Average Error: 11.5 → 6.8
Time: 10.6s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -8.468172603848108220180232219464164813833 \cdot 10^{70}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.72354558279401312652197739398469610707 \cdot 10^{-110}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -7.879034991079353473964043405516768477683 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le -4.802620426891657887802294832116642457905 \cdot 10^{-258}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.073716929976885382619318839914274699583 \cdot 10^{-232}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.599269239471855967164130916465496234962 \cdot 10^{63}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -8.468172603848108220180232219464164813833 \cdot 10^{70}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -6.72354558279401312652197739398469610707 \cdot 10^{-110}:\\
\;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -7.879034991079353473964043405516768477683 \cdot 10^{-195}:\\
\;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\

\mathbf{elif}\;b1 \cdot b2 \le -4.802620426891657887802294832116642457905 \cdot 10^{-258}:\\
\;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.073716929976885382619318839914274699583 \cdot 10^{-232}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.599269239471855967164130916465496234962 \cdot 10^{63}:\\
\;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r103920 = a1;
        double r103921 = a2;
        double r103922 = r103920 * r103921;
        double r103923 = b1;
        double r103924 = b2;
        double r103925 = r103923 * r103924;
        double r103926 = r103922 / r103925;
        return r103926;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r103927 = b1;
        double r103928 = b2;
        double r103929 = r103927 * r103928;
        double r103930 = -8.468172603848108e+70;
        bool r103931 = r103929 <= r103930;
        double r103932 = a1;
        double r103933 = r103932 / r103927;
        double r103934 = a2;
        double r103935 = r103934 / r103928;
        double r103936 = r103933 * r103935;
        double r103937 = -6.723545582794013e-110;
        bool r103938 = r103929 <= r103937;
        double r103939 = r103934 * r103932;
        double r103940 = r103939 / r103929;
        double r103941 = -7.879034991079353e-195;
        bool r103942 = r103929 <= r103941;
        double r103943 = r103935 / r103927;
        double r103944 = r103943 * r103932;
        double r103945 = -4.802620426891658e-258;
        bool r103946 = r103929 <= r103945;
        double r103947 = 1.0737169299768854e-232;
        bool r103948 = r103929 <= r103947;
        double r103949 = 1.599269239471856e+63;
        bool r103950 = r103929 <= r103949;
        double r103951 = r103950 ? r103940 : r103944;
        double r103952 = r103948 ? r103936 : r103951;
        double r103953 = r103946 ? r103940 : r103952;
        double r103954 = r103942 ? r103944 : r103953;
        double r103955 = r103938 ? r103940 : r103954;
        double r103956 = r103931 ? r103936 : r103955;
        return r103956;
}

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

Original11.5
Target11.3
Herbie6.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -8.468172603848108e+70 or -4.802620426891658e-258 < (* b1 b2) < 1.0737169299768854e-232

    1. Initial program 20.1

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

    2. Simplified20.4

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

    4. Applied *-un-lft-identity20.4

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

    5. Applied times-frac12.2

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

    6. Applied associate-*r*8.0

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

    7. Simplified8.0

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

    if -8.468172603848108e+70 < (* b1 b2) < -6.723545582794013e-110 or -7.879034991079353e-195 < (* b1 b2) < -4.802620426891658e-258 or 1.0737169299768854e-232 < (* b1 b2) < 1.599269239471856e+63

    1. Initial program 4.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

    2. Simplified4.4

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

    4. Applied associate-*r/4.2

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

    5. Simplified4.2

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

    if -6.723545582794013e-110 < (* b1 b2) < -7.879034991079353e-195 or 1.599269239471856e+63 < (* b1 b2)

    1. Initial program 11.4

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

    2. Simplified10.9

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

    4. Applied *-un-lft-identity10.9

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

    5. Applied times-frac9.0

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

    6. Applied associate-*r*8.6

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

    7. Simplified8.6

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

    9. Applied div-inv8.6

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

    10. Applied associate-*l*9.0

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

    11. Simplified9.0

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -8.468172603848108220180232219464164813833 \cdot 10^{70}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -6.72354558279401312652197739398469610707 \cdot 10^{-110}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -7.879034991079353473964043405516768477683 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \mathbf{elif}\;b1 \cdot b2 \le -4.802620426891657887802294832116642457905 \cdot 10^{-258}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.073716929976885382619318839914274699583 \cdot 10^{-232}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.599269239471855967164130916465496234962 \cdot 10^{63}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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