Quotient of products

Average Error: 11.4 → 2.3

Time: 27.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.70720944490362918 \cdot 10^{298}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.8228617853424107 \cdot 10^{-289}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.263698258097746 \cdot 10^{299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

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)))) <= -2.707209444903629e+298)) {
		VAR = ((double) (((double) (a2 / b1)) * ((double) (a1 / b2))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -4.822861785342411e-289)) {
			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)))) <= 1.263698258097746e+299)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (((double) (a2 / b1)) * ((double) (a1 / 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	11.4
Target	11.5
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -2.707209444903629e+298 or 1.263698258097746e+299 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 60.0

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

   3. Applied *-commutative 60.0

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

   4. Applied times-frac 8.9

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

   if -2.707209444903629e+298 < (/ (* a1 a2) (* b1 b2)) < -4.822861785342411e-289 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.263698258097746e+299

   1. Initial program 0.7

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

   if -4.822861785342411e-289 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 13.7

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

   3. Applied times-frac 2.8

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.70720944490362918 \cdot 10^{298}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.8228617853424107 \cdot 10^{-289}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.263698258097746 \cdot 10^{299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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