Average Error: 3.5 → 1.5
Time: 1.8m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.082362444610107843014659033646971699324 \cdot 10^{139}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{\alpha}{\beta} + 2\right) + \frac{\beta}{\alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 4.082362444610107843014659033646971699324 \cdot 10^{139}:\\
\;\;\;\;\frac{\sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

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

\end{array}
double f(double alpha, double beta) {
        double r229884 = alpha;
        double r229885 = beta;
        double r229886 = r229884 + r229885;
        double r229887 = r229885 * r229884;
        double r229888 = r229886 + r229887;
        double r229889 = 1.0;
        double r229890 = r229888 + r229889;
        double r229891 = 2.0;
        double r229892 = r229891 * r229889;
        double r229893 = r229886 + r229892;
        double r229894 = r229890 / r229893;
        double r229895 = r229894 / r229893;
        double r229896 = r229893 + r229889;
        double r229897 = r229895 / r229896;
        return r229897;
}
↓
double f(double alpha, double beta) {
        double r229898 = beta;
        double r229899 = 4.082362444610108e+139;
        bool r229900 = r229898 <= r229899;
        double r229901 = 1.0;
        double r229902 = alpha;
        double r229903 = r229902 + r229898;
        double r229904 = 2.0;
        double r229905 = 1.0;
        double r229906 = r229904 * r229905;
        double r229907 = r229903 + r229906;
        double r229908 = r229902 * r229898;
        double r229909 = r229898 + r229908;
        double r229910 = r229902 + r229909;
        double r229911 = r229910 + r229905;
        double r229912 = r229911 / r229907;
        double r229913 = r229907 / r229912;
        double r229914 = r229901 / r229913;
        double r229915 = sqrt(r229914);
        double r229916 = r229915 * r229915;
        double r229917 = r229907 + r229905;
        double r229918 = r229916 / r229917;
        double r229919 = r229902 / r229898;
        double r229920 = 2.0;
        double r229921 = r229919 + r229920;
        double r229922 = r229898 / r229902;
        double r229923 = r229921 + r229922;
        double r229924 = r229901 / r229923;
        double r229925 = r229924 / r229917;
        double r229926 = r229900 ? r229918 : r229925;
        return r229926;
}

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 < 4.082362444610108e+139

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied *-un-lft-identity0.9

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

    5. Applied times-frac0.9

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

    6. Applied associate-/l*0.9

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

    7. Simplified0.9

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

    9. Applied add-sqr-sqrt1.0

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

    if 4.082362444610108e+139 < beta

    1. Initial program 15.3

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

    3. Applied *-un-lft-identity15.3

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

    4. Applied *-un-lft-identity15.3

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

    5. Applied times-frac15.3

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

    6. Applied associate-/l*15.3

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

    7. Simplified15.3

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

    8. Taylor expanded around inf 3.9

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.082362444610107843014659033646971699324 \cdot 10^{139}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}} \cdot \sqrt{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\left(\alpha + \left(\beta + \alpha \cdot \beta\right)\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{\alpha}{\beta} + 2\right) + \frac{\beta}{\alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))