Quotient of products

Average Error: 11.7 → 2.5

Time: 8.2s

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 r2633837 = a1;
        double r2633838 = a2;
        double r2633839 = r2633837 * r2633838;
        double r2633840 = b1;
        double r2633841 = b2;
        double r2633842 = r2633840 * r2633841;
        double r2633843 = r2633839 / r2633842;
        return r2633843;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r2633844 = a1;
        double r2633845 = a2;
        double r2633846 = r2633844 * r2633845;
        double r2633847 = 4.966960556448252e-69;
        bool r2633848 = r2633846 <= r2633847;
        double r2633849 = cbrt(r2633845);
        double r2633850 = b2;
        double r2633851 = cbrt(r2633850);
        double r2633852 = r2633849 / r2633851;
        double r2633853 = b1;
        double r2633854 = cbrt(r2633853);
        double r2633855 = r2633852 / r2633854;
        double r2633856 = r2633855 * r2633855;
        double r2633857 = r2633856 * r2633844;
        double r2633858 = r2633855 * r2633857;
        double r2633859 = 8.502441457836898e+166;
        bool r2633860 = r2633846 <= r2633859;
        double r2633861 = r2633846 / r2633853;
        double r2633862 = r2633861 / r2633850;
        double r2633863 = r2633860 ? r2633862 : r2633858;
        double r2633864 = r2633848 ? r2633858 : r2633863;
        return r2633864;
}

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 div-inv 10.3

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

   6. Applied associate-*l* 10.4

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

   7. Simplified 10.4

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

   9. Applied add-cube-cbrt 11.0

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

   10. Applied add-cube-cbrt 11.1

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

   11. Applied add-cube-cbrt 11.2

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

   12. Applied times-frac 11.2

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

   13. Applied times-frac 7.5

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

   14. Applied associate-*r* 2.8

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

   15. 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 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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