Average Error: 16.4 → 6.2
Time: 3.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3210135597332015.5:\\ \;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3210135597332015.5:\\
\;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r80048 = beta;
        double r80049 = alpha;
        double r80050 = r80048 - r80049;
        double r80051 = r80049 + r80048;
        double r80052 = 2.0;
        double r80053 = r80051 + r80052;
        double r80054 = r80050 / r80053;
        double r80055 = 1.0;
        double r80056 = r80054 + r80055;
        double r80057 = r80056 / r80052;
        return r80057;
}
↓
double f(double alpha, double beta) {
        double r80058 = alpha;
        double r80059 = 3210135597332015.5;
        bool r80060 = r80058 <= r80059;
        double r80061 = beta;
        double r80062 = r80058 + r80061;
        double r80063 = 2.0;
        double r80064 = r80062 + r80063;
        double r80065 = r80061 / r80064;
        double r80066 = r80058 / r80064;
        double r80067 = r80065 - r80066;
        double r80068 = 1.0;
        double r80069 = -r80068;
        double r80070 = r80067 - r80069;
        double r80071 = r80070 / r80063;
        double r80072 = 4.0;
        double r80073 = r80072 / r80058;
        double r80074 = r80073 / r80058;
        double r80075 = 8.0;
        double r80076 = -r80075;
        double r80077 = 3.0;
        double r80078 = pow(r80058, r80077);
        double r80079 = r80076 / r80078;
        double r80080 = r80074 + r80079;
        double r80081 = -r80063;
        double r80082 = r80081 / r80058;
        double r80083 = r80080 + r80082;
        double r80084 = r80065 - r80083;
        double r80085 = r80084 / r80063;
        double r80086 = r80060 ? r80071 : r80085;
        return r80086;
}

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 < 3210135597332015.5

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied sub-neg0.4

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

    7. Applied associate--r+0.4

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

    if 3210135597332015.5 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-48.9

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

    5. Taylor expanded around inf 18.5

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

    6. Simplified18.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3210135597332015.5:\\ \;\;\;\;\frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) - \left(-1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))