Quotient of products

Average Error: 11.3 → 5.8

Time: 22.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.75066566822266414 \cdot 10^{264}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.68925304174710311 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.31795063732092912 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.6083742448641293 \cdot 10^{256}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

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) (b1 * b2)) <= -4.750665668222664e+264)) {
		VAR = ((double) (((double) (a1 * ((double) (a2 / b1)))) / b2));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -4.689253041747103e-90)) {
			VAR_1 = ((double) (a1 * ((double) (a2 / ((double) (b1 * b2))))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 7.317950637320929e-136)) {
				VAR_2 = ((double) (((double) (a1 / b1)) / ((double) (b2 / a2))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 3.608374244864129e+256)) {
					VAR_3 = ((double) (a1 * ((double) (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	5.8

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -4.75066566822266414e264

   1. Initial program 19.3

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

   3. Applied associate-/l* 18.6

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

   5. Applied div-inv 18.6

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

   6. Simplified 18.5

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

   8. Applied associate-/r* 7.7

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

   10. Applied associate-*r/ 3.6

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

   if -4.75066566822266414e264 < (* b1 b2) < -4.68925304174710311e-90 or 7.31795063732092912e-136 < (* b1 b2) < 3.6083742448641293e256

   1. Initial program 4.0

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

   3. Applied associate-/l* 4.2

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

   5. Applied div-inv 4.5

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

   6. Simplified 4.1

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

   if -4.68925304174710311e-90 < (* b1 b2) < 7.31795063732092912e-136 or 3.6083742448641293e256 < (* b1 b2)

   1. Initial program 21.1

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

   3. Applied associate-/l* 20.2

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

   5. Applied *-un-lft-identity 20.2

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

   6. Applied times-frac 12.0

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

   7. Applied associate-/r* 9.2

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

   8. Simplified 9.2

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

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.75066566822266414 \cdot 10^{264}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.68925304174710311 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.31795063732092912 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.6083742448641293 \cdot 10^{256}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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