Average Error: 10.9 → 4.9
Time: 30.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.8341689569487066 \cdot 10^{+275}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.019091421874218 \cdot 10^{-217}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le 2.455350833773267 \cdot 10^{-192}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.1689482458211706 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b2} \cdot \left(a2 \cdot \frac{a1}{b1}\right)\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -3.8341689569487066 \cdot 10^{+275}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -2.019091421874218 \cdot 10^{-217}:\\
\;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\

\mathbf{elif}\;b1 \cdot b2 \le 2.455350833773267 \cdot 10^{-192}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.1689482458211706 \cdot 10^{+282}:\\
\;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b2} \cdot \left(a2 \cdot \frac{a1}{b1}\right)\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5588630 = a1;
        double r5588631 = a2;
        double r5588632 = r5588630 * r5588631;
        double r5588633 = b1;
        double r5588634 = b2;
        double r5588635 = r5588633 * r5588634;
        double r5588636 = r5588632 / r5588635;
        return r5588636;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5588637 = b1;
        double r5588638 = b2;
        double r5588639 = r5588637 * r5588638;
        double r5588640 = -3.8341689569487066e+275;
        bool r5588641 = r5588639 <= r5588640;
        double r5588642 = a1;
        double r5588643 = r5588642 / r5588637;
        double r5588644 = a2;
        double r5588645 = r5588644 / r5588638;
        double r5588646 = r5588643 * r5588645;
        double r5588647 = -2.019091421874218e-217;
        bool r5588648 = r5588639 <= r5588647;
        double r5588649 = r5588644 / r5588639;
        double r5588650 = r5588649 * r5588642;
        double r5588651 = 2.455350833773267e-192;
        bool r5588652 = r5588639 <= r5588651;
        double r5588653 = 1.1689482458211706e+282;
        bool r5588654 = r5588639 <= r5588653;
        double r5588655 = 1.0;
        double r5588656 = r5588655 / r5588638;
        double r5588657 = r5588644 * r5588643;
        double r5588658 = r5588656 * r5588657;
        double r5588659 = r5588654 ? r5588650 : r5588658;
        double r5588660 = r5588652 ? r5588646 : r5588659;
        double r5588661 = r5588648 ? r5588650 : r5588660;
        double r5588662 = r5588641 ? r5588646 : r5588661;
        return r5588662;
}

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
Herbie4.9
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -3.8341689569487066e+275 or -2.019091421874218e-217 < (* b1 b2) < 2.455350833773267e-192

    1. Initial program 26.3

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

    3. Applied times-frac5.8

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

    if -3.8341689569487066e+275 < (* b1 b2) < -2.019091421874218e-217 or 2.455350833773267e-192 < (* b1 b2) < 1.1689482458211706e+282

    1. Initial program 4.4

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

    3. Applied times-frac13.4

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

    5. Applied div-inv13.5

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

    6. Applied associate-*l*11.0

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

    7. Simplified4.9

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

    if 1.1689482458211706e+282 < (* b1 b2)

    1. Initial program 19.9

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

    3. Applied times-frac3.0

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

    5. Applied div-inv3.0

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

    6. Applied associate-*r*3.0

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.8341689569487066 \cdot 10^{+275}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.019091421874218 \cdot 10^{-217}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le 2.455350833773267 \cdot 10^{-192}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.1689482458211706 \cdot 10^{+282}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b2} \cdot \left(a2 \cdot \frac{a1}{b1}\right)\\ \end{array}\]

Reproduce

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

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

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