Quotient of products

Average Error: 11.7 → 5.8

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.58996739274386669 \cdot 10^{194}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.90662871796825084 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.2918227959486676 \cdot 10^{-234}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.5918748189749155 \cdot 10^{172}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]

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)) <= -1.5899673927438667e+194)) {
		VAR = ((double) (((double) (1.0 / b1)) * ((double) (a1 / ((double) (b2 / a2))))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -6.906628717968251e-106)) {
			VAR_1 = ((double) (a1 / ((double) (((double) (b1 * b2)) / a2))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 8.291822795948668e-234)) {
				VAR_2 = ((double) (((double) (a1 * ((double) (a2 / b1)))) / b2));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 3.5918748189749155e+172)) {
					VAR_3 = ((double) (a1 / ((double) (((double) (b1 * b2)) / a2))));
				} else {
					VAR_3 = ((double) (((double) (1.0 / b1)) * ((double) (a1 / ((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	11.7
Target	11.3
Herbie	5.8

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -1.5899673927438667e+194 or 3.5918748189749155e+172 < (* b1 b2)

   1. Initial program 15.4

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

   3. Applied associate-/l* 14.8

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

   5. Applied *-un-lft-identity 14.8

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

   6. Applied times-frac 8.4

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

   7. Applied *-un-lft-identity 8.4

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

   8. Applied times-frac 5.4

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

   9. Simplified 5.4

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

   if -1.5899673927438667e+194 < (* b1 b2) < -6.906628717968251e-106 or 8.291822795948668e-234 < (* b1 b2) < 3.5918748189749155e+172

   1. Initial program 4.4

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

   3. Applied associate-/l* 4.4

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

   if -6.906628717968251e-106 < (* b1 b2) < 8.291822795948668e-234

   1. Initial program 26.3

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

   3. Applied associate-/l* 26.9

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

   5. Applied div-inv 27.3

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

   6. Simplified 27.2

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

   8. Applied associate-/r* 15.7

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

   10. Applied associate-*r/ 10.7

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

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.58996739274386669 \cdot 10^{194}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.90662871796825084 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.2918227959486676 \cdot 10^{-234}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.5918748189749155 \cdot 10^{172}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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