Quotient of products

Average Error: 11.0 → 4.8

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.7227674015649386 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right)} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.51533523845605849 \cdot 10^{-253}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.6759054616252046 \cdot 10^{246}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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) (a1 * a2)) <= -2.7227674015649386e-288)) {
		VAR = ((double) (((double) (((double) (a1 * a2)) / ((double) (((double) (((double) cbrt(b2)) * ((double) cbrt(b2)))) * ((double) (((double) cbrt(b1)) * ((double) cbrt(b1)))))))) * ((double) (((double) (1.0 / ((double) cbrt(b1)))) / ((double) cbrt(b2))))));
	} else {
		double VAR_1;
		if ((((double) (a1 * a2)) <= 2.5153352384560585e-253)) {
			VAR_1 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
		} else {
			double VAR_2;
			if ((((double) (a1 * a2)) <= 1.6759054616252046e+246)) {
				VAR_2 = ((double) (((double) (a1 * a2)) * ((double) (((double) (1.0 / b1)) / b2))));
			} else {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			}
			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	11.0
Target	11.2
Herbie	4.8

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -2.7227674015649386e-288

   1. Initial program 9.2

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

   3. Applied div-inv 9.5

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

   5. Applied associate-/r* 9.3

      \[\leadsto \left(a1 \cdot a2\right) \cdot \color{blue}{\frac{\frac{1}{b1}}{b2}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 10.0

      \[\leadsto \left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{\color{blue}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}}\]

   8. Applied add-cube-cbrt 10.2

      \[\leadsto \left(a1 \cdot a2\right) \cdot \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}\]

   9. Applied *-un-lft-identity 10.2

      \[\leadsto \left(a1 \cdot a2\right) \cdot \frac{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}\]

   10. Applied times-frac 10.2

       \[\leadsto \left(a1 \cdot a2\right) \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{1}{\sqrt[3]{b1}}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}\]

   11. Applied times-frac 10.2

       \[\leadsto \left(a1 \cdot a2\right) \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\right)}\]

   12. Applied associate-*r* 5.8

       \[\leadsto \color{blue}{\left(\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}\right) \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}}\]

   13. Simplified 5.8

       \[\leadsto \color{blue}{\frac{a1 \cdot a2}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right)}} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\]

   if -2.7227674015649386e-288 < (* a1 a2) < 2.51533523845605849e-253 or 1.6759054616252046e246 < (* a1 a2)

   1. Initial program 22.7

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

   3. Applied times-frac 3.7

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

   if 2.51533523845605849e-253 < (* a1 a2) < 1.6759054616252046e246

   1. Initial program 4.5

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

   3. Applied div-inv 4.8

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

   5. Applied associate-/r* 4.6

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

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.7227674015649386 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right)} \cdot \frac{\frac{1}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.51533523845605849 \cdot 10^{-253}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.6759054616252046 \cdot 10^{246}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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