Average Error: 11.3 → 2.4
Time: 13.9s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r80905 = a1;
        double r80906 = a2;
        double r80907 = r80905 * r80906;
        double r80908 = b1;
        double r80909 = b2;
        double r80910 = r80908 * r80909;
        double r80911 = r80907 / r80910;
        return r80911;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r80912 = a1;
        double r80913 = a2;
        double r80914 = r80912 * r80913;
        double r80915 = b1;
        double r80916 = b2;
        double r80917 = r80915 * r80916;
        double r80918 = r80914 / r80917;
        double r80919 = -inf.0;
        bool r80920 = r80918 <= r80919;
        double r80921 = r80913 / r80916;
        double r80922 = r80921 / r80915;
        double r80923 = r80912 * r80922;
        double r80924 = -1.4141443626392e-314;
        bool r80925 = r80918 <= r80924;
        double r80926 = 0.0;
        bool r80927 = r80918 <= r80926;
        double r80928 = !r80927;
        double r80929 = 1.4318024192509232e+298;
        bool r80930 = r80918 <= r80929;
        bool r80931 = r80928 && r80930;
        bool r80932 = r80925 || r80931;
        double r80933 = r80912 / r80915;
        double r80934 = r80933 * r80921;
        double r80935 = r80932 ? r80918 : r80934;
        double r80936 = r80920 ? r80923 : r80935;
        return r80936;
}

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
Target11.2
Herbie2.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

    1. Initial program 64.0

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    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-inv11.3

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

    6. Applied associate-*l*19.0

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

    7. Simplified18.9

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.4141443626392e-314 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4318024192509232e+298

    1. Initial program 3.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

    if -1.4141443626392e-314 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 1.4318024192509232e+298 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 28.6

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

    3. Applied times-frac3.7

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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