Average Error: 16.4 → 6.3
Time: 22.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.0657930263664172 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.0657930263664172 \cdot 10^{+31}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\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 r3924958 = beta;
        double r3924959 = alpha;
        double r3924960 = r3924958 - r3924959;
        double r3924961 = r3924959 + r3924958;
        double r3924962 = 2.0;
        double r3924963 = r3924961 + r3924962;
        double r3924964 = r3924960 / r3924963;
        double r3924965 = 1.0;
        double r3924966 = r3924964 + r3924965;
        double r3924967 = r3924966 / r3924962;
        return r3924967;
}
↓
double f(double alpha, double beta) {
        double r3924968 = alpha;
        double r3924969 = 2.0657930263664172e+31;
        bool r3924970 = r3924968 <= r3924969;
        double r3924971 = beta;
        double r3924972 = 2.0;
        double r3924973 = r3924971 + r3924968;
        double r3924974 = r3924972 + r3924973;
        double r3924975 = r3924971 / r3924974;
        double r3924976 = sqrt(r3924974);
        double r3924977 = r3924968 / r3924976;
        double r3924978 = r3924977 / r3924976;
        double r3924979 = 1.0;
        double r3924980 = r3924978 - r3924979;
        double r3924981 = r3924975 - r3924980;
        double r3924982 = r3924981 / r3924972;
        double r3924983 = 4.0;
        double r3924984 = r3924968 * r3924968;
        double r3924985 = r3924983 / r3924984;
        double r3924986 = r3924972 / r3924968;
        double r3924987 = r3924985 - r3924986;
        double r3924988 = 8.0;
        double r3924989 = r3924968 * r3924984;
        double r3924990 = r3924988 / r3924989;
        double r3924991 = r3924987 - r3924990;
        double r3924992 = r3924975 - r3924991;
        double r3924993 = r3924992 / r3924972;
        double r3924994 = r3924970 ? r3924982 : r3924993;
        return r3924994;
}

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 < 2.0657930263664172e+31

    1. Initial program 1.3

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

    3. Applied div-sub1.3

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

    4. Applied associate-+l-1.3

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

    6. Applied add-sqr-sqrt1.3

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

    7. Applied associate-/r*1.3

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

    if 2.0657930263664172e+31 < alpha

    1. Initial program 51.2

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around inf 18.0

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

    6. Simplified18.0

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.0657930263664172 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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))