Average Error: 16.1 → 6.0
Time: 5.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1188863810682620.75:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1188863810682620.75:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r113560 = beta;
        double r113561 = alpha;
        double r113562 = r113560 - r113561;
        double r113563 = r113561 + r113560;
        double r113564 = 2.0;
        double r113565 = r113563 + r113564;
        double r113566 = r113562 / r113565;
        double r113567 = 1.0;
        double r113568 = r113566 + r113567;
        double r113569 = r113568 / r113564;
        return r113569;
}
↓
double f(double alpha, double beta) {
        double r113570 = alpha;
        double r113571 = 1188863810682620.8;
        bool r113572 = r113570 <= r113571;
        double r113573 = beta;
        double r113574 = r113570 + r113573;
        double r113575 = 2.0;
        double r113576 = r113574 + r113575;
        double r113577 = r113573 / r113576;
        double r113578 = 3.0;
        double r113579 = pow(r113577, r113578);
        double r113580 = cbrt(r113579);
        double r113581 = r113570 / r113576;
        double r113582 = 1.0;
        double r113583 = r113581 - r113582;
        double r113584 = r113580 - r113583;
        double r113585 = r113584 / r113575;
        double r113586 = 4.0;
        double r113587 = r113586 / r113570;
        double r113588 = r113587 / r113570;
        double r113589 = r113575 / r113570;
        double r113590 = 8.0;
        double r113591 = -r113590;
        double r113592 = pow(r113570, r113578);
        double r113593 = r113591 / r113592;
        double r113594 = r113589 - r113593;
        double r113595 = r113588 - r113594;
        double r113596 = r113577 - r113595;
        double r113597 = r113596 / r113575;
        double r113598 = r113572 ? r113585 : r113597;
        return r113598;
}

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 < 1188863810682620.8

    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-cbrt-cube11.2

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

    7. Applied add-cbrt-cube13.7

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

    8. Applied cbrt-undiv13.7

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

    9. Simplified0.3

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

    if 1188863810682620.8 < alpha

    1. Initial program 50.2

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

    3. Applied div-sub50.2

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1188863810682620.75:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))