Average Error: 11.1 → 4.1
Time: 27.5s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.2640381336687406 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.7958098716047684 \cdot 10^{-250}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.738750803985727 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.2640381336687406 \cdot 10^{-291}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.7958098716047684 \cdot 10^{-250}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.738750803985727 \cdot 10^{+303}:\\
\;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5576364 = a1;
        double r5576365 = a2;
        double r5576366 = r5576364 * r5576365;
        double r5576367 = b1;
        double r5576368 = b2;
        double r5576369 = r5576367 * r5576368;
        double r5576370 = r5576366 / r5576369;
        return r5576370;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5576371 = a1;
        double r5576372 = a2;
        double r5576373 = r5576371 * r5576372;
        double r5576374 = b1;
        double r5576375 = b2;
        double r5576376 = r5576374 * r5576375;
        double r5576377 = r5576373 / r5576376;
        double r5576378 = -2.2640381336687406e-291;
        bool r5576379 = r5576377 <= r5576378;
        double r5576380 = 4.7958098716047684e-250;
        bool r5576381 = r5576377 <= r5576380;
        double r5576382 = r5576371 / r5576374;
        double r5576383 = r5576372 / r5576375;
        double r5576384 = r5576382 * r5576383;
        double r5576385 = 5.738750803985727e+303;
        bool r5576386 = r5576377 <= r5576385;
        double r5576387 = 1.0;
        double r5576388 = r5576376 / r5576373;
        double r5576389 = r5576387 / r5576388;
        double r5576390 = r5576372 / r5576374;
        double r5576391 = r5576371 / r5576375;
        double r5576392 = r5576390 * r5576391;
        double r5576393 = r5576386 ? r5576389 : r5576392;
        double r5576394 = r5576381 ? r5576384 : r5576393;
        double r5576395 = r5576379 ? r5576377 : r5576394;
        return r5576395;
}

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

Original11.1
Target10.8
Herbie4.1
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -2.2640381336687406e-291

    1. Initial program 6.8

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

    if -2.2640381336687406e-291 < (/ (* a1 a2) (* b1 b2)) < 4.7958098716047684e-250

    1. Initial program 12.6

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

    3. Applied times-frac3.8

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

    if 4.7958098716047684e-250 < (/ (* a1 a2) (* b1 b2)) < 5.738750803985727e+303

    1. Initial program 0.8

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

    3. Applied clear-num0.8

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

    4. Taylor expanded around -inf 0.8

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

    if 5.738750803985727e+303 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 60.8

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

    3. Applied clear-num60.8

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

    5. Applied div-inv60.8

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

    6. Simplified45.4

      \[\leadsto 1 \cdot \color{blue}{\left(a2 \cdot \frac{a1}{b1 \cdot b2}\right)}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity45.4

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

    9. Applied times-frac14.0

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

    10. Applied associate-*r*6.2

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

    11. Simplified6.0

      \[\leadsto 1 \cdot \left(\color{blue}{\frac{a2}{b1}} \cdot \frac{a1}{b2}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.2640381336687406 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.7958098716047684 \cdot 10^{-250}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.738750803985727 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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