Quotient of products

Average Error: 10.9 → 3.1

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.0953428171912087 \cdot 10^{282}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.7932051454 \cdot 10^{-314}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.7389682791900477 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}} \cdot \sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}\right) \cdot \sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return ((a1 * a2) / (b1 * b2));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((a1 * a2) / (b1 * b2)) <= -9.095342817191209e+282)) {
		VAR = (((a1 / b1) * a2) / b2);
	} else {
		double VAR_1;
		if ((((a1 * a2) / (b1 * b2)) <= -4.7932051454338e-314)) {
			VAR_1 = ((a1 * a2) / (b1 * b2));
		} else {
			double VAR_2;
			if ((((a1 * a2) / (b1 * b2)) <= -0.0)) {
				VAR_2 = ((a1 * (a2 / b2)) / b1);
			} else {
				double VAR_3;
				if ((((a1 * a2) / (b1 * b2)) <= 6.738968279190048e+297)) {
					VAR_3 = ((a1 * a2) / (b1 * b2));
				} else {
					VAR_3 = ((cbrt((((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2)))) * cbrt((((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2))))) * cbrt((((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2)))));
				}
				VAR_2 = VAR_3;
			}
			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	10.9
Target	11.7
Herbie	3.1

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -9.095342817191209e+282

   1. Initial program 51.3

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

   3. Applied times-frac 15.9

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

   5. Applied associate-*r/ 17.1

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

   if -9.095342817191209e+282 < (/ (* a1 a2) (* b1 b2)) < -4.7932051454338e-314 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 6.738968279190048e+297

   1. Initial program 0.8

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

   if -4.7932051454338e-314 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 13.1

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

   3. Applied times-frac 2.5

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

   5. Applied associate-*l/ 3.8

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

   if 6.738968279190048e+297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 61.8

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

   3. Applied times-frac 6.5

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

   5. Applied add-cube-cbrt 7.5

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

   6. Applied *-un-lft-identity 7.5

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

   7. Applied times-frac 7.5

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

   8. Applied associate-*r* 10.3

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

   9. Simplified 10.3

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

   11. Applied add-cube-cbrt 10.5

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.0953428171912087 \cdot 10^{282}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.7932051454 \cdot 10^{-314}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.7389682791900477 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}} \cdot \sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}\right) \cdot \sqrt[3]{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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