Quotient of products

Average Error: 11.2 → 5.6

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -7.08416398127913 \cdot 10^{204}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.09602219250447997 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1 \cdot b2} \cdot \left(a2 \cdot \sqrt[3]{a1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 2.2243731052410934 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.3444595232338455 \cdot 10^{228}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array}\]

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 (((b1 * b2) <= -7.08416398127913e+204)) {
		VAR = ((a1 / b1) * (a2 / b2));
	} else {
		double VAR_1;
		if (((b1 * b2) <= -4.09602219250448e-100)) {
			VAR_1 = (((cbrt(a1) * cbrt(a1)) / (b1 * b2)) * (a2 * cbrt(a1)));
		} else {
			double VAR_2;
			if (((b1 * b2) <= 2.2243731052410934e-300)) {
				VAR_2 = ((1.0 / b1) * (a1 / (b2 / a2)));
			} else {
				double VAR_3;
				if (((b1 * b2) <= 1.3444595232338455e+228)) {
					VAR_3 = (a1 / ((b1 * b2) / a2));
				} else {
					VAR_3 = (((a1 * a2) / b1) / 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	11.2
Target	11.7
Herbie	5.6

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

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -7.08416398127913e+204

   1. Initial program 14.9

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

   3. Applied times-frac 4.5

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

   if -7.08416398127913e+204 < (* b1 b2) < -4.09602219250448e-100

   1. Initial program 3.9

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

   3. Applied associate-/l* 3.5

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

   5. Applied div-inv 3.6

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

   6. Applied add-cube-cbrt 4.3

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

   7. Applied times-frac 2.3

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

   8. Simplified 2.3

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

   if -4.09602219250448e-100 < (* b1 b2) < 2.2243731052410934e-300

   1. Initial program 27.3

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

   3. Applied associate-/l* 28.4

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

   5. Applied *-un-lft-identity 28.4

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

   6. Applied times-frac 16.2

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

   7. Applied *-un-lft-identity 16.2

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

   8. Applied times-frac 11.6

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

   9. Simplified 11.6

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

   if 2.2243731052410934e-300 < (* b1 b2) < 1.3444595232338455e+228

   1. Initial program 5.7

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

   3. Applied associate-/l* 5.1

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

   if 1.3444595232338455e+228 < (* b1 b2)

   1. Initial program 16.8

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

   3. Applied associate-/r* 7.5

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -7.08416398127913 \cdot 10^{204}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.09602219250447997 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1 \cdot b2} \cdot \left(a2 \cdot \sqrt[3]{a1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 2.2243731052410934 \cdot 10^{-300}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.3444595232338455 \cdot 10^{228}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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