Quotient of products

Average Error: 11.5 → 6.6

Time: 2.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.72679722760626064 \cdot 10^{83}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.79742884607077612 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a1 \cdot a2}{b1}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 7.9250296375044805 \cdot 10^{-177}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.61105070741571186 \cdot 10^{243}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a1 \cdot a2}{b1}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

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 (((a1 * a2) <= -6.726797227606261e+83)) {
		VAR = (a1 / ((b1 * b2) / a2));
	} else {
		double VAR_1;
		if (((a1 * a2) <= -2.797428846070776e-182)) {
			VAR_1 = (((((a1 * a2) / b1) / cbrt(b2)) / cbrt(b2)) / cbrt(b2));
		} else {
			double VAR_2;
			if (((a1 * a2) <= 7.92502963750448e-177)) {
				VAR_2 = ((a1 / b1) * (a2 / b2));
			} else {
				double VAR_3;
				if (((a1 * a2) <= 5.611050707415712e+243)) {
					VAR_3 = (((((a1 * a2) / b1) / cbrt(b2)) / cbrt(b2)) / cbrt(b2));
				} 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	11.5
Target	11.5
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -6.726797227606261e+83

   1. Initial program 21.4

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

   3. Applied associate-/l* 15.1

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

   if -6.726797227606261e+83 < (* a1 a2) < -2.797428846070776e-182 or 7.92502963750448e-177 < (* a1 a2) < 5.611050707415712e+243

   1. Initial program 4.0

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

   3. Applied associate-/r* 4.0

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

   5. Applied div-inv 4.1

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}}{b2}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 4.8

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

   8. Applied associate-/r* 4.8

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

   9. Simplified 4.8

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

   if -2.797428846070776e-182 < (* a1 a2) < 7.92502963750448e-177

   1. Initial program 13.9

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

   3. Applied times-frac 5.1

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

   if 5.611050707415712e+243 < (* a1 a2)

   1. Initial program 45.2

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

   3. Applied associate-/r* 48.0

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

   5. Applied *-un-lft-identity 48.0

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

   6. Applied *-un-lft-identity 48.0

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

   7. Applied times-frac 21.3

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

   8. Applied times-frac 9.9

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

   9. Simplified 9.9

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.72679722760626064 \cdot 10^{83}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.79742884607077612 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a1 \cdot a2}{b1}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 7.9250296375044805 \cdot 10^{-177}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.61105070741571186 \cdot 10^{243}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a1 \cdot a2}{b1}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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