Quotient of products

Average Error: 11.2 → 2.7

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.40588661831116472 \cdot 10^{-296}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.37991482612767615 \cdot 10^{284}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

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 ((((a1 * a2) / (b1 * b2)) <= -inf.0)) {
		VAR = ((a1 / b1) / (b2 / a2));
	} else {
		double VAR_1;
		if ((((a1 * a2) / (b1 * b2)) <= -6.405886618311165e-296)) {
			VAR_1 = ((a1 * a2) / (b1 * b2));
		} else {
			double VAR_2;
			if ((((a1 * a2) / (b1 * b2)) <= -0.0)) {
				VAR_2 = (a1 / ((b1 / a2) * b2));
			} else {
				double VAR_3;
				if ((((a1 * a2) / (b1 * b2)) <= 2.379914826127676e+284)) {
					VAR_3 = ((a1 * a2) / (b1 * b2));
				} else {
					VAR_3 = ((a1 / b1) / (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	11.1
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or 2.379914826127676e+284 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 60.3

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

   3. Applied associate-/l* 40.1

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

   5. Applied *-un-lft-identity 40.1

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

   6. Applied times-frac 15.9

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

   7. Applied associate-/r* 9.0

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

   8. Simplified 9.0

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -6.405886618311165e-296 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 2.379914826127676e+284

   1. Initial program 0.8

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

   if -6.405886618311165e-296 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 13.1

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

   3. Applied associate-/l* 6.8

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

   5. Applied associate-/l* 3.9

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

   7. Applied associate-/r/ 3.8

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.40588661831116472 \cdot 10^{-296}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.37991482612767615 \cdot 10^{284}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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