Average Error: 11.1 → 5.2
Time: 2.3s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\
\;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r143973 = a1;
        double r143974 = a2;
        double r143975 = r143973 * r143974;
        double r143976 = b1;
        double r143977 = b2;
        double r143978 = r143976 * r143977;
        double r143979 = r143975 / r143978;
        return r143979;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r143980 = a1;
        double r143981 = a2;
        double r143982 = r143980 * r143981;
        double r143983 = b1;
        double r143984 = b2;
        double r143985 = r143983 * r143984;
        double r143986 = r143982 / r143985;
        double r143987 = -3.949560208634886e-267;
        bool r143988 = r143986 <= r143987;
        double r143989 = 0.0;
        bool r143990 = r143986 <= r143989;
        double r143991 = r143984 / r143981;
        double r143992 = r143980 / r143991;
        double r143993 = r143992 / r143983;
        double r143994 = 1.4604994925847344e+301;
        bool r143995 = r143986 <= r143994;
        double r143996 = cbrt(r143983);
        double r143997 = r143996 * r143996;
        double r143998 = r143980 / r143997;
        double r143999 = r143981 / r143996;
        double r144000 = r143998 * r143999;
        double r144001 = r144000 / r143984;
        double r144002 = r143995 ? r143986 : r144001;
        double r144003 = r143990 ? r143993 : r144002;
        double r144004 = r143988 ? r143986 : r144003;
        return r144004;
}

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

Original11.1
Target11.2
Herbie5.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -3.949560208634886e-267 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4604994925847344e+301

    1. Initial program 4.5

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

    3. Applied associate-/r*9.4

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

    5. Applied clear-num9.6

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

    7. Applied associate-/r/9.8

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

    8. Applied associate-/r*9.9

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

    9. Simplified9.5

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

    10. Taylor expanded around 0 4.5

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

    if -3.949560208634886e-267 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program 12.1

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

    3. Applied associate-/r*6.8

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

    5. Applied clear-num7.3

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

    7. Applied associate-/r/7.3

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

    8. Applied associate-/r*6.8

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

    9. Simplified6.8

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

    11. Applied associate-/l*4.5

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

    if 1.4604994925847344e+301 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 62.4

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

    3. Applied associate-/r*48.4

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

    5. Applied add-cube-cbrt48.6

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

    6. Applied times-frac14.1

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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