Average Error: 10.9 → 5.8
Time: 2.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 = -\infty:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r162840 = a1;
        double r162841 = a2;
        double r162842 = r162840 * r162841;
        double r162843 = b1;
        double r162844 = b2;
        double r162845 = r162843 * r162844;
        double r162846 = r162842 / r162845;
        return r162846;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r162847 = b1;
        double r162848 = b2;
        double r162849 = r162847 * r162848;
        double r162850 = -inf.0;
        bool r162851 = r162849 <= r162850;
        double r162852 = a1;
        double r162853 = r162852 / r162847;
        double r162854 = a2;
        double r162855 = r162854 / r162848;
        double r162856 = r162853 * r162855;
        double r162857 = -4.233204880322462e-264;
        bool r162858 = r162849 <= r162857;
        double r162859 = r162849 / r162854;
        double r162860 = r162852 / r162859;
        double r162861 = 1.3072119399973387e-147;
        bool r162862 = r162849 <= r162861;
        double r162863 = 1.0;
        double r162864 = r162847 / r162852;
        double r162865 = r162864 / r162854;
        double r162866 = r162863 / r162865;
        double r162867 = r162866 / r162848;
        double r162868 = 3.653183490075264e+92;
        bool r162869 = r162849 <= r162868;
        double r162870 = r162869 ? r162860 : r162867;
        double r162871 = r162862 ? r162867 : r162870;
        double r162872 = r162858 ? r162860 : r162871;
        double r162873 = r162851 ? r162856 : r162872;
        return r162873;
}

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

Original10.9
Target11.1
Herbie5.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 20.9

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

    3. Applied times-frac2.0

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

    if -inf.0 < (* b1 b2) < -4.233204880322462e-264 or 1.3072119399973387e-147 < (* b1 b2) < 3.653183490075264e+92

    1. Initial program 4.6

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

    3. Applied associate-/l*4.9

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

    if -4.233204880322462e-264 < (* b1 b2) < 1.3072119399973387e-147 or 3.653183490075264e+92 < (* b1 b2)

    1. Initial program 18.8

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

    3. Applied associate-/r*11.6

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

    5. Applied clear-num11.8

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

    7. Applied associate-/r*8.3

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{b1}{a1}}{a2}}}}{b2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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