Average Error: 3.6 → 1.0
Time: 1.3m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.113517558266252110199189669887066617032 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 \cdot 1 + \left(\alpha + \beta\right)}{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 \cdot 1 + \left(\alpha + \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.113517558266252110199189669887066617032 \cdot 10^{159}:\\
\;\;\;\;\frac{\frac{1}{\frac{2 \cdot 1 + \left(\alpha + \beta\right)}{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 \cdot 1 + \left(\alpha + \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r186888 = alpha;
        double r186889 = beta;
        double r186890 = r186888 + r186889;
        double r186891 = r186889 * r186888;
        double r186892 = r186890 + r186891;
        double r186893 = 1.0;
        double r186894 = r186892 + r186893;
        double r186895 = 2.0;
        double r186896 = r186895 * r186893;
        double r186897 = r186890 + r186896;
        double r186898 = r186894 / r186897;
        double r186899 = r186898 / r186897;
        double r186900 = r186897 + r186893;
        double r186901 = r186899 / r186900;
        return r186901;
}
↓
double f(double alpha, double beta) {
        double r186902 = beta;
        double r186903 = 1.1135175582662521e+159;
        bool r186904 = r186902 <= r186903;
        double r186905 = 1.0;
        double r186906 = 2.0;
        double r186907 = 1.0;
        double r186908 = r186906 * r186907;
        double r186909 = alpha;
        double r186910 = r186909 + r186902;
        double r186911 = r186908 + r186910;
        double r186912 = r186902 * r186909;
        double r186913 = r186910 + r186912;
        double r186914 = r186907 + r186913;
        double r186915 = r186914 / r186911;
        double r186916 = r186911 / r186915;
        double r186917 = r186905 / r186916;
        double r186918 = r186910 + r186908;
        double r186919 = r186918 + r186907;
        double r186920 = r186917 / r186919;
        double r186921 = 2.0;
        double r186922 = r186902 / r186909;
        double r186923 = r186909 / r186902;
        double r186924 = r186922 + r186923;
        double r186925 = r186921 + r186924;
        double r186926 = r186905 / r186925;
        double r186927 = r186926 / r186919;
        double r186928 = r186904 ? r186920 : r186927;
        return r186928;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.1135175582662521e+159

    1. Initial program 1.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    4. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Applied times-frac1.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Applied associate-/l*1.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Simplified1.1

      \[\leadsto \frac{\frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot 1 + \left(\alpha + \beta\right)}{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 \cdot 1 + \left(\alpha + \beta\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    if 1.1135175582662521e+159 < beta

    1. Initial program 16.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity16.4

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    4. Applied *-un-lft-identity16.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Applied times-frac16.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    6. Applied associate-/l*16.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Simplified16.4

      \[\leadsto \frac{\frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot 1 + \left(\alpha + \beta\right)}{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 \cdot 1 + \left(\alpha + \beta\right)}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    8. Taylor expanded around inf 0.1

      \[\leadsto \frac{\frac{\frac{1}{1}}{\color{blue}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.113517558266252110199189669887066617032 \cdot 10^{159}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 \cdot 1 + \left(\alpha + \beta\right)}{\frac{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}{2 \cdot 1 + \left(\alpha + \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))