied add-cube-cbrt 11.0

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

   8. Applied times-frac 4.1

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

   10. Applied pow1 4.1

       \[\leadsto \frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \color{blue}{{\left(\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}\right)}^{1}}\]

   11. Applied pow1 4.1

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\right)}^{1}} \cdot {\left(\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}\right)}^{1}\]

   12. Applied pow-prod-down 4.1

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}\right)}^{1}}\]

   13. Simplified 11.4

       \[\leadsto {\color{blue}{\left(\frac{a2}{\frac{b2}{\frac{a1}{b1}}}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied associate-/r/ 11.3

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

   16. Applied *-un-lft-identity 11.3

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

   17. Applied times-frac 5.5

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

   18. Simplified 5.4

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

   if -2.95890435295476721e211 < (* b1 b2) < -9.2498994217787959e-213 or -0.0 < (* b1 b2) < 1.023298139213398e209

   1. Initial program 5.3

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

   3. Applied associate-/l* 5.1

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

   5. Applied div-inv 5.1

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

   6. Applied associate-/r* 5.5

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.95890435295476721 \cdot 10^{211}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.2498994217787959 \cdot 10^{-213} \lor \neg \left(b1 \cdot b2 \le -0.0\right) \land b1 \cdot b2 \le 1.023298139213398 \cdot 10^{209}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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