Average Error: 11.5 → 6.2
Time: 10.5s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.428461434113379888690634738591477366935 \cdot 10^{241}:\\ \;\;\;\;\frac{1}{\frac{b2}{a1 \cdot \frac{a2}{b1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.173735067615666191078493755628268490435 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 4.720017116358327401313860013731916291401 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.631581414509871450239539618490595928788 \cdot 10^{66}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -1.428461434113379888690634738591477366935 \cdot 10^{241}:\\
\;\;\;\;\frac{1}{\frac{b2}{a1 \cdot \frac{a2}{b1}}}\\

\mathbf{elif}\;b1 \cdot b2 \le -1.173735067615666191078493755628268490435 \cdot 10^{-115}:\\
\;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\

\mathbf{elif}\;b1 \cdot b2 \le 4.720017116358327401313860013731916291401 \cdot 10^{-317}:\\
\;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 4.631581414509871450239539618490595928788 \cdot 10^{66}:\\
\;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r5499627 = a1;
        double r5499628 = a2;
        double r5499629 = r5499627 * r5499628;
        double r5499630 = b1;
        double r5499631 = b2;
        double r5499632 = r5499630 * r5499631;
        double r5499633 = r5499629 / r5499632;
        return r5499633;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r5499634 = b1;
        double r5499635 = b2;
        double r5499636 = r5499634 * r5499635;
        double r5499637 = -1.42846143411338e+241;
        bool r5499638 = r5499636 <= r5499637;
        double r5499639 = 1.0;
        double r5499640 = a1;
        double r5499641 = a2;
        double r5499642 = r5499641 / r5499634;
        double r5499643 = r5499640 * r5499642;
        double r5499644 = r5499635 / r5499643;
        double r5499645 = r5499639 / r5499644;
        double r5499646 = -1.1737350676156662e-115;
        bool r5499647 = r5499636 <= r5499646;
        double r5499648 = r5499639 / r5499636;
        double r5499649 = r5499641 * r5499640;
        double r5499650 = r5499648 * r5499649;
        double r5499651 = 4.7200171163583e-317;
        bool r5499652 = r5499636 <= r5499651;
        double r5499653 = r5499640 / r5499634;
        double r5499654 = r5499653 * r5499641;
        double r5499655 = r5499654 / r5499635;
        double r5499656 = 4.6315814145098715e+66;
        bool r5499657 = r5499636 <= r5499656;
        double r5499658 = r5499634 / r5499641;
        double r5499659 = r5499640 / r5499658;
        double r5499660 = r5499659 / r5499635;
        double r5499661 = r5499657 ? r5499650 : r5499660;
        double r5499662 = r5499652 ? r5499655 : r5499661;
        double r5499663 = r5499647 ? r5499650 : r5499662;
        double r5499664 = r5499638 ? r5499645 : r5499663;
        return r5499664;
}

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
Target11.1
Herbie6.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* b1 b2) < -1.42846143411338e+241

    1. Initial program 19.0

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

    3. Applied times-frac3.3

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

    5. Applied associate-*r/3.4

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

    7. Applied div-inv3.4

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

    8. Applied associate-*l*3.7

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

    9. Simplified3.7

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

    11. Applied clear-num4.0

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

    if -1.42846143411338e+241 < (* b1 b2) < -1.1737350676156662e-115 or 4.7200171163583e-317 < (* b1 b2) < 4.6315814145098715e+66

    1. Initial program 4.6

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

    3. Applied div-inv4.8

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

    if -1.1737350676156662e-115 < (* b1 b2) < 4.7200171163583e-317

    1. Initial program 30.2

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

    3. Applied times-frac10.5

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

    5. Applied associate-*r/11.0

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

    if 4.6315814145098715e+66 < (* b1 b2)

    1. Initial program 11.5

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

    3. Applied times-frac7.8

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

    5. Applied associate-*r/7.7

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

    7. Applied *-un-lft-identity7.7

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

    8. Applied associate-/r*7.7

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

    9. Simplified7.6

      \[\leadsto \frac{\color{blue}{\frac{a1}{\frac{b1}{a2}}}}{b2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.428461434113379888690634738591477366935 \cdot 10^{241}:\\ \;\;\;\;\frac{1}{\frac{b2}{a1 \cdot \frac{a2}{b1}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.173735067615666191078493755628268490435 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 4.720017116358327401313860013731916291401 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.631581414509871450239539618490595928788 \cdot 10^{66}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a2 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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