Average Error: 16.5 → 6.2
Time: 15.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 19821353888989220:\\ \;\;\;\;\frac{\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) - 2} \cdot \left(\left(\beta + \alpha\right) - 2\right) - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 19821353888989220:\\
\;\;\;\;\frac{\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) - 2} \cdot \left(\left(\beta + \alpha\right) - 2\right) - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\beta \cdot \frac{1}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r77537 = beta;
        double r77538 = alpha;
        double r77539 = r77537 - r77538;
        double r77540 = r77538 + r77537;
        double r77541 = 2.0;
        double r77542 = r77540 + r77541;
        double r77543 = r77539 / r77542;
        double r77544 = 1.0;
        double r77545 = r77543 + r77544;
        double r77546 = r77545 / r77541;
        return r77546;
}
↓
double f(double alpha, double beta) {
        double r77547 = alpha;
        double r77548 = 1.982135388898922e+16;
        bool r77549 = r77547 <= r77548;
        double r77550 = beta;
        double r77551 = 2.0;
        double r77552 = r77550 + r77547;
        double r77553 = r77551 + r77552;
        double r77554 = r77550 / r77553;
        double r77555 = r77552 - r77551;
        double r77556 = r77554 / r77555;
        double r77557 = r77556 * r77555;
        double r77558 = r77551 + r77547;
        double r77559 = r77550 + r77558;
        double r77560 = r77547 / r77559;
        double r77561 = 1.0;
        double r77562 = r77560 - r77561;
        double r77563 = r77557 - r77562;
        double r77564 = r77563 / r77551;
        double r77565 = 1.0;
        double r77566 = r77565 / r77553;
        double r77567 = r77550 * r77566;
        double r77568 = 4.0;
        double r77569 = r77568 / r77547;
        double r77570 = r77569 / r77547;
        double r77571 = r77551 / r77547;
        double r77572 = r77570 - r77571;
        double r77573 = 8.0;
        double r77574 = 3.0;
        double r77575 = pow(r77547, r77574);
        double r77576 = r77573 / r77575;
        double r77577 = r77572 - r77576;
        double r77578 = r77567 - r77577;
        double r77579 = r77578 / r77551;
        double r77580 = r77549 ? r77564 : r77579;
        return r77580;
}

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 < 1.982135388898922e+16

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied div-sub0.4

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

    5. Applied associate-+l-0.4

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

    6. Simplified0.4

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

    8. Applied div-inv0.4

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

    9. Simplified0.4

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

    11. Applied flip-+8.8

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

    12. Applied associate-/r/8.8

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

    13. Applied associate-*r*8.8

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

    14. Simplified0.4

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

    if 1.982135388898922e+16 < alpha

    1. Initial program 50.5

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

    2. Simplified50.5

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

    4. Applied div-sub50.5

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

    5. Applied associate-+l-48.9

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

    6. Simplified48.9

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

    8. Applied div-inv48.9

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

    9. Simplified48.9

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

    11. Applied add-cbrt-cube51.4

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

    12. Applied add-cbrt-cube58.4

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

    13. Applied cbrt-undiv58.4

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

    14. Simplified48.9

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

    15. Taylor expanded around inf 18.6

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

    16. Simplified18.6

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 19821353888989220:\\ \;\;\;\;\frac{\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)}}{\left(\beta + \alpha\right) - 2} \cdot \left(\left(\beta + \alpha\right) - 2\right) - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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))