Average Error: 3.8 → 2.4
Time: 19.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 6.632569412339182309516867344517319586661 \cdot 10^{173}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 6.632569412339182309516867344517319586661 \cdot 10^{173}:\\
\;\;\;\;\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}}{\alpha + \left(3 + \beta\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r79059 = alpha;
        double r79060 = beta;
        double r79061 = r79059 + r79060;
        double r79062 = r79060 * r79059;
        double r79063 = r79061 + r79062;
        double r79064 = 1.0;
        double r79065 = r79063 + r79064;
        double r79066 = 2.0;
        double r79067 = r79066 * r79064;
        double r79068 = r79061 + r79067;
        double r79069 = r79065 / r79068;
        double r79070 = r79069 / r79068;
        double r79071 = r79068 + r79064;
        double r79072 = r79070 / r79071;
        return r79072;
}
↓
double f(double alpha, double beta) {
        double r79073 = beta;
        double r79074 = 6.632569412339182e+173;
        bool r79075 = r79073 <= r79074;
        double r79076 = alpha;
        double r79077 = r79076 + r79073;
        double r79078 = r79073 * r79076;
        double r79079 = r79077 + r79078;
        double r79080 = 1.0;
        double r79081 = r79079 + r79080;
        double r79082 = 2.0;
        double r79083 = r79082 * r79080;
        double r79084 = r79077 + r79083;
        double r79085 = r79081 / r79084;
        double r79086 = r79085 / r79084;
        double r79087 = 3.0;
        double r79088 = r79087 + r79073;
        double r79089 = r79076 + r79088;
        double r79090 = r79086 / r79089;
        double r79091 = 0.0;
        double r79092 = r79075 ? r79090 : r79091;
        return r79092;
}

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 < 6.632569412339182e+173

    1. Initial program 1.6

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

    2. Taylor expanded around 0 1.6

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\alpha + \left(3 + \beta\right)}}\]

    if 6.632569412339182e+173 < beta

    1. Initial program 17.2

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

    2. Taylor expanded around 0 17.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\alpha + \left(3 + \beta\right)}}\]

    3. Taylor expanded around inf 6.9

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.632569412339182309516867344517319586661 \cdot 10^{173}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\alpha + \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))