Average Error: 11.0 → 5.2
Time: 18.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.4533836186065774 \cdot 10^{+266}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -5.538350108544492 \cdot 10^{-165}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 4.167031796800163 \cdot 10^{-112}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 1.387016818262132 \cdot 10^{+154}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -1.4533836186065774 \cdot 10^{+266}:\\
\;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \le -5.538350108544492 \cdot 10^{-165}:\\
\;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\

\mathbf{elif}\;b1 \cdot b2 \le 4.167031796800163 \cdot 10^{-112}:\\
\;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\

\mathbf{elif}\;b1 \cdot b2 \le 1.387016818262132 \cdot 10^{+154}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5595715 = a1;
        double r5595716 = a2;
        double r5595717 = r5595715 * r5595716;
        double r5595718 = b1;
        double r5595719 = b2;
        double r5595720 = r5595718 * r5595719;
        double r5595721 = r5595717 / r5595720;
        return r5595721;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5595722 = b1;
        double r5595723 = b2;
        double r5595724 = r5595722 * r5595723;
        double r5595725 = -1.4533836186065774e+266;
        bool r5595726 = r5595724 <= r5595725;
        double r5595727 = a1;
        double r5595728 = r5595727 / r5595722;
        double r5595729 = a2;
        double r5595730 = r5595723 / r5595729;
        double r5595731 = r5595728 / r5595730;
        double r5595732 = -5.538350108544492e-165;
        bool r5595733 = r5595724 <= r5595732;
        double r5595734 = r5595727 / r5595724;
        double r5595735 = r5595734 * r5595729;
        double r5595736 = 4.167031796800163e-112;
        bool r5595737 = r5595724 <= r5595736;
        double r5595738 = cbrt(r5595729);
        double r5595739 = r5595723 / r5595738;
        double r5595740 = r5595727 / r5595739;
        double r5595741 = r5595738 / r5595722;
        double r5595742 = r5595738 * r5595741;
        double r5595743 = r5595740 * r5595742;
        double r5595744 = 1.387016818262132e+154;
        bool r5595745 = r5595724 <= r5595744;
        double r5595746 = r5595724 / r5595729;
        double r5595747 = r5595727 / r5595746;
        double r5595748 = r5595745 ? r5595747 : r5595743;
        double r5595749 = r5595737 ? r5595743 : r5595748;
        double r5595750 = r5595733 ? r5595735 : r5595749;
        double r5595751 = r5595726 ? r5595731 : r5595750;
        return r5595751;
}

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

Original11.0
Target10.8
Herbie5.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* b1 b2) < -1.4533836186065774e+266

    1. Initial program 19.4

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

    3. Applied associate-/l*19.0

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

    5. Applied *-un-lft-identity19.0

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

    6. Applied times-frac8.4

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

    7. Simplified8.4

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

    9. Applied associate-/r*3.5

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

    if -1.4533836186065774e+266 < (* b1 b2) < -5.538350108544492e-165

    1. Initial program 4.6

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

    3. Applied associate-/l*4.8

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

    5. Applied *-un-lft-identity4.8

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

    6. Applied times-frac11.1

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

    7. Simplified11.1

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

    9. Applied associate-*r/4.8

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

    10. Applied associate-/r/4.4

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

    if -5.538350108544492e-165 < (* b1 b2) < 4.167031796800163e-112 or 1.387016818262132e+154 < (* b1 b2)

    1. Initial program 18.7

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

    3. Applied associate-/l*18.4

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

    5. Applied add-cube-cbrt18.9

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

    6. Applied times-frac11.7

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

    7. Applied *-un-lft-identity11.7

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

    8. Applied times-frac7.7

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

    9. Simplified7.5

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

    if 4.167031796800163e-112 < (* b1 b2) < 1.387016818262132e+154

    1. Initial program 3.0

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

    3. Applied associate-/l*3.1

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

    5. Applied *-un-lft-identity3.1

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

    6. Applied times-frac10.5

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

    7. Simplified10.5

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

    9. Applied associate-*r/3.1

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1 \cdot b2}{a2}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.4533836186065774 \cdot 10^{+266}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -5.538350108544492 \cdot 10^{-165}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{elif}\;b1 \cdot b2 \le 4.167031796800163 \cdot 10^{-112}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 1.387016818262132 \cdot 10^{+154}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \left(\sqrt[3]{a2} \cdot \frac{\sqrt[3]{a2}}{b1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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