Quotient of products

Average Error: 11.1 → 5.4

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.37606304837535754 \cdot 10^{275}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{\frac{a1}{b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.4049694003619378 \cdot 10^{-139}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.397144190531792 \cdot 10^{-173}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.51943875701004492 \cdot 10^{245}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{\frac{a1}{b2}}{\frac{1}{a2}}\\ \end{array}\]

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)) <= -3.3760630483753575e+275)) {
		VAR = ((double) (((double) (1.0 / b1)) * ((double) (((double) (a1 / b2)) / ((double) (1.0 / a2))))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -3.404969400361938e-139)) {
			VAR_1 = ((double) (a1 * ((double) (a2 / ((double) (b1 * b2))))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 1.397144190531792e-173)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 1.519438757010045e+245)) {
					VAR_3 = ((double) (a1 * ((double) (a2 / ((double) (b1 * b2))))));
				} else {
					VAR_3 = ((double) (((double) (1.0 / b1)) * ((double) (((double) (a1 / b2)) / ((double) (1.0 / 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	11.1
Target	11.3
Herbie	5.4

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -3.37606304837535754e275 or 1.51943875701004492e245 < (* b1 b2)

   1. Initial program 17.2

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

   3. Applied associate-/l* 17.2

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

   5. Applied *-un-lft-identity 17.2

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

   6. Applied times-frac 7.5

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

   7. Applied *-un-lft-identity 7.5

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

   8. Applied times-frac 3.9

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

   9. Simplified 3.9

      \[\leadsto \color{blue}{\frac{1}{b1}} \cdot \frac{a1}{\frac{b2}{a2}}\]
   10. Using strategy rm

   11. Applied div-inv 4.0

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

   12. Applied associate-/r* 3.3

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

   if -3.37606304837535754e275 < (* b1 b2) < -3.4049694003619378e-139 or 1.397144190531792e-173 < (* b1 b2) < 1.51943875701004492e245

   1. Initial program 4.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   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 4.4

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

   6. Simplified 4.2

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

   if -3.4049694003619378e-139 < (* b1 b2) < 1.397144190531792e-173

   1. Initial program 26.5

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

   3. Applied times-frac 12.3

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.37606304837535754 \cdot 10^{275}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{\frac{a1}{b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.4049694003619378 \cdot 10^{-139}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.397144190531792 \cdot 10^{-173}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.51943875701004492 \cdot 10^{245}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{\frac{a1}{b2}}{\frac{1}{a2}}\\ \end{array}\]

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)))