Quotient of products

Average Error: 11.3 → 2.4

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.34772011203715 \cdot 10^{-311}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.60497432469669966 \cdot 10^{297}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -inf.0)) {
		VAR = ((double) (((double) (a1 * ((double) (a2 / b1)))) * ((double) (1.0 / b2))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.3477201120371e-311)) {
			VAR_1 = ((double) (((double) (a1 * a2)) * ((double) (1.0 / ((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)))) <= 4.6049743246966997e+297)) {
					VAR_3 = ((double) (((double) (a1 * a2)) * ((double) (1.0 / ((double) (b1 * b2))))));
				} else {
					VAR_3 = ((double) (((double) (a1 / b1)) * ((double) (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	11.3
Target	11.5
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied times-frac 11.3

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

   5. Applied div-inv 11.4

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

   6. Applied associate-*r* 18.2

      \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}}\]
   7. Using strategy rm

   8. Applied div-inv 18.2

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

   9. Applied associate-*l* 19.0

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

   10. Simplified 18.9

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.3477201120371e-311 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 4.6049743246966997e+297

   1. Initial program 0.8

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

   3. Applied div-inv 1.2

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

   if -1.3477201120371e-311 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 4.6049743246966997e+297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 22.2

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

   3. Applied times-frac 2.9

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.34772011203715 \cdot 10^{-311}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.60497432469669966 \cdot 10^{297}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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