Average Error: 15.9 → 3.1
Time: 26.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999996507407:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999996507407:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \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}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3515272 = beta;
        double r3515273 = alpha;
        double r3515274 = r3515272 - r3515273;
        double r3515275 = r3515273 + r3515272;
        double r3515276 = 2.0;
        double r3515277 = r3515275 + r3515276;
        double r3515278 = r3515274 / r3515277;
        double r3515279 = 1.0;
        double r3515280 = r3515278 + r3515279;
        double r3515281 = r3515280 / r3515276;
        return r3515281;
}
↓
double f(double alpha, double beta) {
        double r3515282 = beta;
        double r3515283 = alpha;
        double r3515284 = r3515282 - r3515283;
        double r3515285 = r3515283 + r3515282;
        double r3515286 = 2.0;
        double r3515287 = r3515285 + r3515286;
        double r3515288 = r3515284 / r3515287;
        double r3515289 = -0.9999999996507407;
        bool r3515290 = r3515288 <= r3515289;
        double r3515291 = r3515282 / r3515287;
        double r3515292 = 4.0;
        double r3515293 = r3515283 * r3515283;
        double r3515294 = r3515292 / r3515293;
        double r3515295 = r3515286 / r3515283;
        double r3515296 = r3515294 - r3515295;
        double r3515297 = 8.0;
        double r3515298 = r3515293 * r3515283;
        double r3515299 = r3515297 / r3515298;
        double r3515300 = r3515296 - r3515299;
        double r3515301 = r3515291 - r3515300;
        double r3515302 = r3515301 / r3515286;
        double r3515303 = r3515283 / r3515287;
        double r3515304 = 1.0;
        double r3515305 = r3515303 - r3515304;
        double r3515306 = r3515291 - r3515305;
        double r3515307 = log(r3515306);
        double r3515308 = exp(r3515307);
        double r3515309 = r3515308 / r3515286;
        double r3515310 = r3515290 ? r3515302 : r3515309;
        return r3515310;
}

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 (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999996507407

    1. Initial program 60.0

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

    3. Applied div-sub60.0

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

    4. Applied associate-+l-58.1

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

    5. Taylor expanded around -inf 11.3

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

    6. Simplified11.3

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

    if -0.9999999996507407 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    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-exp-log0.2

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999996507407:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))