Average Error: 16.2 → 6.0
Time: 4.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6665734284871154:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6665734284871154:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{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 r84766 = beta;
        double r84767 = alpha;
        double r84768 = r84766 - r84767;
        double r84769 = r84767 + r84766;
        double r84770 = 2.0;
        double r84771 = r84769 + r84770;
        double r84772 = r84768 / r84771;
        double r84773 = 1.0;
        double r84774 = r84772 + r84773;
        double r84775 = r84774 / r84770;
        return r84775;
}
↓
double f(double alpha, double beta) {
        double r84776 = alpha;
        double r84777 = 6665734284871154.0;
        bool r84778 = r84776 <= r84777;
        double r84779 = beta;
        double r84780 = r84776 + r84779;
        double r84781 = 2.0;
        double r84782 = r84780 + r84781;
        double r84783 = r84779 / r84782;
        double r84784 = 1.0;
        double r84785 = r84784 * r84784;
        double r84786 = -r84785;
        double r84787 = r84776 / r84782;
        double r84788 = r84787 * r84776;
        double r84789 = r84788 / r84782;
        double r84790 = r84786 + r84789;
        double r84791 = 1.0;
        double r84792 = r84791 / r84782;
        double r84793 = r84776 * r84792;
        double r84794 = r84793 + r84784;
        double r84795 = r84790 / r84794;
        double r84796 = r84783 - r84795;
        double r84797 = r84796 / r84781;
        double r84798 = 4.0;
        double r84799 = r84798 / r84776;
        double r84800 = r84799 / r84776;
        double r84801 = 8.0;
        double r84802 = -r84801;
        double r84803 = 3.0;
        double r84804 = pow(r84776, r84803);
        double r84805 = r84802 / r84804;
        double r84806 = r84800 + r84805;
        double r84807 = -r84781;
        double r84808 = r84807 / r84776;
        double r84809 = r84806 + r84808;
        double r84810 = r84783 - r84809;
        double r84811 = r84810 / r84781;
        double r84812 = r84778 ? r84797 : r84811;
        return r84812;
}

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 < 6665734284871154.0

    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 div-inv0.4

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

    8. Applied flip--0.4

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

    9. Simplified0.4

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

    if 6665734284871154.0 < alpha

    1. Initial program 50.8

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

    3. Applied div-sub50.7

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

    4. Applied associate-+l-49.1

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

    5. Taylor expanded around inf 18.2

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

    6. Simplified18.2

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6665734284871154:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))