Average Error: 15.7 → 6.2
Time: 19.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1634825657092984630335766528:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r4613083 = beta;
        double r4613084 = alpha;
        double r4613085 = r4613083 - r4613084;
        double r4613086 = r4613084 + r4613083;
        double r4613087 = 2.0;
        double r4613088 = r4613086 + r4613087;
        double r4613089 = r4613085 / r4613088;
        double r4613090 = 1.0;
        double r4613091 = r4613089 + r4613090;
        double r4613092 = r4613091 / r4613087;
        return r4613092;
}
↓
double f(double alpha, double beta) {
        double r4613093 = alpha;
        double r4613094 = 1.6348256570929846e+27;
        bool r4613095 = r4613093 <= r4613094;
        double r4613096 = beta;
        double r4613097 = 2.0;
        double r4613098 = r4613096 + r4613093;
        double r4613099 = r4613097 + r4613098;
        double r4613100 = r4613096 / r4613099;
        double r4613101 = r4613093 / r4613099;
        double r4613102 = 1.0;
        double r4613103 = r4613101 - r4613102;
        double r4613104 = r4613100 - r4613103;
        double r4613105 = log(r4613104);
        double r4613106 = exp(r4613105);
        double r4613107 = r4613106 / r4613097;
        double r4613108 = 4.0;
        double r4613109 = r4613093 * r4613093;
        double r4613110 = r4613108 / r4613109;
        double r4613111 = r4613097 / r4613093;
        double r4613112 = r4613110 - r4613111;
        double r4613113 = 8.0;
        double r4613114 = r4613093 * r4613109;
        double r4613115 = r4613113 / r4613114;
        double r4613116 = r4613112 - r4613115;
        double r4613117 = r4613100 - r4613116;
        double r4613118 = r4613117 / r4613097;
        double r4613119 = r4613095 ? r4613107 : r4613118;
        return r4613119;
}

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 < 1.6348256570929846e+27

    1. Initial program 1.1

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

    3. Applied div-sub1.1

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

    4. Applied associate-+l-1.1

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

    6. Applied add-exp-log1.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{2}\]

    if 1.6348256570929846e+27 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    6. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))