Quotient of products

Average Error: 11.2 → 5.4

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -5.9344259756136052 \cdot 10^{255}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.76975887556352704 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 7.17232861347048113 \cdot 10^{-178}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.2395081457445914 \cdot 10^{191}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

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) (a1 * a2)) <= -5.934425975613605e+255)) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		double VAR_1;
		if ((((double) (a1 * a2)) <= -3.769758875563527e-76)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (b1 * b2)) / ((double) (a1 * a2))))));
		} else {
			double VAR_2;
			if ((((double) (a1 * a2)) <= 7.172328613470481e-178)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (a1 * a2)) <= 1.2395081457445914e+191)) {
					VAR_3 = ((double) (((double) (((double) (a1 * a2)) * ((double) (1.0 / b2)))) / b1));
				} else {
					VAR_3 = ((double) (a1 * ((double) (((double) (a2 / b2)) / b1))));
				}
				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	11.0
Herbie	5.4

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -5.9344259756136052e255 or -3.76975887556352704e-76 < (* a1 a2) < 7.17232861347048113e-178

   1. Initial program 15.4

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

   3. Applied times-frac 6.1

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

   if -5.9344259756136052e255 < (* a1 a2) < -3.76975887556352704e-76

   1. Initial program 4.9

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

   3. Applied clear-num 5.0

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

   if 7.17232861347048113e-178 < (* a1 a2) < 1.2395081457445914e191

   1. Initial program 4.1

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

   3. Applied times-frac 16.0

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

   5. Applied associate-*l/ 11.8

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

   7. Applied div-inv 11.9

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

   8. Applied associate-*r* 3.6

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

   if 1.2395081457445914e191 < (* a1 a2)

   1. Initial program 32.3

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

   3. Applied times-frac 10.3

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

   5. Applied div-inv 10.4

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

   6. Applied associate-*l* 9.1

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

   7. Simplified 9.0

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -5.9344259756136052 \cdot 10^{255}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.76975887556352704 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 7.17232861347048113 \cdot 10^{-178}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.2395081457445914 \cdot 10^{191}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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