Quotient of products

Average Error: 11.3 → 2.6

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1}}{b1} \cdot \left(\sqrt[3]{a1} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right)\right) \cdot \frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.87152 \cdot 10^{-320}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.133778318688682 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.71035497095794815 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

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) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -inf.0)) {
		VAR = ((double) (((double) (((double) (((double) cbrt(a1)) / b1)) * ((double) (((double) cbrt(a1)) * ((double) (((double) cbrt(a2)) * ((double) cbrt(a2)))))))) * ((double) (((double) cbrt(a1)) / ((double) (b2 / ((double) cbrt(a2))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.8715206664466e-320)) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 2.133778318688682e-226)) {
				VAR_2 = ((double) (((double) (a1 / b1)) / ((double) (b2 / a2))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 1.7103549709579482e+297)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (((double) (a1 / b1)) / ((double) (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	11.3
Target	11.3
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 31.3

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

   5. Applied add-cube-cbrt 31.8

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

   6. Applied times-frac 15.5

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

   7. Applied add-cube-cbrt 15.8

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

   8. Applied times-frac 7.8

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

   10. Applied div-inv 7.8

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

   11. Applied times-frac 7.8

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

   12. Simplified 7.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.8715206664466e-320 or 2.133778318688682e-226 < (/ (* a1 a2) (* b1 b2)) < 1.7103549709579482e+297

   1. Initial program 0.8

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

   if -1.8715206664466e-320 < (/ (* a1 a2) (* b1 b2)) < 2.133778318688682e-226 or 1.7103549709579482e+297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 20.9

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

   3. Applied associate-/l* 14.5

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

   5. Applied *-un-lft-identity 14.5

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

   6. Applied times-frac 6.6

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

   7. Applied associate-/r* 4.4

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

   8. Simplified 4.4

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1}}{b1} \cdot \left(\sqrt[3]{a1} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right)\right) \cdot \frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.87152 \cdot 10^{-320}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.133778318688682 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.71035497095794815 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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