Average Error: 3.9 → 2.4
Time: 37.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 3.097558604616753343996040534278213161537 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 3.097558604616753343996040534278213161537 \cdot 10^{192}:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r103921 = alpha;
        double r103922 = beta;
        double r103923 = r103921 + r103922;
        double r103924 = r103922 * r103921;
        double r103925 = r103923 + r103924;
        double r103926 = 1.0;
        double r103927 = r103925 + r103926;
        double r103928 = 2.0;
        double r103929 = r103928 * r103926;
        double r103930 = r103923 + r103929;
        double r103931 = r103927 / r103930;
        double r103932 = r103931 / r103930;
        double r103933 = r103930 + r103926;
        double r103934 = r103932 / r103933;
        return r103934;
}
↓
double f(double alpha, double beta) {
        double r103935 = beta;
        double r103936 = 3.0975586046167533e+192;
        bool r103937 = r103935 <= r103936;
        double r103938 = alpha;
        double r103939 = r103938 + r103935;
        double r103940 = r103935 * r103938;
        double r103941 = r103939 + r103940;
        double r103942 = 1.0;
        double r103943 = r103941 + r103942;
        double r103944 = 2.0;
        double r103945 = r103944 * r103942;
        double r103946 = r103939 + r103945;
        double r103947 = r103943 / r103946;
        double r103948 = r103947 / r103946;
        double r103949 = r103946 + r103942;
        double r103950 = r103948 / r103949;
        double r103951 = 0.0;
        double r103952 = r103937 ? r103950 : r103951;
        return r103952;
}

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 < 3.0975586046167533e+192

    1. Initial program 1.7

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

    3. Applied *-un-lft-identity1.7

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

    4. Applied *-un-lft-identity1.7

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

    5. Applied *-un-lft-identity1.7

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

    6. Applied times-frac1.7

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

    7. Applied times-frac1.7

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

    8. Simplified1.7

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

    if 3.0975586046167533e+192 < beta

    1. Initial program 19.7

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

    2. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 3.097558604616753343996040534278213161537 \cdot 10^{192}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))