Quotient of products

Average Error: 11.2 → 6.2

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.68668093917740342 \cdot 10^{275}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.08718180476195655 \cdot 10^{-263}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.1382336163668068 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{b2 \cdot \frac{b1}{a2 \cdot a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.24251358724885461 \cdot 10^{126}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((double) (b1 * b2)) <= -6.686680939177403e+275)) {
		VAR = ((double) ((a2 / b1) * (a1 / b2)));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -2.0871818047619565e-263)) {
			VAR_1 = ((double) (a2 * (a1 / ((double) (b1 * b2)))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 3.138233616366807e-212)) {
				VAR_2 = (1.0 / ((double) (b2 * (b1 / ((double) (a2 * a1))))));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 9.242513587248855e+126)) {
					VAR_3 = ((double) (a2 * (a1 / ((double) (b1 * b2)))));
				} else {
					VAR_3 = ((double) ((a2 / b1) * (a1 / 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	11.2
Target	11.0
Herbie	6.2

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -6.68668093917740342e275 or 9.24251358724885461e126 < (* b1 b2)

   1. Initial program 15.8

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

   2. Taylor expanded around 0 15.8

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

   3. Simplified 15.5

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

   5. Applied *-un-lft-identity 15.5

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

   6. Applied times-frac 8.0

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

   7. Applied associate-*r* 5.3

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

   8. Simplified 5.3

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

   if -6.68668093917740342e275 < (* b1 b2) < -2.08718180476195655e-263 or 3.1382336163668068e-212 < (* b1 b2) < 9.24251358724885461e126

   1. Initial program 4.8

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

   2. Taylor expanded around 0 4.8

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

   3. Simplified 4.8

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

   if -2.08718180476195655e-263 < (* b1 b2) < 3.1382336163668068e-212

   1. Initial program 37.1

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

   3. Applied clear-num 37.1

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

   4. Simplified 17.9

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.68668093917740342 \cdot 10^{275}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.08718180476195655 \cdot 10^{-263}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.1382336163668068 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{b2 \cdot \frac{b1}{a2 \cdot a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 9.24251358724885461 \cdot 10^{126}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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