Quotient of products

Average Error: 10.7 → 6.4

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.04801940866511539 \cdot 10^{67}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.21348959676995967 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.21004619463227181 \cdot 10^{-222}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.48851413857169847 \cdot 10^{-277}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.5417218424085528 \cdot 10^{174}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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) (b1 * b2)) <= -4.0480194086651154e+67)) {
		VAR = ((double) (a1 / ((double) (((double) (b1 * b2)) / a2))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -1.2134895967699597e-198)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (b1 * b2)) / ((double) (a1 * a2))))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= -6.210046194632272e-222)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 1.4885141385716985e-277)) {
					VAR_3 = ((double) (((double) (a1 * ((double) (a2 / b2)))) / b1));
				} else {
					double VAR_4;
					if ((((double) (b1 * b2)) <= 9.541721842408553e+174)) {
						VAR_4 = ((double) (a1 / ((double) (((double) (b1 * b2)) / a2))));
					} else {
						VAR_4 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
					}
					VAR_3 = VAR_4;
				}
				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	10.7
Target	11.6
Herbie	6.4

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

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -4.04801940866511539e67 or 1.48851413857169847e-277 < (* b1 b2) < 9.5417218424085528e174

   1. Initial program 7.4

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

   3. Applied associate-/l* 7.1

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

   if -4.04801940866511539e67 < (* b1 b2) < -1.21348959676995967e-198

   1. Initial program 3.4

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

   3. Applied clear-num 3.6

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

   if -1.21348959676995967e-198 < (* b1 b2) < -6.21004619463227181e-222 or 9.5417218424085528e174 < (* b1 b2)

   1. Initial program 13.9

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

   3. Applied times-frac 5.8

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

   if -6.21004619463227181e-222 < (* b1 b2) < 1.48851413857169847e-277

   1. Initial program 40.8

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

   3. Applied times-frac 10.8

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

   5. Applied associate-*l/ 8.7

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b2}}{b1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.04801940866511539 \cdot 10^{67}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.21348959676995967 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.21004619463227181 \cdot 10^{-222}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.48851413857169847 \cdot 10^{-277}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.5417218424085528 \cdot 10^{174}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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