Quotient of products

Average Error: 11.2 → 6.0

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.21410678211229495 \cdot 10^{303}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.740095961455516 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.02145443568749372 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.24212309305567693 \cdot 10^{277}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \end{array}\]

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 VAR;
	if (((b1 * b2) <= -4.214106782112295e+303)) {
		VAR = ((a1 / b1) * (a2 / b2));
	} else {
		double VAR_1;
		if (((b1 * b2) <= -4.740095961455516e-226)) {
			VAR_1 = (1.0 / (((b1 * b2) / a1) / a2));
		} else {
			double VAR_2;
			if (((b1 * b2) <= 5.021454435687494e-211)) {
				VAR_2 = (((a1 * a2) / b1) / b2);
			} else {
				double VAR_3;
				if (((b1 * b2) <= 1.242123093055677e+277)) {
					VAR_3 = ((a1 * a2) * (1.0 / (b1 * b2)));
				} else {
					VAR_3 = (1.0 / ((b1 / a1) * (b2 / a2)));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.6
Herbie	6.0

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

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -4.214106782112295e+303

   1. Initial program 21.1

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

   3. Applied times-frac 2.0

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

   if -4.214106782112295e+303 < (* b1 b2) < -4.740095961455516e-226

   1. Initial program 4.9

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

   3. Applied clear-num 5.3

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

   5. Applied associate-/r* 5.6

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

   if -4.740095961455516e-226 < (* b1 b2) < 5.021454435687494e-211

   1. Initial program 35.2

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

   3. Applied associate-/r* 16.2

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

   if 5.021454435687494e-211 < (* b1 b2) < 1.242123093055677e+277

   1. Initial program 4.6

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

   3. Applied div-inv 4.7

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

   if 1.242123093055677e+277 < (* b1 b2)

   1. Initial program 20.6

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

   3. Applied clear-num 20.7

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

   5. Applied times-frac 2.8

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{a1} \cdot \frac{b2}{a2}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -4.21410678211229495 \cdot 10^{303}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.740095961455516 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.02145443568749372 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.24212309305567693 \cdot 10^{277}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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