Average Error: 11.4 → 3.3
Time: 10.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.510660628541362523085079504704820761096 \cdot 10^{-304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.658174826189562104084513448440762983031 \cdot 10^{285}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.510660628541362523085079504704820761096 \cdot 10^{-304}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.658174826189562104084513448440762983031 \cdot 10^{285}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5293819 = a1;
        double r5293820 = a2;
        double r5293821 = r5293819 * r5293820;
        double r5293822 = b1;
        double r5293823 = b2;
        double r5293824 = r5293822 * r5293823;
        double r5293825 = r5293821 / r5293824;
        return r5293825;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5293826 = a1;
        double r5293827 = a2;
        double r5293828 = r5293826 * r5293827;
        double r5293829 = b1;
        double r5293830 = b2;
        double r5293831 = r5293829 * r5293830;
        double r5293832 = r5293828 / r5293831;
        double r5293833 = -inf.0;
        bool r5293834 = r5293832 <= r5293833;
        double r5293835 = r5293827 / r5293830;
        double r5293836 = r5293829 / r5293835;
        double r5293837 = r5293826 / r5293836;
        double r5293838 = -1.5106606285413625e-304;
        bool r5293839 = r5293832 <= r5293838;
        double r5293840 = 0.0;
        bool r5293841 = r5293832 <= r5293840;
        double r5293842 = r5293827 / r5293829;
        double r5293843 = r5293842 / r5293830;
        double r5293844 = r5293826 * r5293843;
        double r5293845 = 1.658174826189562e+285;
        bool r5293846 = r5293832 <= r5293845;
        double r5293847 = r5293846 ? r5293832 : r5293837;
        double r5293848 = r5293841 ? r5293844 : r5293847;
        double r5293849 = r5293839 ? r5293832 : r5293848;
        double r5293850 = r5293834 ? r5293837 : r5293849;
        return r5293850;
}

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

Original11.4
Target11.1
Herbie3.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or 1.658174826189562e+285 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 60.8

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

    3. Applied associate-/l*40.4

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

    5. Applied associate-/l*13.7

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.5106606285413625e-304 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.658174826189562e+285

    1. Initial program 0.9

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

    if -1.5106606285413625e-304 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program 13.2

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

    3. Applied associate-/r*5.9

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

    5. Applied *-un-lft-identity5.9

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

    6. Applied *-un-lft-identity5.9

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

    7. Applied times-frac3.2

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

    8. Applied times-frac3.9

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

    9. Simplified3.9

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.510660628541362523085079504704820761096 \cdot 10^{-304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.658174826189562104084513448440762983031 \cdot 10^{285}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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