Quotient of products

Average Error: 11.4 → 2.5

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.6844398141843 \cdot 10^{-312}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.55317884451173137 \cdot 10^{276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return ((a1 * a2) / (b1 * b2));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double temp;
	if ((((a1 * a2) / (b1 * b2)) <= -inf.0)) {
		temp = ((a1 / b1) * (a2 / b2));
	} else {
		double temp_1;
		if ((((a1 * a2) / (b1 * b2)) <= -6.6844398141843e-312)) {
			temp_1 = ((a1 * a2) * ((1.0 / b1) / b2));
		} else {
			double temp_2;
			if ((((a1 * a2) / (b1 * b2)) <= 0.0)) {
				temp_2 = ((a1 / b1) * (a2 / b2));
			} else {
				double temp_3;
				if ((((a1 * a2) / (b1 * b2)) <= 8.553178844511731e+276)) {
					temp_3 = ((a1 * a2) / (b1 * b2));
				} else {
					temp_3 = ((a1 / b1) * (a2 / b2));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -6.6844398141843e-312 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 8.553178844511731e+276 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.7

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

   3. Applied times-frac 4.3

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -6.6844398141843e-312

   1. Initial program 0.9

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

   3. Applied associate-/r* 8.9

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

   5. Applied *-un-lft-identity 8.9

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

   6. Applied div-inv 9.0

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

   7. Applied times-frac 1.4

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

   8. Simplified 1.4

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

   if 0.0 < (/ (* a1 a2) (* b1 b2)) < 8.553178844511731e+276

   1. Initial program 0.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.6844398141843 \cdot 10^{-312}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.55317884451173137 \cdot 10^{276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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