Average Error: 11.1 → 5.2
Time: 2.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\
\;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r137580 = a1;
        double r137581 = a2;
        double r137582 = r137580 * r137581;
        double r137583 = b1;
        double r137584 = b2;
        double r137585 = r137583 * r137584;
        double r137586 = r137582 / r137585;
        return r137586;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r137587 = a1;
        double r137588 = a2;
        double r137589 = r137587 * r137588;
        double r137590 = b1;
        double r137591 = b2;
        double r137592 = r137590 * r137591;
        double r137593 = r137589 / r137592;
        double r137594 = -3.949560208634886e-267;
        bool r137595 = r137593 <= r137594;
        double r137596 = 0.0;
        bool r137597 = r137593 <= r137596;
        double r137598 = r137591 / r137588;
        double r137599 = r137587 / r137598;
        double r137600 = r137599 / r137590;
        double r137601 = 1.4604994925847344e+301;
        bool r137602 = r137593 <= r137601;
        double r137603 = cbrt(r137590);
        double r137604 = r137603 * r137603;
        double r137605 = r137587 / r137604;
        double r137606 = r137588 / r137603;
        double r137607 = r137605 * r137606;
        double r137608 = r137607 / r137591;
        double r137609 = r137602 ? r137593 : r137608;
        double r137610 = r137597 ? r137600 : r137609;
        double r137611 = r137595 ? r137593 : r137610;
        return r137611;
}

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

Original11.1
Target11.2
Herbie5.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -3.949560208634886e-267 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4604994925847344e+301

    1. Initial program 4.5

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

    3. Applied associate-/r*9.4

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

    5. Applied clear-num9.6

      \[\leadsto \color{blue}{\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}}\]
    6. Using strategy rm

    7. Applied associate-/r/9.8

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

    8. Applied associate-/r*9.9

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

    9. Simplified9.5

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

    10. Taylor expanded around 0 4.5

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

    if -3.949560208634886e-267 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program 12.1

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

    3. Applied associate-/r*6.8

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

    5. Applied clear-num7.3

      \[\leadsto \color{blue}{\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}}\]
    6. Using strategy rm

    7. Applied associate-/r/7.3

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

    8. Applied associate-/r*6.8

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

    9. Simplified6.8

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

    11. Applied associate-/l*4.5

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

    if 1.4604994925847344e+301 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 62.4

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

    3. Applied associate-/r*48.4

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

    5. Applied add-cube-cbrt48.6

      \[\leadsto \frac{\frac{a1 \cdot a2}{\color{blue}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}}{b2}\]

    6. Applied times-frac14.1

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.949560208634886 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{\frac{b2}{a2}}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.46049949258473444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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