Average Error: 15.7 → 6.2
Time: 16.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{1}{\beta}} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1634825657092984630335766528:\\
\;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{1}{\beta}} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r88629 = beta;
        double r88630 = alpha;
        double r88631 = r88629 - r88630;
        double r88632 = r88630 + r88629;
        double r88633 = 2.0;
        double r88634 = r88632 + r88633;
        double r88635 = r88631 / r88634;
        double r88636 = 1.0;
        double r88637 = r88635 + r88636;
        double r88638 = r88637 / r88633;
        return r88638;
}
↓
double f(double alpha, double beta) {
        double r88639 = alpha;
        double r88640 = 1.6348256570929846e+27;
        bool r88641 = r88639 <= r88640;
        double r88642 = 1.0;
        double r88643 = 2.0;
        double r88644 = beta;
        double r88645 = r88643 + r88644;
        double r88646 = r88639 + r88645;
        double r88647 = r88642 / r88646;
        double r88648 = r88642 / r88644;
        double r88649 = r88647 / r88648;
        double r88650 = r88643 + r88639;
        double r88651 = r88644 + r88650;
        double r88652 = r88639 / r88651;
        double r88653 = 1.0;
        double r88654 = r88652 - r88653;
        double r88655 = r88649 - r88654;
        double r88656 = log(r88655);
        double r88657 = exp(r88656);
        double r88658 = r88657 / r88643;
        double r88659 = r88644 / r88651;
        double r88660 = 4.0;
        double r88661 = r88639 * r88639;
        double r88662 = r88660 / r88661;
        double r88663 = r88643 / r88639;
        double r88664 = 8.0;
        double r88665 = 3.0;
        double r88666 = pow(r88639, r88665);
        double r88667 = r88664 / r88666;
        double r88668 = r88663 + r88667;
        double r88669 = r88662 - r88668;
        double r88670 = r88659 - r88669;
        double r88671 = r88670 / r88643;
        double r88672 = r88641 ? r88658 : r88671;
        return r88672;
}

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 < 1.6348256570929846e+27

    1. Initial program 1.1

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

    2. Simplified1.1

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

    4. Applied div-sub1.1

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

    5. Applied associate-+l-1.1

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

    6. Simplified1.1

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

    8. Applied add-exp-log1.1

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

    9. Simplified1.1

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

    11. Applied clear-num1.1

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

    12. Simplified1.1

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

    14. Applied div-inv1.1

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

    15. Applied associate-/r*1.1

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

    16. Simplified1.1

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

    if 1.6348256570929846e+27 < alpha

    1. Initial program 50.4

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

    2. Simplified50.4

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

    4. Applied div-sub50.3

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

    5. Applied associate-+l-48.7

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

    6. Simplified48.7

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

    7. Taylor expanded around inf 18.4

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

    8. Simplified18.4

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{1}{\beta}} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))