Quotient of products

Average Error: 11.3 → 5.0

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

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 temp;
	if (((a1 * a2) <= -6.087120586968689e+188)) {
		temp = (a1 * ((a2 / b2) / b1));
	} else {
		double temp_1;
		if (((a1 * a2) <= -3.0178565167744215e-307)) {
			temp_1 = ((a2 * a1) * ((1.0 / b2) / b1));
		} else {
			double temp_2;
			if (((a1 * a2) <= 5.3291942180717e-313)) {
				temp_2 = (((cbrt(a1) * cbrt(a1)) / 1.0) * ((cbrt(a1) / b1) * (a2 / b2)));
			} else {
				double temp_3;
				if (((a1 * a2) <= 4.488164189797612e+154)) {
					temp_3 = ((a2 * a1) * ((1.0 / b2) / b1));
				} else {
					temp_3 = (a1 * ((a2 / b2) / b1));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.7
Herbie	5.0

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -6.087120586968689e+188 or 4.488164189797612e+154 < (* a1 a2)

   1. Initial program 31.5

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

   3. Applied times-frac 11.2

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

   5. Applied div-inv 11.2

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

   6. Applied associate-*l* 9.8

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

   7. Simplified 9.8

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

   if -6.087120586968689e+188 < (* a1 a2) < -3.0178565167744215e-307 or 5.3291942180717e-313 < (* a1 a2) < 4.488164189797612e+154

   1. Initial program 4.5

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

   3. Applied times-frac 13.9

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

   5. Applied div-inv 13.9

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

   6. Applied associate-*l* 13.6

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

   7. Simplified 13.5

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

   9. Applied *-un-lft-identity 13.5

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

   10. Applied div-inv 13.6

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

   11. Applied times-frac 10.3

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

   12. Applied associate-*r* 4.6

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

   13. Simplified 4.6

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

   if -3.0178565167744215e-307 < (* a1 a2) < 5.3291942180717e-313

   1. Initial program 21.6

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

   3. Applied times-frac 2.9

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

   5. Applied *-un-lft-identity 2.9

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

   6. Applied add-cube-cbrt 3.2

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

   7. Applied times-frac 3.2

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

   8. Applied associate-*l* 2.4

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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