Quotient of products

Average Error: 11.7 → 5.3

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.1847928602155 \cdot 10^{222} \lor \neg \left(b1 \cdot b2 \le -2.65012901835582308 \cdot 10^{-171} \lor \neg \left(b1 \cdot b2 \le 1.9618991491095908 \cdot 10^{-204} \lor \neg \left(b1 \cdot b2 \le 4.8487651961466998 \cdot 10^{178}\right)\right)\right):\\ \;\;\;\;\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \end{array}\]

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) (b1 * b2)) <= -1.1847928602155e+222) || !((((double) (b1 * b2)) <= -2.650129018355823e-171) || !((((double) (b1 * b2)) <= 1.9618991491095908e-204) || !(((double) (b1 * b2)) <= 4.8487651961467e+178))))) {
		VAR = ((double) (((double) (((double) (a1 / b1)) * a2)) * ((double) (1.0 / b2))));
	} else {
		VAR = ((double) (a1 / ((double) (((double) (b1 * b2)) / a2))));
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.7
Target	11.0
Herbie	5.3

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

Derivation

1. Split input into 2 regimes

2. if (* b1 b2) < -1.1847928602155e222 or -2.65012901835582308e-171 < (* b1 b2) < 1.9618991491095908e-204 or 4.8487651961466998e178 < (* b1 b2)

   1. Initial program 21.3

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

   3. Applied times-frac 6.3

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

   5. Applied div-inv 6.3

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

   6. Applied associate-*r* 6.3

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

   if -1.1847928602155e222 < (* b1 b2) < -2.65012901835582308e-171 or 1.9618991491095908e-204 < (* b1 b2) < 4.8487651961466998e178

   1. Initial program 4.4

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

   3. Applied associate-/l* 4.4

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

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.1847928602155 \cdot 10^{222} \lor \neg \left(b1 \cdot b2 \le -2.65012901835582308 \cdot 10^{-171} \lor \neg \left(b1 \cdot b2 \le 1.9618991491095908 \cdot 10^{-204} \lor \neg \left(b1 \cdot b2 \le 4.8487651961466998 \cdot 10^{178}\right)\right)\right):\\ \;\;\;\;\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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