Quotient of products

Average Error: 10.7 → 5.4

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.7340483331695302 \cdot 10^{161}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0416429465285198 \cdot 10^{-239}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.4009849960185782 \cdot 10^{-242}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.1437013582095321 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

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) (a1 * a2)) <= -1.7340483331695302e+161)) {
		VAR = ((double) (a1 * ((double) (((double) (a2 / b1)) / b2))));
	} else {
		double VAR_1;
		if ((((double) (a1 * a2)) <= -3.04164294652852e-239)) {
			VAR_1 = ((double) (((double) (((double) (a1 * a2)) / b1)) * ((double) (1.0 / b2))));
		} else {
			double VAR_2;
			if ((((double) (a1 * a2)) <= 3.4009849960185782e-242)) {
				VAR_2 = ((double) (a1 * ((double) (((double) (a2 / b1)) / b2))));
			} else {
				double VAR_3;
				if ((((double) (a1 * a2)) <= 3.143701358209532e+86)) {
					VAR_3 = ((double) (((double) (((double) (a1 * a2)) / b2)) / b1));
				} 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	10.7
Target	11.0
Herbie	5.4

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -1.7340483331695302e161 or -3.0416429465285198e-239 < (* a1 a2) < 3.4009849960185782e-242 or 3.1437013582095321e86 < (* a1 a2)

   1. Initial program 19.2

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

   3. Applied associate-/r* 19.4

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

   5. Applied *-un-lft-identity 19.4

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

   6. Applied *-un-lft-identity 19.4

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

   7. Applied times-frac 11.5

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

   8. Applied times-frac 7.3

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

   9. Simplified 7.3

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

   if -1.7340483331695302e161 < (* a1 a2) < -3.0416429465285198e-239

   1. Initial program 3.8

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

   3. Applied associate-/r* 4.3

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

   5. Applied div-inv 4.4

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

   if 3.4009849960185782e-242 < (* a1 a2) < 3.1437013582095321e86

   1. Initial program 3.3

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

   3. Applied associate-/r* 2.9

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

   5. Applied div-inv 3.0

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

   7. Applied associate-*l/ 3.1

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

   8. Simplified 3.1

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.7340483331695302 \cdot 10^{161}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0416429465285198 \cdot 10^{-239}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.4009849960185782 \cdot 10^{-242}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.1437013582095321 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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