Average Error: 16.5 → 6.1
Time: 21.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 12052311324865314.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{2.0 + \left(\beta + \alpha\right)}, 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 12052311324865314.0:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{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}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3234257 = beta;
        double r3234258 = alpha;
        double r3234259 = r3234257 - r3234258;
        double r3234260 = r3234258 + r3234257;
        double r3234261 = 2.0;
        double r3234262 = r3234260 + r3234261;
        double r3234263 = r3234259 / r3234262;
        double r3234264 = 1.0;
        double r3234265 = r3234263 + r3234264;
        double r3234266 = r3234265 / r3234261;
        return r3234266;
}
↓
double f(double alpha, double beta) {
        double r3234267 = alpha;
        double r3234268 = 12052311324865314.0;
        bool r3234269 = r3234267 <= r3234268;
        double r3234270 = beta;
        double r3234271 = r3234270 - r3234267;
        double r3234272 = 1.0;
        double r3234273 = 2.0;
        double r3234274 = r3234270 + r3234267;
        double r3234275 = r3234273 + r3234274;
        double r3234276 = r3234272 / r3234275;
        double r3234277 = 1.0;
        double r3234278 = fma(r3234271, r3234276, r3234277);
        double r3234279 = r3234278 / r3234273;
        double r3234280 = r3234270 / r3234275;
        double r3234281 = 4.0;
        double r3234282 = r3234267 * r3234267;
        double r3234283 = r3234281 / r3234282;
        double r3234284 = r3234273 / r3234267;
        double r3234285 = r3234283 - r3234284;
        double r3234286 = 8.0;
        double r3234287 = r3234282 * r3234267;
        double r3234288 = r3234286 / r3234287;
        double r3234289 = r3234285 - r3234288;
        double r3234290 = r3234280 - r3234289;
        double r3234291 = r3234290 / r3234273;
        double r3234292 = r3234269 ? r3234279 : r3234291;
        return r3234292;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 12052311324865314.0

    1. Initial program 0.4

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

    3. Applied div-inv0.4

      \[\leadsto \frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\]

    4. Applied fma-def0.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2.0}, 1.0\right)}}{2.0}\]

    if 12052311324865314.0 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.8

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

    6. Applied add-log-exp48.8

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

    7. Taylor expanded around inf 18.0

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

    8. Simplified18.0

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 12052311324865314.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{2.0 + \left(\beta + \alpha\right)}, 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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))