Quotient of products

Average Error: 11.4 → 6.2

Time: 7.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -inf.0:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a1}{b2} \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le -3.16685254877843521 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.707841562366554 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.2191177261132069 \cdot 10^{193}:\\ \;\;\;\;\frac{a1}{\left(b1 \cdot b2\right) \cdot \frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

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) (b1 * b2)) <= -inf.0)) {
		VAR = ((double) (((double) (1.0 / b1)) * ((double) (((double) (a1 / b2)) * a2))));
	} else {
		double VAR_1;
		if ((((double) (b1 * b2)) <= -3.1668525487784352e-232)) {
			VAR_1 = ((double) (((double) (a1 / ((double) (b1 * b2)))) / ((double) (1.0 / a2))));
		} else {
			double VAR_2;
			if ((((double) (b1 * b2)) <= 1.707841562366554e-256)) {
				VAR_2 = ((double) (((double) (((double) (a1 * a2)) / b1)) / b2));
			} else {
				double VAR_3;
				if ((((double) (b1 * b2)) <= 1.219117726113207e+193)) {
					VAR_3 = ((double) (a1 / ((double) (((double) (b1 * b2)) * ((double) (1.0 / a2))))));
				} else {
					VAR_3 = ((double) (((double) (a1 / 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	11.4
Target	11.2
Herbie	6.2

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 23.0

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

   3. Applied associate-/l* 22.8

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

   5. Applied *-un-lft-identity 22.8

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

   6. Applied times-frac 9.0

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

   7. Applied *-un-lft-identity 9.0

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

   8. Applied times-frac 4.4

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

   9. Simplified 4.4

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

   11. Applied associate-/r/ 3.1

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

   if -inf.0 < (* b1 b2) < -3.16685254877843521e-232

   1. Initial program 5.2

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

   3. Applied associate-/l* 5.2

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

   5. Applied div-inv 5.2

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

   6. Applied associate-/r* 5.3

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

   if -3.16685254877843521e-232 < (* b1 b2) < 1.707841562366554e-256

   1. Initial program 41.7

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

   3. Applied associate-/r* 17.4

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

   if 1.707841562366554e-256 < (* b1 b2) < 1.2191177261132069e193

   1. Initial program 4.8

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

   3. Applied associate-/l* 5.2

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

   5. Applied div-inv 5.2

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

   if 1.2191177261132069e193 < (* b1 b2)

   1. Initial program 13.9

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

   3. Applied associate-/l* 14.2

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

   5. Applied *-un-lft-identity 14.2

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

   6. Applied times-frac 7.7

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

   7. Applied associate-/r* 4.5

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

   8. Simplified 4.5

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -inf.0:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a1}{b2} \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le -3.16685254877843521 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.707841562366554 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.2191177261132069 \cdot 10^{193}:\\ \;\;\;\;\frac{a1}{\left(b1 \cdot b2\right) \cdot \frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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