Quotient of products

Average Error: 11.3 → 2.4

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.1561895240699571 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a1 \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.0924393730667472 \cdot 10^{-296} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 3.2485315446963486 \cdot 10^{304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{b1}\right) \cdot \left(a2 \cdot \frac{\sqrt[3]{a1}}{b2}\right)\\ \end{array}\]

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)))) <= -4.156189524069957e+296)) {
		VAR = ((double) (((double) (1.0 / b1)) * ((double) (a1 * ((double) (a2 / b2))))));
	} else {
		double VAR_1;
		if (((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -3.092439373066747e-296) || (!(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 0.0) && (((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 3.2485315446963486e+304)))) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(a1)) * ((double) (((double) cbrt(a1)) / b1)))) * ((double) (a2 * ((double) (((double) cbrt(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	11.3
Target	11.4
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -4.1561895240699571e296

   1. Initial program 59.0

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

   3. Applied associate-/l* 32.7

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

   4. Simplified 19.1

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

   6. Applied *-un-lft-identity 19.1

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

   7. Applied times-frac 17.2

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

   8. Simplified 17.6

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

   if -4.1561895240699571e296 < (/ (* a1 a2) (* b1 b2)) < -3.0924393730667472e-296 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 3.2485315446963486e304

   1. Initial program 0.9

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

   if -3.0924393730667472e-296 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 3.2485315446963486e304 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 22.1

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

   3. Applied associate-/l* 15.0

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

   4. Simplified 5.7

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

   6. Applied add-cube-cbrt 6.0

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

   7. Applied times-frac 3.6

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

   8. Simplified 3.6

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

   9. Simplified 3.3

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.1561895240699571 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a1 \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.0924393730667472 \cdot 10^{-296} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 3.2485315446963486 \cdot 10^{304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{b1}\right) \cdot \left(a2 \cdot \frac{\sqrt[3]{a1}}{b2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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