Average Error: 11.3 → 5.5
Time: 6.2s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\
\;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\

\mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\
\;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r121476 = a1;
        double r121477 = a2;
        double r121478 = r121476 * r121477;
        double r121479 = b1;
        double r121480 = b2;
        double r121481 = r121479 * r121480;
        double r121482 = r121478 / r121481;
        return r121482;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r121483 = b1;
        double r121484 = b2;
        double r121485 = r121483 * r121484;
        double r121486 = -1.512192312330299e+168;
        bool r121487 = r121485 <= r121486;
        double r121488 = a1;
        double r121489 = a2;
        double r121490 = r121489 / r121484;
        double r121491 = r121488 * r121490;
        double r121492 = r121491 / r121483;
        double r121493 = -1.7831160215664477e-257;
        bool r121494 = r121485 <= r121493;
        double r121495 = r121488 * r121489;
        double r121496 = 1.0;
        double r121497 = r121496 / r121485;
        double r121498 = r121495 * r121497;
        double r121499 = 5.8922441633555e-314;
        bool r121500 = r121485 <= r121499;
        double r121501 = r121484 / r121489;
        double r121502 = r121501 / r121488;
        double r121503 = r121502 * r121483;
        double r121504 = r121496 / r121503;
        double r121505 = 1.7927717050604466e+257;
        bool r121506 = r121485 <= r121505;
        double r121507 = r121506 ? r121498 : r121492;
        double r121508 = r121500 ? r121504 : r121507;
        double r121509 = r121494 ? r121498 : r121508;
        double r121510 = r121487 ? r121492 : r121509;
        return r121510;
}

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
Target11.3
Herbie5.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -1.512192312330299e+168 or 1.7927717050604466e+257 < (* b1 b2)

    1. Initial program 15.6

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

    3. Applied associate-/r*7.8

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

    5. Applied clear-num8.4

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

    7. Applied associate-/r/8.6

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

    8. Applied associate-/r*8.0

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

    9. Simplified4.6

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

    if -1.512192312330299e+168 < (* b1 b2) < -1.7831160215664477e-257 or 5.8922441633555e-314 < (* b1 b2) < 1.7927717050604466e+257

    1. Initial program 5.4

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

    3. Applied div-inv5.5

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

    if -1.7831160215664477e-257 < (* b1 b2) < 5.8922441633555e-314

    1. Initial program 50.6

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

    3. Applied associate-/r*20.2

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

    5. Applied clear-num20.3

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

    7. Applied associate-/r/20.3

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

    8. Simplified7.9

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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