Quotient of products

Average Error: 11.4 → 2.3

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.4671499395447483 \cdot 10^{-308} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.3804268269775264 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

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) / (b1 * b2)) <= -inf.0) || !((((a1 * a2) / (b1 * b2)) <= -1.4671499395447483e-308) || !((((a1 * a2) / (b1 * b2)) <= 0.0) || !(((a1 * a2) / (b1 * b2)) <= 1.3804268269775264e+294))))) {
		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	2.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -1.4671499395447483e-308 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 1.3804268269775264e+294 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.6

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

   3. Applied times-frac 3.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.4671499395447483e-308 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.3804268269775264e+294

   1. Initial program 0.9

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

   3. Applied div-inv 1.3

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

   5. Applied associate-/r* 1.3

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.4671499395447483 \cdot 10^{-308} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.3804268269775264 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

Reproduce

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

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

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