Average Error: 16.1 → 6.1
Time: 22.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 10272132967178968:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3593481 = beta;
        double r3593482 = alpha;
        double r3593483 = r3593481 - r3593482;
        double r3593484 = r3593482 + r3593481;
        double r3593485 = 2.0;
        double r3593486 = r3593484 + r3593485;
        double r3593487 = r3593483 / r3593486;
        double r3593488 = 1.0;
        double r3593489 = r3593487 + r3593488;
        double r3593490 = r3593489 / r3593485;
        return r3593490;
}
↓
double f(double alpha, double beta) {
        double r3593491 = alpha;
        double r3593492 = 10272132967178968.0;
        bool r3593493 = r3593491 <= r3593492;
        double r3593494 = beta;
        double r3593495 = 2.0;
        double r3593496 = r3593494 + r3593491;
        double r3593497 = r3593495 + r3593496;
        double r3593498 = r3593494 / r3593497;
        double r3593499 = r3593498 * r3593498;
        double r3593500 = r3593498 * r3593499;
        double r3593501 = cbrt(r3593500);
        double r3593502 = r3593491 / r3593497;
        double r3593503 = 1.0;
        double r3593504 = r3593502 - r3593503;
        double r3593505 = r3593501 - r3593504;
        double r3593506 = r3593505 / r3593495;
        double r3593507 = 1.0;
        double r3593508 = sqrt(r3593497);
        double r3593509 = r3593507 / r3593508;
        double r3593510 = r3593494 / r3593508;
        double r3593511 = r3593509 * r3593510;
        double r3593512 = 4.0;
        double r3593513 = r3593491 * r3593491;
        double r3593514 = r3593512 / r3593513;
        double r3593515 = r3593495 / r3593491;
        double r3593516 = 8.0;
        double r3593517 = r3593491 * r3593513;
        double r3593518 = r3593516 / r3593517;
        double r3593519 = r3593515 + r3593518;
        double r3593520 = r3593514 - r3593519;
        double r3593521 = r3593511 - r3593520;
        double r3593522 = r3593521 / r3593495;
        double r3593523 = r3593493 ? r3593506 : r3593522;
        return r3593523;
}

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 < 10272132967178968.0

    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 add-cbrt-cube0.5

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

    if 10272132967178968.0 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.4

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

    6. Applied add-sqr-sqrt48.4

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

    7. Applied *-un-lft-identity48.4

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

    8. Applied times-frac48.4

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

    9. Taylor expanded around inf 18.4

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

    10. Simplified18.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))