Quotient of products

Average Error: 11.4 → 5.2

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty \lor \neg \left(a1 \cdot a2 \le -4.0878819804843486 \cdot 10^{-254} \lor \neg \left(a1 \cdot a2 \le 1.7944304405375 \cdot 10^{-310} \lor \neg \left(a1 \cdot a2 \le 3.0403037297418217 \cdot 10^{246}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double temp;
	if ((((a1 * a2) <= -inf.0) || !(((a1 * a2) <= -4.0878819804843486e-254) || !(((a1 * a2) <= 1.7944304405375e-310) || !((a1 * a2) <= 3.0403037297418217e+246))))) {
		temp = ((a1 / b1) * (a2 / b2));
	} else {
		temp = ((a1 * a2) * ((1.0 / b1) / b2));
	}
	return temp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.0
Herbie	5.2

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

Derivation

1. Split input into 2 regimes

2. if (* a1 a2) < -inf.0 or -4.0878819804843486e-254 < (* a1 a2) < 1.7944304405375e-310 or 3.0403037297418217e+246 < (* a1 a2)

   1. Initial program 27.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   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) < -4.0878819804843486e-254 or 1.7944304405375e-310 < (* a1 a2) < 3.0403037297418217e+246

   1. Initial program 5.5

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

   3. Applied div-inv 5.8

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

   5. Applied associate-/r* 5.5

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty \lor \neg \left(a1 \cdot a2 \le -4.0878819804843486 \cdot 10^{-254} \lor \neg \left(a1 \cdot a2 \le 1.7944304405375 \cdot 10^{-310} \lor \neg \left(a1 \cdot a2 \le 3.0403037297418217 \cdot 10^{246}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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