Quotient of products

Average Error: 11.6 → 5.5

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.720837941461105 \cdot 10^{286}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.51097602999619632 \cdot 10^{-213}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.43806413534 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{b2 \cdot \left(\frac{1}{a1} \cdot \frac{b1}{a2}\right)}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.30808618925979213 \cdot 10^{220}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

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)) <= -3.720837941461105e+286)) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -2.5109760299961963e-213)) {
			VAR_1 = ((double) (a2 * ((double) (a1 / ((double) (b1 * b2))))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 2.4380641353373e-314)) {
				VAR_2 = ((double) (1.0 / ((double) (b2 * ((double) (((double) (1.0 / a1)) * ((double) (b1 / a2))))))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 8.308086189259792e+220)) {
					VAR_3 = ((double) (a2 * ((double) (a1 / ((double) (b1 * b2))))));
				} else {
					VAR_3 = ((double) (((double) (((double) (a1 / b1)) * a2)) / 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	11.6
Target	11.9
Herbie	5.5

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

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -3.720837941461105e286

   1. Initial program 20.1

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

   3. Applied times-frac 3.6

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

   if -3.720837941461105e286 < (* b1 b2) < -2.51097602999619632e-213 or 2.43806413534e-314 < (* b1 b2) < 8.30808618925979213e220

   1. Initial program 5.1

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

   2. Taylor expanded around 0 5.1

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

   3. Simplified 5.3

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

   if -2.51097602999619632e-213 < (* b1 b2) < 2.43806413534e-314

   1. Initial program 43.1

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

   3. Applied clear-num 43.1

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

   4. Simplified 18.2

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

   6. Applied *-un-lft-identity 18.2

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

   7. Applied times-frac 8.7

      \[\leadsto \frac{1}{b2 \cdot \color{blue}{\left(\frac{1}{a1} \cdot \frac{b1}{a2}\right)}}\]

   if 8.30808618925979213e220 < (* b1 b2)

   1. Initial program 17.2

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

   3. Applied associate-/r* 9.1

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

   4. Simplified 5.1

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.720837941461105 \cdot 10^{286}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.51097602999619632 \cdot 10^{-213}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.43806413534 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{b2 \cdot \left(\frac{1}{a1} \cdot \frac{b1}{a2}\right)}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.30808618925979213 \cdot 10^{220}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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