Quotient of products

Average Error: 10.9 → 2.9

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.111168099783512 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.99364642502751304 \cdot 10^{245}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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) / (b1 * b2)) <= -inf.0)) {
		VAR = (((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2)));
	} else {
		double VAR_1;
		if ((((a1 * a2) / (b1 * b2)) <= -2.11116809978351e-309)) {
			VAR_1 = ((a1 * a2) / (b1 * b2));
		} else {
			double VAR_2;
			if ((((a1 * a2) / (b1 * b2)) <= -0.0)) {
				VAR_2 = (((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2)));
			} else {
				double VAR_3;
				if ((((a1 * a2) / (b1 * b2)) <= 1.993646425027513e+245)) {
					VAR_3 = ((a1 * a2) / (b1 * b2));
				} else {
					VAR_3 = ((a1 / b1) * (a2 / 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	10.9
Target	11.5
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -2.11116809978351e-309 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 17.5

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

   3. Applied times-frac 3.2

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

   5. Applied add-cube-cbrt 3.4

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

   6. Applied *-un-lft-identity 3.4

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

   7. Applied times-frac 3.4

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

   8. Applied associate-*r* 4.2

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

   9. Simplified 4.2

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -2.11116809978351e-309 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.993646425027513e+245

   1. Initial program 0.8

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

   if 1.993646425027513e+245 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 49.1

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

   3. Applied times-frac 10.8

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.111168099783512 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.99364642502751304 \cdot 10^{245}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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