Quotient of products

Average Error: 10.8 → 3.1

Time: 4.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.63422559640946159 \cdot 10^{286}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.2077081312051486 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.4317747321825792 \cdot 10^{250}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

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) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -2.6342255964094616e+286)) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.2077081312051486e-288)) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -0.0)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 8.431774732182579e+250)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (a1 * ((double) (((double) (a2 / b1)) / b2))));
				}
				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	10.8
Target	11.4
Herbie	3.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -2.63422559640946159e286 or -1.2077081312051486e-288 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 17.0

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

   3. Applied times-frac 3.8

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

   if -2.63422559640946159e286 < (/ (* a1 a2) (* b1 b2)) < -1.2077081312051486e-288 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 8.4317747321825792e250

   1. Initial program 0.7

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

   if 8.4317747321825792e250 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 50.7

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

   3. Applied associate-/r* 41.1

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

   5. Applied *-un-lft-identity 41.1

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

   6. Applied *-un-lft-identity 41.1

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

   7. Applied times-frac 16.3

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

   8. Applied times-frac 16.0

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

   9. Simplified 16.0

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.63422559640946159 \cdot 10^{286}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.2077081312051486 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.4317747321825792 \cdot 10^{250}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

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)))