Average Error: 10.9 → 5.1
Time: 14.2s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.162408852142661 \cdot 10^{+116}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.9643938750475 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.9503724066455816 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.712692769698527 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -3.162408852142661 \cdot 10^{+116}:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \le -2.9643938750475 \cdot 10^{-323}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 3.9503724066455816 \cdot 10^{-293}:\\
\;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\

\mathbf{elif}\;a1 \cdot a2 \le 9.712692769698527 \cdot 10^{+194}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r6065529 = a1;
        double r6065530 = a2;
        double r6065531 = r6065529 * r6065530;
        double r6065532 = b1;
        double r6065533 = b2;
        double r6065534 = r6065532 * r6065533;
        double r6065535 = r6065531 / r6065534;
        return r6065535;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r6065536 = a1;
        double r6065537 = a2;
        double r6065538 = r6065536 * r6065537;
        double r6065539 = -3.162408852142661e+116;
        bool r6065540 = r6065538 <= r6065539;
        double r6065541 = b1;
        double r6065542 = b2;
        double r6065543 = r6065537 / r6065542;
        double r6065544 = r6065541 / r6065543;
        double r6065545 = r6065536 / r6065544;
        double r6065546 = -2.9643938750475e-323;
        bool r6065547 = r6065538 <= r6065546;
        double r6065548 = r6065538 / r6065541;
        double r6065549 = r6065548 / r6065542;
        double r6065550 = 3.9503724066455816e-293;
        bool r6065551 = r6065538 <= r6065550;
        double r6065552 = 9.712692769698527e+194;
        bool r6065553 = r6065538 <= r6065552;
        double r6065554 = r6065543 / r6065541;
        double r6065555 = r6065536 * r6065554;
        double r6065556 = r6065553 ? r6065549 : r6065555;
        double r6065557 = r6065551 ? r6065545 : r6065556;
        double r6065558 = r6065547 ? r6065549 : r6065557;
        double r6065559 = r6065540 ? r6065545 : r6065558;
        return r6065559;
}

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
Target10.7
Herbie5.1
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -3.162408852142661e+116 or -2.9643938750475e-323 < (* a1 a2) < 3.9503724066455816e-293

    1. Initial program 21.4

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

    3. Applied times-frac5.8

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

    5. Applied associate-*l/10.2

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

    7. Applied associate-/l*5.8

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

    if -3.162408852142661e+116 < (* a1 a2) < -2.9643938750475e-323 or 3.9503724066455816e-293 < (* a1 a2) < 9.712692769698527e+194

    1. Initial program 4.4

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

    3. Applied associate-/r*4.3

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

    if 9.712692769698527e+194 < (* a1 a2)

    1. Initial program 32.7

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

    3. Applied times-frac12.4

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

    5. Applied div-inv12.5

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

    6. Applied associate-*l*10.9

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

    7. Simplified10.8

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.162408852142661 \cdot 10^{+116}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.9643938750475 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.9503724066455816 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 9.712692769698527 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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