Average Error: 16.3 → 6.0
Time: 17.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8237268244209.631:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 8237268244209.631:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r4505554 = beta;
        double r4505555 = alpha;
        double r4505556 = r4505554 - r4505555;
        double r4505557 = r4505555 + r4505554;
        double r4505558 = 2.0;
        double r4505559 = r4505557 + r4505558;
        double r4505560 = r4505556 / r4505559;
        double r4505561 = 1.0;
        double r4505562 = r4505560 + r4505561;
        double r4505563 = r4505562 / r4505558;
        return r4505563;
}
↓
double f(double alpha, double beta) {
        double r4505564 = alpha;
        double r4505565 = 8237268244209.631;
        bool r4505566 = r4505564 <= r4505565;
        double r4505567 = beta;
        double r4505568 = 2.0;
        double r4505569 = r4505567 + r4505564;
        double r4505570 = r4505568 + r4505569;
        double r4505571 = r4505567 / r4505570;
        double r4505572 = sqrt(r4505570);
        double r4505573 = r4505564 / r4505572;
        double r4505574 = r4505573 / r4505572;
        double r4505575 = 1.0;
        double r4505576 = r4505574 - r4505575;
        double r4505577 = r4505571 - r4505576;
        double r4505578 = r4505577 / r4505568;
        double r4505579 = 4.0;
        double r4505580 = r4505564 * r4505564;
        double r4505581 = r4505579 / r4505580;
        double r4505582 = r4505568 / r4505564;
        double r4505583 = r4505581 - r4505582;
        double r4505584 = 8.0;
        double r4505585 = r4505564 * r4505580;
        double r4505586 = r4505584 / r4505585;
        double r4505587 = r4505583 - r4505586;
        double r4505588 = r4505571 - r4505587;
        double r4505589 = r4505588 / r4505568;
        double r4505590 = r4505566 ? r4505578 : r4505589;
        return r4505590;
}

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 < 8237268244209.631

    1. Initial program 0.2

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.2

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-0.2

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt{\left(\alpha + \beta\right) + 2.0}}} - 1.0\right)}{2.0}\]

    7. Applied associate-/r*0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\frac{\frac{\alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} - 1.0\right)}{2.0}\]

    if 8237268244209.631 < alpha

    1. Initial program 50.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub50.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-48.6

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around -inf 18.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    6. Simplified18.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8237268244209.631:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{\sqrt{2.0 + \left(\beta + \alpha\right)}}}{\sqrt{2.0 + \left(\beta + \alpha\right)}} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))