Quotient of products

Average Error: 11.2 → 6.1

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.2501052912353021 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.6793275492381075 \cdot 10^{-220}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 0.0:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.90397960473963782 \cdot 10^{260}:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

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)) <= -2.250105291235302e+162)) {
		VAR = ((double) (((double) (a1 / b1)) / ((double) (b2 / a2))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -4.6793275492381075e-220)) {
			VAR_1 = ((double) (((double) (a1 / ((double) (b1 * b2)))) * a2));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 0.0)) {
				VAR_2 = ((double) (((double) (((double) (a1 * a2)) / b1)) / b2));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 1.9039796047396378e+260)) {
					VAR_3 = ((double) (1.0 / ((double) (((double) (((double) (b1 * b2)) / a2)) / a1))));
				} else {
					VAR_3 = ((double) (((double) (a1 / b1)) / ((double) (b2 / 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	11.2
Target	11.5
Herbie	6.1

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

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -2.250105291235302e+162 or 1.9039796047396378e+260 < (* b1 b2)

   1. Initial program 15.5

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

   3. Applied associate-/l* 15.4

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

   5. Applied *-un-lft-identity 15.4

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

   6. Applied times-frac 7.9

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

   7. Applied associate-/r* 4.6

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

   8. Simplified 4.6

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

   if -2.250105291235302e+162 < (* b1 b2) < -4.6793275492381075e-220

   1. Initial program 4.4

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

   3. Applied associate-/l* 4.3

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

   5. Applied associate-/r/ 4.4

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

   if -4.6793275492381075e-220 < (* b1 b2) < 0.0

   1. Initial program 44.5

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

   3. Applied associate-/r* 17.1

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

   if 0.0 < (* b1 b2) < 1.9039796047396378e+260

   1. Initial program 5.5

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

   3. Applied associate-/l* 5.7

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

   5. Applied clear-num 6.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.2501052912353021 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.6793275492381075 \cdot 10^{-220}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 0.0:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.90397960473963782 \cdot 10^{260}:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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