Average Error: 11.0 → 2.8
Time: 2.2s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1}{\frac{b2}{a2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.768910935847288 \cdot 10^{-283}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.0774205845335064 \cdot 10^{273}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;\frac{1}{\frac{b1}{\frac{a1}{\frac{b2}{a2}}}}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.768910935847288 \cdot 10^{-283}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.0774205845335064 \cdot 10^{273}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r123642 = a1;
        double r123643 = a2;
        double r123644 = r123642 * r123643;
        double r123645 = b1;
        double r123646 = b2;
        double r123647 = r123645 * r123646;
        double r123648 = r123644 / r123647;
        return r123648;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r123649 = a1;
        double r123650 = a2;
        double r123651 = r123649 * r123650;
        double r123652 = b1;
        double r123653 = b2;
        double r123654 = r123652 * r123653;
        double r123655 = r123651 / r123654;
        double r123656 = -inf.0;
        bool r123657 = r123655 <= r123656;
        double r123658 = 1.0;
        double r123659 = r123653 / r123650;
        double r123660 = r123649 / r123659;
        double r123661 = r123652 / r123660;
        double r123662 = r123658 / r123661;
        double r123663 = -8.768910935847288e-283;
        bool r123664 = r123655 <= r123663;
        double r123665 = 0.0;
        bool r123666 = r123655 <= r123665;
        double r123667 = r123649 / r123652;
        double r123668 = r123650 / r123653;
        double r123669 = r123667 * r123668;
        double r123670 = 7.077420584533506e+273;
        bool r123671 = r123655 <= r123670;
        double r123672 = r123671 ? r123655 : r123669;
        double r123673 = r123666 ? r123669 : r123672;
        double r123674 = r123664 ? r123655 : r123673;
        double r123675 = r123657 ? r123662 : r123674;
        return r123675;
}

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

Original11.0
Target11.1
Herbie2.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

    1. Initial program 64.0

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

    3. Applied clear-num64.0

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

    5. Applied associate-/l*35.5

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

    7. Applied associate-/l*21.3

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -8.768910935847288e-283 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 7.077420584533506e+273

    1. Initial program 0.8

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

    3. Applied clear-num1.1

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

    5. Applied associate-/l*8.0

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

    7. Applied div-inv8.4

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

    8. Applied add-cube-cbrt8.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{b1 \cdot \frac{1}{\frac{a1 \cdot a2}{b2}}}\]

    9. Applied times-frac8.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b1} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{a1 \cdot a2}{b2}}}}\]

    10. Simplified8.2

      \[\leadsto \color{blue}{\frac{1}{b1}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{a1 \cdot a2}{b2}}}\]

    11. Simplified7.8

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

    13. Applied frac-times0.8

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

    14. Simplified0.8

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

    if -8.768910935847288e-283 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 7.077420584533506e+273 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 21.3

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

    3. Applied clear-num21.6

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

    5. Applied associate-/l*15.3

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

    7. Applied div-inv15.3

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

    8. Applied add-cube-cbrt15.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{b1 \cdot \frac{1}{\frac{a1 \cdot a2}{b2}}}\]

    9. Applied times-frac14.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b1} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{a1 \cdot a2}{b2}}}}\]

    10. Simplified14.9

      \[\leadsto \color{blue}{\frac{1}{b1}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{a1 \cdot a2}{b2}}}\]

    11. Simplified14.8

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

    13. Applied *-un-lft-identity14.8

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

    14. Applied times-frac6.3

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

    15. Applied associate-*r*4.3

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

    16. Simplified4.3

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1}{\frac{b2}{a2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.768910935847288 \cdot 10^{-283}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.0774205845335064 \cdot 10^{273}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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