Average Error: 11.3 → 5.0
Time: 3.1s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]
\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}
double f(double a1, double a2, double b1, double b2) {
        double r142689 = a1;
        double r142690 = a2;
        double r142691 = r142689 * r142690;
        double r142692 = b1;
        double r142693 = b2;
        double r142694 = r142692 * r142693;
        double r142695 = r142691 / r142694;
        return r142695;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r142696 = a1;
        double r142697 = a2;
        double r142698 = r142696 * r142697;
        double r142699 = -6.087120586968689e+188;
        bool r142700 = r142698 <= r142699;
        double r142701 = b2;
        double r142702 = r142697 / r142701;
        double r142703 = b1;
        double r142704 = r142702 / r142703;
        double r142705 = r142696 * r142704;
        double r142706 = -3.0178565167744215e-307;
        bool r142707 = r142698 <= r142706;
        double r142708 = r142697 * r142696;
        double r142709 = 1.0;
        double r142710 = r142709 / r142701;
        double r142711 = r142710 / r142703;
        double r142712 = r142708 * r142711;
        double r142713 = 5.3291942180717e-313;
        bool r142714 = r142698 <= r142713;
        double r142715 = cbrt(r142696);
        double r142716 = r142715 * r142715;
        double r142717 = r142716 / r142709;
        double r142718 = r142715 / r142703;
        double r142719 = r142718 * r142702;
        double r142720 = r142717 * r142719;
        double r142721 = 4.488164189797612e+154;
        bool r142722 = r142698 <= r142721;
        double r142723 = r142722 ? r142712 : r142705;
        double r142724 = r142714 ? r142720 : r142723;
        double r142725 = r142707 ? r142712 : r142724;
        double r142726 = r142700 ? r142705 : r142725;
        return r142726;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.3
Target11.7
Herbie5.0
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

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-frac11.2

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

    5. Applied div-inv11.2

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

    6. Applied associate-*l*9.8

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

    7. Simplified9.8

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

    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-frac13.9

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

    5. Applied div-inv13.9

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

    6. Applied associate-*l*13.6

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

    7. Simplified13.5

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

    9. Applied *-un-lft-identity13.5

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

    10. Applied div-inv13.6

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

    11. Applied times-frac10.3

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

    12. Applied associate-*r*4.6

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

    13. Simplified4.6

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

    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-frac2.9

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

    5. Applied *-un-lft-identity2.9

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

    6. Applied add-cube-cbrt3.2

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

    7. Applied times-frac3.2

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

    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 simplification5.0

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

Reproduce

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

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

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