Quotient of products

Average Error: 11.4 → 2.5

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.33947 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.4104011596369226 \cdot 10^{-267} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.5963299920435087 \cdot 10^{290}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

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) || !((((a1 * a2) / (b1 * b2)) <= -5.3394662477355e-319) || !((((a1 * a2) / (b1 * b2)) <= 8.410401159636923e-267) || !(((a1 * a2) / (b1 * b2)) <= 7.596329992043509e+290))))) {
		temp = ((a1 / b2) * (a2 / b1));
	} else {
		temp = ((a1 * a2) / (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	11.6
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -5.3394662477355e-319 < (/ (* a1 a2) (* b1 b2)) < 8.410401159636923e-267 or 7.596329992043509e+290 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.3

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

   3. Applied *-commutative 25.3

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

   4. Applied times-frac 4.6

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -5.3394662477355e-319 or 8.410401159636923e-267 < (/ (* a1 a2) (* b1 b2)) < 7.596329992043509e+290

   1. Initial program 0.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.33947 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.4104011596369226 \cdot 10^{-267} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.5963299920435087 \cdot 10^{290}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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