Average Error: 15.9 → 6.5
Time: 42.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 640404480216242093242897854889984:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \sqrt[3]{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 640404480216242093242897854889984:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \sqrt[3]{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \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{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r4235048 = beta;
        double r4235049 = alpha;
        double r4235050 = r4235048 - r4235049;
        double r4235051 = r4235049 + r4235048;
        double r4235052 = 2.0;
        double r4235053 = r4235051 + r4235052;
        double r4235054 = r4235050 / r4235053;
        double r4235055 = 1.0;
        double r4235056 = r4235054 + r4235055;
        double r4235057 = r4235056 / r4235052;
        return r4235057;
}
↓
double f(double alpha, double beta) {
        double r4235058 = alpha;
        double r4235059 = 6.404044802162421e+32;
        bool r4235060 = r4235058 <= r4235059;
        double r4235061 = beta;
        double r4235062 = 2.0;
        double r4235063 = r4235061 + r4235058;
        double r4235064 = r4235062 + r4235063;
        double r4235065 = r4235061 / r4235064;
        double r4235066 = r4235058 / r4235064;
        double r4235067 = 1.0;
        double r4235068 = r4235066 - r4235067;
        double r4235069 = r4235068 * r4235068;
        double r4235070 = r4235068 * r4235069;
        double r4235071 = cbrt(r4235070);
        double r4235072 = r4235065 - r4235071;
        double r4235073 = r4235072 / r4235062;
        double r4235074 = 4.0;
        double r4235075 = r4235058 * r4235058;
        double r4235076 = r4235074 / r4235075;
        double r4235077 = 8.0;
        double r4235078 = r4235058 * r4235075;
        double r4235079 = r4235077 / r4235078;
        double r4235080 = r4235076 - r4235079;
        double r4235081 = r4235062 / r4235058;
        double r4235082 = r4235080 - r4235081;
        double r4235083 = r4235065 - r4235082;
        double r4235084 = r4235083 / r4235062;
        double r4235085 = r4235060 ? r4235073 : r4235084;
        return r4235085;
}

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 < 6.404044802162421e+32

    1. Initial program 1.6

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

    3. Applied div-sub1.6

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

    4. Applied associate-+l-1.5

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

    6. Applied add-cbrt-cube1.6

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

    if 6.404044802162421e+32 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.0

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

    6. Applied add-cbrt-cube49.0

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

    7. Taylor expanded around inf 18.4

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

    8. Simplified18.4

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 640404480216242093242897854889984:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \sqrt[3]{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))