Average Error: 16.1 → 6.4
Time: 1.0m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r181496 = beta;
        double r181497 = alpha;
        double r181498 = r181496 - r181497;
        double r181499 = r181497 + r181496;
        double r181500 = 2.0;
        double r181501 = r181499 + r181500;
        double r181502 = r181498 / r181501;
        double r181503 = 1.0;
        double r181504 = r181502 + r181503;
        double r181505 = r181504 / r181500;
        return r181505;
}
↓
double f(double alpha, double beta) {
        double r181506 = alpha;
        double r181507 = 1.3056787770045593e+40;
        bool r181508 = r181506 <= r181507;
        double r181509 = beta;
        double r181510 = r181506 + r181509;
        double r181511 = 2.0;
        double r181512 = r181510 + r181511;
        double r181513 = r181509 / r181512;
        double r181514 = cbrt(r181513);
        double r181515 = 6.0;
        double r181516 = pow(r181514, r181515);
        double r181517 = cbrt(r181516);
        double r181518 = exp(r181517);
        double r181519 = log(r181518);
        double r181520 = r181519 * r181514;
        double r181521 = r181506 / r181512;
        double r181522 = 1.0;
        double r181523 = r181521 - r181522;
        double r181524 = r181520 - r181523;
        double r181525 = r181524 / r181511;
        double r181526 = 4.0;
        double r181527 = r181506 * r181506;
        double r181528 = r181526 / r181527;
        double r181529 = r181511 / r181506;
        double r181530 = r181528 - r181529;
        double r181531 = 8.0;
        double r181532 = 3.0;
        double r181533 = pow(r181506, r181532);
        double r181534 = r181531 / r181533;
        double r181535 = r181530 - r181534;
        double r181536 = r181513 - r181535;
        double r181537 = r181536 / r181511;
        double r181538 = r181508 ? r181525 : r181537;
        return r181538;
}

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 < 1.3056787770045593e+40

    1. Initial program 1.9

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

    3. Applied div-sub1.9

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

    4. Applied associate-+l-1.9

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

    6. Applied add-cube-cbrt1.9

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

    8. Applied pow1/321.7

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

    9. Applied pow1/321.7

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

    10. Applied pow-prod-down1.9

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

    11. Simplified1.9

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    12. Using strategy rm

    13. Applied add-log-exp1.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}^{\frac{1}{3}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    14. Simplified1.9

      \[\leadsto \frac{\log \color{blue}{\left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    if 1.3056787770045593e+40 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around inf 17.5

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

    6. Simplified17.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))