Average Error: 11.4 → 3.4
Time: 2.1s
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 r129802 = a1;
        double r129803 = a2;
        double r129804 = r129802 * r129803;
        double r129805 = b1;
        double r129806 = b2;
        double r129807 = r129805 * r129806;
        double r129808 = r129804 / r129807;
        return r129808;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r129809 = a1;
        double r129810 = a2;
        double r129811 = r129809 * r129810;
        double r129812 = b1;
        double r129813 = b2;
        double r129814 = r129812 * r129813;
        double r129815 = r129811 / r129814;
        double r129816 = -1.6828003633594403e+301;
        bool r129817 = r129815 <= r129816;
        double r129818 = r129810 / r129812;
        double r129819 = r129809 * r129818;
        double r129820 = 1.0;
        double r129821 = r129820 / r129813;
        double r129822 = r129819 * r129821;
        double r129823 = -4.520444799089932e-307;
        bool r129824 = r129815 <= r129823;
        double r129825 = 0.0;
        bool r129826 = r129815 <= r129825;
        double r129827 = r129810 / r129813;
        double r129828 = r129812 / r129827;
        double r129829 = r129809 / r129828;
        double r129830 = 4.517334051010812e+288;
        bool r129831 = r129815 <= r129830;
        double r129832 = r129831 ? r129815 : r129822;
        double r129833 = r129826 ? r129829 : r129832;
        double r129834 = r129824 ? r129815 : r129833;
        double r129835 = r129817 ? r129822 : r129834;
        return r129835;
}

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*13.7

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

    9. Taylor expanded around 0 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)))