Quotient of products

Average Error: 11.3 → 5.1

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.38546161768463811 \cdot 10^{243}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.9578465031717975 \cdot 10^{-230}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.5655575476826919 \cdot 10^{-168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.9091311228388088 \cdot 10^{248}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \end{array}\]

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) (b1 * b2)) <= -1.3854616176846381e+243)) {
		VAR = ((double) (((double) (((double) (a1 / ((double) cbrt(b2)))) / ((double) cbrt(b2)))) * ((double) (((double) (a2 / b1)) / ((double) cbrt(b2))))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -3.9578465031717975e-230)) {
			VAR_1 = ((double) (((double) (a1 * a2)) * ((double) (((double) (1.0 / b1)) / b2))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 1.565557547682692e-168)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 2.909131122838809e+248)) {
					VAR_3 = ((double) (((double) (a1 * a2)) * ((double) (((double) (1.0 / b1)) / b2))));
				} else {
					VAR_3 = ((double) (((double) (((double) (a1 / ((double) cbrt(b2)))) / ((double) cbrt(b2)))) * ((double) (((double) (a2 / b1)) / ((double) 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	11.3
Target	11.2
Herbie	5.1

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -1.38546161768463811e243 or 2.9091311228388088e248 < (* b1 b2)

   1. Initial program 18.0

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

   3. Applied associate-/r* 8.6

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

   5. Applied add-cube-cbrt 8.8

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

   6. Applied *-un-lft-identity 8.8

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

   7. Applied times-frac 4.0

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

   8. Applied times-frac 3.7

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

   9. Simplified 3.7

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

   if -1.38546161768463811e243 < (* b1 b2) < -3.9578465031717975e-230 or 1.5655575476826919e-168 < (* b1 b2) < 2.9091311228388088e248

   1. Initial program 4.6

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

   3. Applied associate-/r* 11.5

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

   5. Applied *-un-lft-identity 11.5

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

   6. Applied div-inv 11.6

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

   7. Applied times-frac 4.6

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

   8. Simplified 4.6

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

   if -3.9578465031717975e-230 < (* b1 b2) < 1.5655575476826919e-168

   1. Initial program 31.0

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

   3. Applied times-frac 10.1

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.38546161768463811 \cdot 10^{243}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.9578465031717975 \cdot 10^{-230}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.5655575476826919 \cdot 10^{-168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.9091311228388088 \cdot 10^{248}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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