Quotient of products

Average Error: 11.2 → 7.2

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.02847831670121447 \cdot 10^{130}:\\ \;\;\;\;\frac{a2}{\frac{b1}{\frac{a1}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.69759389374256809 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0020672856766477 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{a2}{b1 \cdot b2}}{\frac{1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.9433517635345511 \cdot 10^{207}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \end{array}\]

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 (((a1 * a2) <= -2.0284783167012145e+130)) {
		VAR = (a2 / (b1 / (a1 / b2)));
	} else {
		double VAR_1;
		if (((a1 * a2) <= -1.697593893742568e-139)) {
			VAR_1 = (((a1 * a2) / b1) / b2);
		} else {
			double VAR_2;
			if (((a1 * a2) <= 1.0020672856766477e-117)) {
				VAR_2 = ((a2 / (b1 * b2)) / (1.0 / a1));
			} else {
				double VAR_3;
				if (((a1 * a2) <= 3.943351763534551e+207)) {
					VAR_3 = (((a1 * a2) / b1) / b2);
				} else {
					VAR_3 = (a2 / ((b1 * b2) / a1));
				}
				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	11.3
Herbie	7.2

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -2.0284783167012145e+130

   1. Initial program 24.5

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

   3. Applied *-commutative 24.5

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

   4. Applied associate-/l* 14.1

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

   6. Applied associate-/l* 13.5

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

   if -2.0284783167012145e+130 < (* a1 a2) < -1.697593893742568e-139 or 1.0020672856766477e-117 < (* a1 a2) < 3.943351763534551e+207

   1. Initial program 3.9

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

   3. Applied associate-/r* 3.7

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

   if -1.697593893742568e-139 < (* a1 a2) < 1.0020672856766477e-117

   1. Initial program 12.3

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

   3. Applied *-commutative 12.3

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

   4. Applied associate-/l* 8.3

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

   6. Applied div-inv 8.3

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

   7. Applied associate-/r* 8.2

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

   if 3.943351763534551e+207 < (* a1 a2)

   1. Initial program 35.1

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

   3. Applied *-commutative 35.1

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

   4. Applied associate-/l* 16.0

      \[\leadsto \color{blue}{\frac{a2}{\frac{b1 \cdot b2}{a1}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.02847831670121447 \cdot 10^{130}:\\ \;\;\;\;\frac{a2}{\frac{b1}{\frac{a1}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.69759389374256809 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0020672856766477 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{a2}{b1 \cdot b2}}{\frac{1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.9433517635345511 \cdot 10^{207}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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