Average Error: 3.7 → 2.3
Time: 18.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.383089087415949391214288371462890131236 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 1.383089087415949391214288371462890131236 \cdot 10^{162}:\\
\;\;\;\;\frac{\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{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}:\\
\;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r202117 = alpha;
        double r202118 = beta;
        double r202119 = r202117 + r202118;
        double r202120 = r202118 * r202117;
        double r202121 = r202119 + r202120;
        double r202122 = 1.0;
        double r202123 = r202121 + r202122;
        double r202124 = 2.0;
        double r202125 = r202124 * r202122;
        double r202126 = r202119 + r202125;
        double r202127 = r202123 / r202126;
        double r202128 = r202127 / r202126;
        double r202129 = r202126 + r202122;
        double r202130 = r202128 / r202129;
        return r202130;
}
↓
double f(double alpha, double beta) {
        double r202131 = beta;
        double r202132 = 1.3830890874159494e+162;
        bool r202133 = r202131 <= r202132;
        double r202134 = alpha;
        double r202135 = r202134 + r202131;
        double r202136 = r202131 * r202134;
        double r202137 = r202135 + r202136;
        double r202138 = 1.0;
        double r202139 = r202137 + r202138;
        double r202140 = 1.0;
        double r202141 = 2.0;
        double r202142 = r202141 * r202138;
        double r202143 = r202135 + r202142;
        double r202144 = r202140 / r202143;
        double r202145 = r202139 * r202144;
        double r202146 = r202145 / r202143;
        double r202147 = r202143 + r202138;
        double r202148 = r202146 / r202147;
        double r202149 = 0.0;
        double r202150 = r202149 / r202147;
        double r202151 = r202133 ? r202148 : r202150;
        return r202151;
}

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 < 1.3830890874159494e+162

    1. Initial program 1.2

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

    3. Applied div-inv1.2

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

    if 1.3830890874159494e+162 < beta

    1. Initial program 17.4

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

    2. Taylor expanded around inf 8.3

      \[\leadsto \frac{\color{blue}{0}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.383089087415949391214288371462890131236 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))