Average Error: 3.6 → 2.3
Time: 26.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\
\;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r5312046 = alpha;
        double r5312047 = beta;
        double r5312048 = r5312046 + r5312047;
        double r5312049 = r5312047 * r5312046;
        double r5312050 = r5312048 + r5312049;
        double r5312051 = 1.0;
        double r5312052 = r5312050 + r5312051;
        double r5312053 = 2.0;
        double r5312054 = 1.0;
        double r5312055 = r5312053 * r5312054;
        double r5312056 = r5312048 + r5312055;
        double r5312057 = r5312052 / r5312056;
        double r5312058 = r5312057 / r5312056;
        double r5312059 = r5312056 + r5312051;
        double r5312060 = r5312058 / r5312059;
        return r5312060;
}
↓
double f(double alpha, double beta) {
        double r5312061 = alpha;
        double r5312062 = 4.0898973557336655e+168;
        bool r5312063 = r5312061 <= r5312062;
        double r5312064 = beta;
        double r5312065 = r5312061 + r5312064;
        double r5312066 = r5312064 * r5312061;
        double r5312067 = r5312065 + r5312066;
        double r5312068 = 1.0;
        double r5312069 = r5312067 + r5312068;
        double r5312070 = 2.0;
        double r5312071 = r5312065 + r5312070;
        double r5312072 = r5312069 / r5312071;
        double r5312073 = r5312072 / r5312071;
        double r5312074 = r5312071 + r5312068;
        double r5312075 = r5312073 / r5312074;
        double r5312076 = 0.0;
        double r5312077 = r5312063 ? r5312075 : r5312076;
        return r5312077;
}

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 alpha < 4.0898973557336655e+168

    1. Initial program 1.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

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

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    if 4.0898973557336655e+168 < alpha

    1. Initial program 16.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Taylor expanded around inf 7.3

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.0898973557336655 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))