Average Error: 10.8 → 5.7
Time: 15.2s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.023233587679621 \cdot 10^{+251}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.1296092868396195 \cdot 10^{-227}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.887172661957492 \cdot 10^{-160}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.1722925108220896 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -3.023233587679621 \cdot 10^{+251}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le -2.1296092868396195 \cdot 10^{-227}:\\
\;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 2.887172661957492 \cdot 10^{-160}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 2.1722925108220896 \cdot 10^{+142}:\\
\;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5875875 = a1;
        double r5875876 = a2;
        double r5875877 = r5875875 * r5875876;
        double r5875878 = b1;
        double r5875879 = b2;
        double r5875880 = r5875878 * r5875879;
        double r5875881 = r5875877 / r5875880;
        return r5875881;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5875882 = a1;
        double r5875883 = a2;
        double r5875884 = r5875882 * r5875883;
        double r5875885 = -3.023233587679621e+251;
        bool r5875886 = r5875884 <= r5875885;
        double r5875887 = b1;
        double r5875888 = r5875882 / r5875887;
        double r5875889 = b2;
        double r5875890 = r5875883 / r5875889;
        double r5875891 = r5875888 * r5875890;
        double r5875892 = -2.1296092868396195e-227;
        bool r5875893 = r5875884 <= r5875892;
        double r5875894 = 1.0;
        double r5875895 = r5875894 / r5875887;
        double r5875896 = r5875884 * r5875895;
        double r5875897 = r5875896 / r5875889;
        double r5875898 = 2.887172661957492e-160;
        bool r5875899 = r5875884 <= r5875898;
        double r5875900 = 2.1722925108220896e+142;
        bool r5875901 = r5875884 <= r5875900;
        double r5875902 = r5875883 / r5875887;
        double r5875903 = r5875882 * r5875902;
        double r5875904 = r5875903 / r5875889;
        double r5875905 = r5875901 ? r5875897 : r5875904;
        double r5875906 = r5875899 ? r5875891 : r5875905;
        double r5875907 = r5875893 ? r5875897 : r5875906;
        double r5875908 = r5875886 ? r5875891 : r5875907;
        return r5875908;
}

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

Original10.8
Target10.7
Herbie5.7
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -3.023233587679621e+251 or -2.1296092868396195e-227 < (* a1 a2) < 2.887172661957492e-160

    1. Initial program 17.9

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

    3. Applied times-frac5.2

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

    if -3.023233587679621e+251 < (* a1 a2) < -2.1296092868396195e-227 or 2.887172661957492e-160 < (* a1 a2) < 2.1722925108220896e+142

    1. Initial program 4.3

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

    3. Applied associate-/r*4.3

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

    5. Applied *-un-lft-identity4.3

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

    6. Applied associate-/r*4.3

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

    7. Simplified11.5

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

    9. Applied div-inv11.6

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

    10. Applied associate-*r*4.4

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

    if 2.1722925108220896e+142 < (* a1 a2)

    1. Initial program 26.0

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

    3. Applied associate-/r*26.9

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

    5. Applied *-un-lft-identity26.9

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

    6. Applied associate-/r*26.9

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

    7. Simplified15.9

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.023233587679621 \cdot 10^{+251}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.1296092868396195 \cdot 10^{-227}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.887172661957492 \cdot 10^{-160}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.1722925108220896 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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