Quotient of products

Average Error: 11.7 → 2.5

Time: 35.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le 4.966960556448252 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \left(\left(\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}\right) \cdot a1\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 8.502441457836898 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \left(\left(\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}\right) \cdot a1\right)\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r8576853 = a1;
        double r8576854 = a2;
        double r8576855 = r8576853 * r8576854;
        double r8576856 = b1;
        double r8576857 = b2;
        double r8576858 = r8576856 * r8576857;
        double r8576859 = r8576855 / r8576858;
        return r8576859;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r8576860 = a1;
        double r8576861 = a2;
        double r8576862 = r8576860 * r8576861;
        double r8576863 = 4.966960556448252e-69;
        bool r8576864 = r8576862 <= r8576863;
        double r8576865 = cbrt(r8576861);
        double r8576866 = b2;
        double r8576867 = cbrt(r8576866);
        double r8576868 = r8576865 / r8576867;
        double r8576869 = b1;
        double r8576870 = cbrt(r8576869);
        double r8576871 = r8576868 / r8576870;
        double r8576872 = r8576871 * r8576871;
        double r8576873 = r8576872 * r8576860;
        double r8576874 = r8576871 * r8576873;
        double r8576875 = 8.502441457836898e+166;
        bool r8576876 = r8576862 <= r8576875;
        double r8576877 = r8576862 / r8576869;
        double r8576878 = r8576877 / r8576866;
        double r8576879 = r8576876 ? r8576878 : r8576874;
        double r8576880 = r8576864 ? r8576874 : r8576879;
        return r8576880;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.7
Target	11.4
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if (* a1 a2) < 4.966960556448252e-69 or 8.502441457836898e+166 < (* a1 a2)

   1. Initial program 13.4

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

   3. Applied times-frac 10.3

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

   5. Applied associate-*l/ 10.9

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

   7. Applied *-un-lft-identity 10.9

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

   8. Applied times-frac 10.4

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

   9. Simplified 10.4

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

   11. Applied add-cube-cbrt 11.0

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

   12. Applied add-cube-cbrt 11.1

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

   13. Applied add-cube-cbrt 11.2

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

   14. Applied times-frac 11.2

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

   15. Applied times-frac 7.5

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

   16. Applied associate-*r* 2.8

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

   17. Simplified 2.2

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

   if 4.966960556448252e-69 < (* a1 a2) < 8.502441457836898e+166

   1. Initial program 3.5

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

   3. Applied associate-/r* 3.9

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le 4.966960556448252 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \left(\left(\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}\right) \cdot a1\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 8.502441457836898 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \left(\left(\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}\right) \cdot a1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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