Quotient of products

Average Error: 11.5 → 5.6

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.9920259308575664 \cdot 10^{151}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.60843877903754066 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0600231449193626 \cdot 10^{-281}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b2} \cdot \frac{1}{b1}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0183147166813423 \cdot 10^{238}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b2} \cdot \frac{1}{b1}\right)\\ \end{array}\]

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)) <= -3.9920259308575664e+151)) {
		VAR = ((double) (((double) (a1 * ((double) (a2 / b2)))) / b1));
	} else {
		double VAR_1;
		if ((((double) (a1 * a2)) <= -4.608438779037541e-266)) {
			VAR_1 = ((double) (((double) (((double) (a1 * a2)) / b1)) / b2));
		} else {
			double VAR_2;
			if ((((double) (a1 * a2)) <= 1.0600231449193626e-281)) {
				VAR_2 = ((double) (a1 * ((double) (((double) (a2 / b2)) * ((double) (1.0 / b1))))));
			} else {
				double VAR_3;
				if ((((double) (a1 * a2)) <= 1.0183147166813423e+238)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (a1 * ((double) (((double) (a2 / b2)) * ((double) (1.0 / 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	11.5
Target	11.2
Herbie	5.6

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -3.9920259308575664e151

   1. Initial program 26.5

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

   3. Applied times-frac 11.4

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

   5. Applied associate-*l/ 15.1

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

   if -3.9920259308575664e151 < (* a1 a2) < -4.60843877903754066e-266

   1. Initial program 4.4

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

   3. Applied associate-/r* 4.2

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

   if -4.60843877903754066e-266 < (* a1 a2) < 1.0600231449193626e-281 or 1.0183147166813423e238 < (* a1 a2)

   1. Initial program 23.5

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

   3. Applied times-frac 4.4

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

   5. Applied associate-*l/ 9.5

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

   7. Applied div-inv 9.5

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

   9. Applied associate-*l* 4.1

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

   if 1.0600231449193626e-281 < (* a1 a2) < 1.0183147166813423e238

   1. Initial program 5.4

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.9920259308575664 \cdot 10^{151}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.60843877903754066 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0600231449193626 \cdot 10^{-281}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b2} \cdot \frac{1}{b1}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0183147166813423 \cdot 10^{238}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b2} \cdot \frac{1}{b1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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