Average Error: 16.1 → 6.0
Time: 2.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 27243826751.57198:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 27243826751.57198:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 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{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r16889330 = beta;
        double r16889331 = alpha;
        double r16889332 = r16889330 - r16889331;
        double r16889333 = r16889331 + r16889330;
        double r16889334 = 2.0;
        double r16889335 = r16889333 + r16889334;
        double r16889336 = r16889332 / r16889335;
        double r16889337 = 1.0;
        double r16889338 = r16889336 + r16889337;
        double r16889339 = r16889338 / r16889334;
        return r16889339;
}
↓
double f(double alpha, double beta) {
        double r16889340 = alpha;
        double r16889341 = 27243826751.57198;
        bool r16889342 = r16889340 <= r16889341;
        double r16889343 = beta;
        double r16889344 = 2.0;
        double r16889345 = r16889343 + r16889340;
        double r16889346 = r16889344 + r16889345;
        double r16889347 = r16889343 / r16889346;
        double r16889348 = r16889340 / r16889346;
        double r16889349 = 3.0;
        double r16889350 = pow(r16889348, r16889349);
        double r16889351 = 1.0;
        double r16889352 = pow(r16889351, r16889349);
        double r16889353 = r16889350 - r16889352;
        double r16889354 = r16889353 * r16889353;
        double r16889355 = r16889354 * r16889353;
        double r16889356 = cbrt(r16889355);
        double r16889357 = r16889348 * r16889348;
        double r16889358 = r16889351 * r16889348;
        double r16889359 = r16889351 * r16889351;
        double r16889360 = r16889358 + r16889359;
        double r16889361 = r16889357 + r16889360;
        double r16889362 = r16889356 / r16889361;
        double r16889363 = r16889347 - r16889362;
        double r16889364 = r16889363 / r16889344;
        double r16889365 = 4.0;
        double r16889366 = r16889340 * r16889340;
        double r16889367 = r16889365 / r16889366;
        double r16889368 = r16889344 / r16889340;
        double r16889369 = r16889367 - r16889368;
        double r16889370 = 8.0;
        double r16889371 = r16889370 / r16889340;
        double r16889372 = r16889371 / r16889366;
        double r16889373 = r16889369 - r16889372;
        double r16889374 = r16889347 - r16889373;
        double r16889375 = r16889374 / r16889344;
        double r16889376 = r16889342 ? r16889364 : r16889375;
        return r16889376;
}

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 < 27243826751.57198

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied flip3--0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube0.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\color{blue}{\sqrt[3]{\left(\left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}{2.0}\]

    if 27243826751.57198 < alpha

    1. Initial program 49.4

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

    3. Applied div-sub49.3

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

    4. Applied associate-+l-47.9

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

    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

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 27243826751.57198:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)\right) \cdot \left({\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3} - {1.0}^{3}\right)}}{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + \left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(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))