Quotient of products

Average Error: 11.3 → 2.7

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.245920718540046 \cdot 10^{283}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.1005372653274905 \cdot 10^{-290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.1783316697451028 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \end{array}\]

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) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.245920718540046e+283)) {
		VAR = ((double) (a1 / ((double) (b1 * ((double) (b2 / a2))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -1.1005372653274905e-290)) {
			VAR_1 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -0.0)) {
				VAR_2 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 2.1783316697451028e+297)) {
					VAR_3 = ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
				} else {
					VAR_3 = ((double) (((double) (a1 * ((double) (((double) (((double) cbrt(a2)) * ((double) cbrt(a2)))) / b1)))) * ((double) (((double) cbrt(a2)) / b2))));
				}
				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	11.3
Target	11.3
Herbie	2.7

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -1.245920718540046e283

   1. Initial program 53.0

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

   3. Applied associate-/l* 26.2

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

   5. Applied *-un-lft-identity 26.2

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

   6. Applied times-frac 15.9

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

   7. Simplified 15.9

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

   if -1.245920718540046e283 < (/ (* a1 a2) (* b1 b2)) < -1.1005372653274905e-290 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 2.1783316697451028e297

   1. Initial program 0.9

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

   if -1.1005372653274905e-290 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 12.5

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

   3. Applied times-frac 2.6

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

   if 2.1783316697451028e297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 60.5

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

   3. Applied associate-/l* 45.2

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

   5. Applied div-inv 45.4

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

   6. Simplified 45.4

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

   8. Applied add-cube-cbrt 45.6

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

   9. Applied times-frac 13.1

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

   10. Applied associate-*r* 9.1

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.245920718540046 \cdot 10^{283}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.1005372653274905 \cdot 10^{-290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.1783316697451028 \cdot 10^{297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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