Average Error: 11.9 → 5.5
Time: 2.1s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\
\;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\
\;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\

\mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\
\;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\
\;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r117621 = a1;
        double r117622 = a2;
        double r117623 = r117621 * r117622;
        double r117624 = b1;
        double r117625 = b2;
        double r117626 = r117624 * r117625;
        double r117627 = r117623 / r117626;
        return r117627;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r117628 = b1;
        double r117629 = b2;
        double r117630 = r117628 * r117629;
        double r117631 = -3.025024538805417e+256;
        bool r117632 = r117630 <= r117631;
        double r117633 = a1;
        double r117634 = r117633 / r117628;
        double r117635 = a2;
        double r117636 = r117634 * r117635;
        double r117637 = r117636 / r117629;
        double r117638 = -4.049627701690918e-94;
        bool r117639 = r117630 <= r117638;
        double r117640 = 1.0;
        double r117641 = r117640 / r117629;
        double r117642 = r117641 / r117628;
        double r117643 = r117635 * r117642;
        double r117644 = r117633 * r117643;
        double r117645 = 3.1880082982352746e-295;
        bool r117646 = r117630 <= r117645;
        double r117647 = 2.2303155390985713e+300;
        bool r117648 = r117630 <= r117647;
        double r117649 = r117635 / r117629;
        double r117650 = r117633 * r117649;
        double r117651 = r117650 / r117628;
        double r117652 = r117648 ? r117644 : r117651;
        double r117653 = r117646 ? r117637 : r117652;
        double r117654 = r117639 ? r117644 : r117653;
        double r117655 = r117632 ? r117637 : r117654;
        return r117655;
}

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

Original11.9
Target11.4
Herbie5.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -3.025024538805417e+256 or -4.049627701690918e-94 < (* b1 b2) < 3.1880082982352746e-295

    1. Initial program 24.1

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

    3. Applied times-frac8.1

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

    5. Applied associate-*r/7.5

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

    if -3.025024538805417e+256 < (* b1 b2) < -4.049627701690918e-94 or 3.1880082982352746e-295 < (* b1 b2) < 2.2303155390985713e+300

    1. Initial program 5.4

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

    3. Applied times-frac13.8

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

    5. Applied div-inv13.8

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

    6. Applied associate-*l*11.1

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

    7. Simplified11.0

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

    9. Applied *-un-lft-identity11.0

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

    10. Applied div-inv11.1

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

    11. Applied times-frac4.9

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

    12. Simplified4.9

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

    if 2.2303155390985713e+300 < (* b1 b2)

    1. Initial program 22.3

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

    3. Applied times-frac2.8

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

    5. Applied associate-*l/3.2

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.0250245388054168 \cdot 10^{256}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.04962770169091819 \cdot 10^{-94}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 3.18800829823527456 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.23031553909857134 \cdot 10^{300}:\\ \;\;\;\;a1 \cdot \left(a2 \cdot \frac{\frac{1}{b2}}{b1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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