Average Error: 11.5 → 5.3
Time: 8.7s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.78185687302351595353600927911780875218 \cdot 10^{144}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.150210671513833606732172966148884455885 \cdot 10^{-228}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.343720045836759282443581250746346844563 \cdot 10^{-182}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a1}{b2} \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 4.542196225721951094431610617285657897859 \cdot 10^{280}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -3.78185687302351595353600927911780875218 \cdot 10^{144}:\\
\;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.150210671513833606732172966148884455885 \cdot 10^{-228}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \le 4.343720045836759282443581250746346844563 \cdot 10^{-182}:\\
\;\;\;\;\frac{1}{b1} \cdot \left(\frac{a1}{b2} \cdot a2\right)\\

\mathbf{elif}\;b1 \cdot b2 \le 4.542196225721951094431610617285657897859 \cdot 10^{280}:\\
\;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r87675 = a1;
        double r87676 = a2;
        double r87677 = r87675 * r87676;
        double r87678 = b1;
        double r87679 = b2;
        double r87680 = r87678 * r87679;
        double r87681 = r87677 / r87680;
        return r87681;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r87682 = b1;
        double r87683 = b2;
        double r87684 = r87682 * r87683;
        double r87685 = -3.781856873023516e+144;
        bool r87686 = r87684 <= r87685;
        double r87687 = 1.0;
        double r87688 = r87687 / r87682;
        double r87689 = a1;
        double r87690 = a2;
        double r87691 = r87683 / r87690;
        double r87692 = r87689 / r87691;
        double r87693 = r87688 * r87692;
        double r87694 = -4.1502106715138336e-228;
        bool r87695 = r87684 <= r87694;
        double r87696 = r87684 / r87690;
        double r87697 = r87689 / r87696;
        double r87698 = 4.343720045836759e-182;
        bool r87699 = r87684 <= r87698;
        double r87700 = r87689 / r87683;
        double r87701 = r87700 * r87690;
        double r87702 = r87688 * r87701;
        double r87703 = 4.542196225721951e+280;
        bool r87704 = r87684 <= r87703;
        double r87705 = r87689 / r87684;
        double r87706 = r87705 * r87690;
        double r87707 = r87704 ? r87706 : r87693;
        double r87708 = r87699 ? r87702 : r87707;
        double r87709 = r87695 ? r87697 : r87708;
        double r87710 = r87686 ? r87693 : r87709;
        return r87710;
}

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
Target11.2
Herbie5.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* b1 b2) < -3.781856873023516e+144 or 4.542196225721951e+280 < (* b1 b2)

    1. Initial program 16.9

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

    3. Applied associate-/l*16.2

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

    5. Applied *-un-lft-identity16.2

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

    6. Applied times-frac8.9

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

    7. Applied *-un-lft-identity8.9

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

    8. Applied times-frac6.1

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

    9. Simplified6.1

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

    if -3.781856873023516e+144 < (* b1 b2) < -4.1502106715138336e-228

    1. Initial program 4.4

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

    3. Applied associate-/l*4.1

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

    5. Applied associate-/r/4.2

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

    7. Applied *-un-lft-identity4.2

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

    8. Applied associate-*l*4.2

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

    9. Simplified4.1

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

    if -4.1502106715138336e-228 < (* b1 b2) < 4.343720045836759e-182

    1. Initial program 31.3

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

    3. Applied associate-/l*31.9

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

    5. Applied associate-/r/31.9

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

    7. Applied *-un-lft-identity31.9

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

    8. Applied times-frac17.4

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

    9. Applied associate-*l*8.9

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

    if 4.343720045836759e-182 < (* b1 b2) < 4.542196225721951e+280

    1. Initial program 5.3

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

    3. Applied associate-/l*4.6

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

    5. Applied associate-/r/4.4

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.78185687302351595353600927911780875218 \cdot 10^{144}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.150210671513833606732172966148884455885 \cdot 10^{-228}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.343720045836759282443581250746346844563 \cdot 10^{-182}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a1}{b2} \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 4.542196225721951094431610617285657897859 \cdot 10^{280}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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