Average Error: 11.5 → 4.7
Time: 3.9s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -8.1576239880029206 \cdot 10^{195}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -7.6760268479022844 \cdot 10^{-278}:\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.3851302523781732 \cdot 10^{-273}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.0507310125992406 \cdot 10^{162}:\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -8.1576239880029206 \cdot 10^{195}:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le -7.6760268479022844 \cdot 10^{-278}:\\
\;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\

\mathbf{elif}\;a1 \cdot a2 \le 1.3851302523781732 \cdot 10^{-273}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 2.0507310125992406 \cdot 10^{162}:\\
\;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r170979 = a1;
        double r170980 = a2;
        double r170981 = r170979 * r170980;
        double r170982 = b1;
        double r170983 = b2;
        double r170984 = r170982 * r170983;
        double r170985 = r170981 / r170984;
        return r170985;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r170986 = a1;
        double r170987 = a2;
        double r170988 = r170986 * r170987;
        double r170989 = -8.157623988002921e+195;
        bool r170990 = r170988 <= r170989;
        double r170991 = b1;
        double r170992 = r170987 / r170991;
        double r170993 = b2;
        double r170994 = r170992 / r170993;
        double r170995 = r170986 * r170994;
        double r170996 = -7.676026847902284e-278;
        bool r170997 = r170988 <= r170996;
        double r170998 = cbrt(r170993);
        double r170999 = r170998 * r170998;
        double r171000 = r170988 / r170999;
        double r171001 = 1.0;
        double r171002 = r171001 / r170991;
        double r171003 = r171002 / r170998;
        double r171004 = r171000 * r171003;
        double r171005 = 1.3851302523781732e-273;
        bool r171006 = r170988 <= r171005;
        double r171007 = r170986 / r170991;
        double r171008 = r170987 / r170993;
        double r171009 = r171007 * r171008;
        double r171010 = 2.0507310125992406e+162;
        bool r171011 = r170988 <= r171010;
        double r171012 = r171011 ? r171004 : r170995;
        double r171013 = r171006 ? r171009 : r171012;
        double r171014 = r170997 ? r171004 : r171013;
        double r171015 = r170990 ? r170995 : r171014;
        return r171015;
}

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

Original11.5
Target11.3
Herbie4.7
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -8.157623988002921e+195 or 2.0507310125992406e+162 < (* a1 a2)

    1. Initial program 30.8

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

    3. Applied associate-/r*31.9

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

    5. Applied *-un-lft-identity31.9

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

    6. Applied *-un-lft-identity31.9

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

    7. Applied times-frac18.2

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

    8. Applied times-frac11.8

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

    9. Simplified11.8

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

    if -8.157623988002921e+195 < (* a1 a2) < -7.676026847902284e-278 or 1.3851302523781732e-273 < (* a1 a2) < 2.0507310125992406e+162

    1. Initial program 5.0

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

    3. Applied associate-/r*4.6

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

    5. Applied add-cube-cbrt5.4

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

    6. Applied div-inv5.4

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}\]

    7. Applied times-frac3.5

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}}\]

    if -7.676026847902284e-278 < (* a1 a2) < 1.3851302523781732e-273

    1. Initial program 18.1

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

    3. Applied times-frac3.2

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

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -8.1576239880029206 \cdot 10^{195}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -7.6760268479022844 \cdot 10^{-278}:\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.3851302523781732 \cdot 10^{-273}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.0507310125992406 \cdot 10^{162}:\\ \;\;\;\;\frac{a1 \cdot a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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