Average Error: 16.5 → 6.0
Time: 4.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 27638105531957988:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 27638105531957988:\\
\;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\beta \cdot \frac{1}{\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 r75509 = beta;
        double r75510 = alpha;
        double r75511 = r75509 - r75510;
        double r75512 = r75510 + r75509;
        double r75513 = 2.0;
        double r75514 = r75512 + r75513;
        double r75515 = r75511 / r75514;
        double r75516 = 1.0;
        double r75517 = r75515 + r75516;
        double r75518 = r75517 / r75513;
        return r75518;
}
↓
double f(double alpha, double beta) {
        double r75519 = alpha;
        double r75520 = 27638105531957988.0;
        bool r75521 = r75519 <= r75520;
        double r75522 = beta;
        double r75523 = 1.0;
        double r75524 = r75519 + r75522;
        double r75525 = 2.0;
        double r75526 = r75524 + r75525;
        double r75527 = r75523 / r75526;
        double r75528 = r75522 * r75527;
        double r75529 = r75519 / r75526;
        double r75530 = 1.0;
        double r75531 = r75529 - r75530;
        double r75532 = exp(r75531);
        double r75533 = log(r75532);
        double r75534 = r75528 - r75533;
        double r75535 = r75534 / r75525;
        double r75536 = 4.0;
        double r75537 = r75536 / r75519;
        double r75538 = r75537 / r75519;
        double r75539 = 8.0;
        double r75540 = -r75539;
        double r75541 = 3.0;
        double r75542 = pow(r75519, r75541);
        double r75543 = r75540 / r75542;
        double r75544 = r75538 + r75543;
        double r75545 = -r75525;
        double r75546 = r75545 / r75519;
        double r75547 = r75544 + r75546;
        double r75548 = r75528 - r75547;
        double r75549 = r75548 / r75525;
        double r75550 = r75521 ? r75535 : r75549;
        return r75550;
}

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 < 27638105531957988.0

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied add-log-exp0.3

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

    7. Applied add-log-exp0.4

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

    8. Applied diff-log0.4

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

    9. Simplified0.3

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

    11. Applied div-inv0.3

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

    if 27638105531957988.0 < alpha

    1. Initial program 50.9

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

    3. Applied div-sub50.9

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

    4. Applied associate-+l-49.2

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

    6. Applied add-log-exp49.2

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

    7. Applied add-log-exp49.3

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

    8. Applied diff-log49.3

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

    9. Simplified49.3

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

    11. Applied div-inv49.3

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

    12. Taylor expanded around inf 18.1

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

    13. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 27638105531957988:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))