Quotient of products

Average Error: 11.2 → 4.0

Time: 7.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.91555786715051108 \cdot 10^{224}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2 \cdot \frac{a1}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.33486512330338852 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\right) \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 9.94465371630532202 \cdot 10^{-306}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.44165590446337528 \cdot 10^{297}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\right) \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

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))) <= -9.915557867150511e+224)) {
		VAR = (1.0 / (b1 / ((double) (a2 * (a1 / b2)))));
	} else {
		double VAR_1;
		if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -5.3348651233033885e-292)) {
			VAR_1 = ((double) (((double) (((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2))))) * ((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2))))))) * ((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2)))))));
		} else {
			double VAR_2;
			if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 9.944653716305322e-306)) {
				VAR_2 = ((double) ((a1 / b1) * (a2 / b2)));
			} else {
				double VAR_3;
				if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 7.441655904463375e+297)) {
					VAR_3 = ((double) (((double) (((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2))))) * ((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2))))))) * ((double) cbrt((((double) (a1 * a2)) / ((double) (b1 * b2)))))));
				} else {
					VAR_3 = (a1 / ((double) (b1 * (b2 / a2))));
				}
				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	11.2
Target	11.0
Herbie	4.0

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -9.91555786715051108e224

   1. Initial program 34.7

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

   3. Applied clear-num 34.7

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

   5. Applied associate-/l* 27.7

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

   6. Simplified 20.5

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

   if -9.91555786715051108e224 < (/ (* a1 a2) (* b1 b2)) < -5.33486512330338852e-292 or 9.94465371630532202e-306 < (/ (* a1 a2) (* b1 b2)) < 7.44165590446337528e297

   1. Initial program 0.9

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

   3. Applied add-cube-cbrt 1.9

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

   if -5.33486512330338852e-292 < (/ (* a1 a2) (* b1 b2)) < 9.94465371630532202e-306

   1. Initial program 13.4

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

   3. Applied times-frac 2.5

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

   if 7.44165590446337528e297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 61.3

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

   3. Applied associate-/l* 45.6

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

   4. Simplified 12.4

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.91555786715051108 \cdot 10^{224}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2 \cdot \frac{a1}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.33486512330338852 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\right) \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 9.94465371630532202 \cdot 10^{-306}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.44165590446337528 \cdot 10^{297}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\right) \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

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)))