Average Error: 11.6 → 4.6
Time: 40.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\
\;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\

\mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\
\;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\

\mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\
\;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\

\mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\
\;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r88649 = a1;
        double r88650 = a2;
        double r88651 = r88649 * r88650;
        double r88652 = b1;
        double r88653 = b2;
        double r88654 = r88652 * r88653;
        double r88655 = r88651 / r88654;
        return r88655;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r88656 = b1;
        double r88657 = b2;
        double r88658 = r88656 * r88657;
        double r88659 = -9.786216370120351e+209;
        bool r88660 = r88658 <= r88659;
        double r88661 = 1.0;
        double r88662 = a2;
        double r88663 = r88657 / r88662;
        double r88664 = a1;
        double r88665 = r88663 / r88664;
        double r88666 = r88656 * r88665;
        double r88667 = r88661 / r88666;
        double r88668 = -9.111280530104685e-184;
        bool r88669 = r88658 <= r88668;
        double r88670 = cbrt(r88664);
        double r88671 = r88670 * r88670;
        double r88672 = r88658 / r88671;
        double r88673 = r88661 / r88672;
        double r88674 = r88661 / r88662;
        double r88675 = r88674 / r88670;
        double r88676 = r88673 / r88675;
        double r88677 = 4.7400658062009e-320;
        bool r88678 = r88658 <= r88677;
        double r88679 = 8.888412293585825e+181;
        bool r88680 = r88658 <= r88679;
        double r88681 = r88664 / r88656;
        double r88682 = r88681 / r88663;
        double r88683 = r88680 ? r88676 : r88682;
        double r88684 = r88678 ? r88667 : r88683;
        double r88685 = r88669 ? r88676 : r88684;
        double r88686 = r88660 ? r88667 : r88685;
        return r88686;
}

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

Original11.6
Target11.4
Herbie4.6
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -9.786216370120351e+209 or -9.111280530104685e-184 < (* b1 b2) < 4.7400658062009e-320

    1. Initial program 25.7

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

    3. Applied associate-/l*25.3

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

    5. Applied clear-num25.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity25.4

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

    8. Applied *-un-lft-identity25.4

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

    9. Applied times-frac12.9

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

    10. Applied times-frac6.7

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

    11. Simplified6.7

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

    if -9.786216370120351e+209 < (* b1 b2) < -9.111280530104685e-184 or 4.7400658062009e-320 < (* b1 b2) < 8.888412293585825e+181

    1. Initial program 5.2

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

    3. Applied associate-/l*5.2

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

    5. Applied clear-num5.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b1 \cdot b2}{a2}}{a1}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt6.2

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

    8. Applied div-inv6.2

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

    9. Applied times-frac3.8

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

    10. Applied associate-/r*3.7

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

    if 8.888412293585825e+181 < (* b1 b2)

    1. Initial program 16.3

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

    3. Applied associate-/l*15.8

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

    5. Applied *-un-lft-identity15.8

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

    6. Applied times-frac8.8

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

    7. Applied associate-/r*5.3

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

    8. Simplified5.3

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.78621637012035120889222133129827862576 \cdot 10^{209}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.111280530104685398727222154888263046467 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.74006580620091934483000099877771584647 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{b1 \cdot \frac{\frac{b2}{a2}}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.888412293585824693935829272063102395365 \cdot 10^{181}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1 \cdot b2}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}}{\frac{\frac{1}{a2}}{\sqrt[3]{a1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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