Quotient of products

Average Error: 11.2 → 2.7

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.5067477990392214 \cdot 10^{290} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5628228799811281 \cdot 10^{-309} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.1836115191722058 \cdot 10^{-304} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.82873126065355613 \cdot 10^{295}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

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) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -2.5067477990392214e+290) || !((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.56282287998113e-309) || !((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 2.1836115191722058e-304) || !(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 1.8287312606535561e+295))))) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		VAR = ((double) (((double) (a1 * a2)) * ((double) (((double) (1.0 / b1)) / b2))));
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.7
Herbie	2.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -2.5067477990392214e+290 or -1.56282287998113e-309 < (/ (* a1 a2) (* b1 b2)) < 2.1836115191722058e-304 or 1.8287312606535561e+295 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.2

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

   3. Applied times-frac 4.8

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

   if -2.5067477990392214e+290 < (/ (* a1 a2) (* b1 b2)) < -1.56282287998113e-309 or 2.1836115191722058e-304 < (/ (* a1 a2) (* b1 b2)) < 1.8287312606535561e+295

   1. Initial program 0.8

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

   3. Applied associate-/r* 8.0

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

   5. Applied *-un-lft-identity 8.0

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

   6. Applied div-inv 8.1

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

   7. Applied times-frac 1.2

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

   8. Simplified 1.2

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right)} \cdot \frac{\frac{1}{b1}}{b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.5067477990392214 \cdot 10^{290} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5628228799811281 \cdot 10^{-309} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.1836115191722058 \cdot 10^{-304} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.82873126065355613 \cdot 10^{295}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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