Average Error: 11.4 → 3.4
Time: 2.3s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\
\;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{else}:\\
\;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r119813 = a1;
        double r119814 = a2;
        double r119815 = r119813 * r119814;
        double r119816 = b1;
        double r119817 = b2;
        double r119818 = r119816 * r119817;
        double r119819 = r119815 / r119818;
        return r119819;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r119820 = a1;
        double r119821 = a2;
        double r119822 = r119820 * r119821;
        double r119823 = b1;
        double r119824 = b2;
        double r119825 = r119823 * r119824;
        double r119826 = r119822 / r119825;
        double r119827 = -1.6828003633594403e+301;
        bool r119828 = r119826 <= r119827;
        double r119829 = r119821 / r119823;
        double r119830 = r119820 * r119829;
        double r119831 = 1.0;
        double r119832 = r119831 / r119824;
        double r119833 = r119830 * r119832;
        double r119834 = -4.520444799089932e-307;
        bool r119835 = r119826 <= r119834;
        double r119836 = 0.0;
        bool r119837 = r119826 <= r119836;
        double r119838 = r119821 / r119824;
        double r119839 = r119823 / r119838;
        double r119840 = r119820 / r119839;
        double r119841 = 4.517334051010812e+288;
        bool r119842 = r119826 <= r119841;
        double r119843 = r119842 ? r119826 : r119833;
        double r119844 = r119837 ? r119840 : r119843;
        double r119845 = r119835 ? r119826 : r119844;
        double r119846 = r119828 ? r119833 : r119845;
        return r119846;
}

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

Original11.4
Target11.5
Herbie3.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -1.6828003633594403e+301 or 4.517334051010812e+288 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 59.3

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

    3. Applied associate-/l*39.8

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

    5. Applied div-inv39.9

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

    6. Simplified39.8

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

    8. Applied associate-/r*15.2

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

    10. Applied div-inv15.3

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

    11. Applied associate-*r*14.9

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

    if -1.6828003633594403e+301 < (/ (* a1 a2) (* b1 b2)) < -4.520444799089932e-307 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 4.517334051010812e+288

    1. Initial program 0.8

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

    3. Applied associate-/l*7.3

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

    5. Applied div-inv7.6

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

    6. Simplified7.3

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

    8. Applied associate-*r/0.8

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

    if -4.520444799089932e-307 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program 13.1

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

    3. Applied associate-/l*7.1

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

    5. Applied associate-/l*3.8

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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