Quotient of products

Average Error: 10.9 → 2.4

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.853268977 \cdot 10^{-316}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 9.60018876186239971 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.0670057991566725 \cdot 10^{282}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -inf.0)) {
		VAR = ((double) (((double) (a2 / b1)) / ((double) (b2 / a1))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -8.8532689766022e-316)) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 9.6001887618624e-294)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 4.0670057991566725e+282)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (a1 / ((double) (b1 * ((double) (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	10.9
Target	11.1
Herbie	2.4

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/r* 31.5

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

   4. Simplified 16.7

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

   6. Applied associate-/l* 17.6

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

   7. Simplified 18.4

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

   9. Applied associate-/r* 10.6

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -8.853268977e-316 or 9.60018876186239971e-294 < (/ (* a1 a2) (* b1 b2)) < 4.0670057991566725e282

   1. Initial program 0.7

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

   if -8.853268977e-316 < (/ (* a1 a2) (* b1 b2)) < 9.60018876186239971e-294

   1. Initial program 12.7

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

   3. Applied times-frac 2.1

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

   if 4.0670057991566725e282 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 57.3

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

   3. Applied associate-/l* 42.6

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

   4. Simplified 12.6

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.853268977 \cdot 10^{-316}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 9.60018876186239971 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.0670057991566725 \cdot 10^{282}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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