Quotient of products

Average Error: 11.4 → 2.5

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.0214919528087893 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1}}{\sqrt[3]{b2}} \cdot \left(\frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.2762806338293389 \cdot 10^{303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1}}{\sqrt[3]{b2}} \cdot \left(\frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\right)\\ \end{array}\]

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) / (b1 * b2)) <= -inf.0)) {
		temp = (((a1 / b1) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2)));
	} else {
		double temp_1;
		if ((((a1 * a2) / (b1 * b2)) <= -1.0214919528087893e-301)) {
			temp_1 = (1.0 / ((b1 * b2) / (a1 * a2)));
		} else {
			double temp_2;
			if ((((a1 * a2) / (b1 * b2)) <= 0.0)) {
				temp_2 = ((((cbrt(a1) * cbrt(a1)) / 1.0) / cbrt(b2)) * (((cbrt(a1) / b1) / cbrt(b2)) * (a2 / cbrt(b2))));
			} else {
				double temp_3;
				if ((((a1 * a2) / (b1 * b2)) <= 2.276280633829339e+303)) {
					temp_3 = (1.0 / ((b1 * b2) / (a1 * a2)));
				} else {
					temp_3 = ((((cbrt(a1) * cbrt(a1)) / 1.0) / cbrt(b2)) * (((cbrt(a1) / b1) / cbrt(b2)) * (a2 / cbrt(b2))));
				}
				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	11.4
Target	11.2
Herbie	2.5

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

Derivation

1. Split input into 3 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 times-frac 13.4

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

   5. Applied add-cube-cbrt 14.4

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

   6. Applied *-un-lft-identity 14.4

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

   7. Applied times-frac 14.3

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

   8. Applied associate-*r* 12.8

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

   9. Simplified 12.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.0214919528087893e-301 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 2.276280633829339e+303

   1. Initial program 0.8

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

   3. Applied clear-num 1.1

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

   if -1.0214919528087893e-301 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 2.276280633829339e+303 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 22.1

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

   3. Applied times-frac 3.3

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

   5. Applied add-cube-cbrt 3.6

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

   6. Applied *-un-lft-identity 3.6

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

   7. Applied times-frac 3.6

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

   8. Applied associate-*r* 4.1

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

   9. Simplified 4.1

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{a2}{\sqrt[3]{b2}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 4.1

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

   12. Applied add-cube-cbrt 4.2

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

   13. Applied times-frac 4.2

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

   14. Applied times-frac 3.6

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

   15. Applied associate-*l* 3.6

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.0214919528087893 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1}}{\sqrt[3]{b2}} \cdot \left(\frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.2762806338293389 \cdot 10^{303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1}}{\sqrt[3]{b2}} \cdot \left(\frac{\frac{\sqrt[3]{a1}}{b1}}{\sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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