Quotient of products

Average Error: 11.2 → 5.7

Time: 4.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -inf.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.08718180476195655 \cdot 10^{-263}:\\ \;\;\;\;\frac{a1}{b2 \cdot b1} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 0.0 \lor \neg \left(b1 \cdot b2 \le 6.20077002672593022 \cdot 10^{229}\right):\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \left(\left(\sqrt[3]{a2} \cdot a1\right) \cdot \frac{1}{b2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \end{array}\]

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)) <= -inf.0)) {
		VAR = ((double) ((a1 / b1) * (a2 / b2)));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -2.0871818047619565e-263)) {
			VAR_1 = ((double) ((a1 / ((double) (b2 * b1))) * a2));
		} else {
			double VAR_2;
			if (((((double) (b1 * b2)) <= 0.0) || !(((double) (b1 * b2)) <= 6.20077002672593e+229))) {
				VAR_2 = ((double) ((((double) (((double) cbrt(a2)) * ((double) cbrt(a2)))) / b1) * ((double) (((double) (((double) cbrt(a2)) * a1)) * (1.0 / b2)))));
			} else {
				VAR_2 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
			}
			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	11.2
Target	11.0
Herbie	5.7

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 23.4

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

   3. Applied times-frac 2.3

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

   if -inf.0 < (* b1 b2) < -2.08718180476195655e-263

   1. Initial program 5.4

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

   3. Applied clear-num 5.8

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 5.8

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}}\]
   6. Using strategy rm

   7. Applied div-inv 5.9

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

   8. Applied add-cube-cbrt 5.9

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

   9. Applied times-frac 5.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{b1 \cdot b2}{a1}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{a2}}}\]

   10. Simplified 5.4

       \[\leadsto \color{blue}{\frac{a1}{b2 \cdot b1}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{a2}}\]

   11. Simplified 5.4

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

   if -2.08718180476195655e-263 < (* b1 b2) < 0.0 or 6.20077002672593022e229 < (* b1 b2)

   1. Initial program 28.3

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

   3. Applied clear-num 28.5

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 28.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 28.4

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

   8. Applied *-un-lft-identity 28.4

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

   9. Applied times-frac 12.7

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

   10. Applied times-frac 5.8

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1}{1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{\frac{b2}{a1}}{\sqrt[3]{a2}}}}\]

   11. Applied add-cube-cbrt 5.8

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

   12. Applied times-frac 5.7

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{b1}{1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{b2}{a1}}{\sqrt[3]{a2}}}}\]

   13. Simplified 5.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{b2}{a1}}{\sqrt[3]{a2}}}\]

   14. Simplified 7.4

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

   if 0.0 < (* b1 b2) < 6.20077002672593022e229

   1. Initial program 5.6

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

   3. Applied div-inv 5.9

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -inf.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.08718180476195655 \cdot 10^{-263}:\\ \;\;\;\;\frac{a1}{b2 \cdot b1} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 0.0 \lor \neg \left(b1 \cdot b2 \le 6.20077002672593022 \cdot 10^{229}\right):\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \left(\left(\sqrt[3]{a2} \cdot a1\right) \cdot \frac{1}{b2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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