Average Error: 15.9 → 6.1
Time: 26.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\
\;\;\;\;\frac{1.0 + \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r1184977 = beta;
        double r1184978 = alpha;
        double r1184979 = r1184977 - r1184978;
        double r1184980 = r1184978 + r1184977;
        double r1184981 = 2.0;
        double r1184982 = r1184980 + r1184981;
        double r1184983 = r1184979 / r1184982;
        double r1184984 = 1.0;
        double r1184985 = r1184983 + r1184984;
        double r1184986 = r1184985 / r1184981;
        return r1184986;
}
↓
double f(double alpha, double beta) {
        double r1184987 = alpha;
        double r1184988 = 2.081929906225012e+19;
        bool r1184989 = r1184987 <= r1184988;
        double r1184990 = 1.0;
        double r1184991 = beta;
        double r1184992 = 2.0;
        double r1184993 = r1184991 + r1184987;
        double r1184994 = r1184992 + r1184993;
        double r1184995 = r1184991 / r1184994;
        double r1184996 = r1184987 / r1184994;
        double r1184997 = r1184995 - r1184996;
        double r1184998 = r1184990 + r1184997;
        double r1184999 = r1184998 / r1184992;
        double r1185000 = 4.0;
        double r1185001 = r1184987 * r1184987;
        double r1185002 = r1185000 / r1185001;
        double r1185003 = r1184992 / r1184987;
        double r1185004 = r1185002 - r1185003;
        double r1185005 = 8.0;
        double r1185006 = r1184987 * r1185001;
        double r1185007 = r1185005 / r1185006;
        double r1185008 = r1185004 - r1185007;
        double r1185009 = r1184995 - r1185008;
        double r1185010 = r1185009 / r1184992;
        double r1185011 = r1184989 ? r1184999 : r1185010;
        return r1185011;
}

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 < 2.081929906225012e+19

    1. Initial program 0.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    if 2.081929906225012e+19 < alpha

    1. Initial program 50.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub50.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-48.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-log-exp48.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}\right)}}{2.0}\]

    7. Taylor expanded around -inf 18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    8. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))