Average Error: 16.6 → 6.1
Time: 7.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6566215071237633024:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6566215071237633024:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r103351 = beta;
        double r103352 = alpha;
        double r103353 = r103351 - r103352;
        double r103354 = r103352 + r103351;
        double r103355 = 2.0;
        double r103356 = r103354 + r103355;
        double r103357 = r103353 / r103356;
        double r103358 = 1.0;
        double r103359 = r103357 + r103358;
        double r103360 = r103359 / r103355;
        return r103360;
}
↓
double f(double alpha, double beta) {
        double r103361 = alpha;
        double r103362 = 6.566215071237633e+18;
        bool r103363 = r103361 <= r103362;
        double r103364 = beta;
        double r103365 = r103361 + r103364;
        double r103366 = 2.0;
        double r103367 = r103365 + r103366;
        double r103368 = r103364 / r103367;
        double r103369 = exp(r103368);
        double r103370 = log(r103369);
        double r103371 = r103361 / r103367;
        double r103372 = 1.0;
        double r103373 = r103371 - r103372;
        double r103374 = exp(r103373);
        double r103375 = log(r103374);
        double r103376 = r103370 - r103375;
        double r103377 = r103376 / r103366;
        double r103378 = 4.0;
        double r103379 = 1.0;
        double r103380 = 2.0;
        double r103381 = pow(r103361, r103380);
        double r103382 = r103379 / r103381;
        double r103383 = r103379 / r103361;
        double r103384 = 8.0;
        double r103385 = 3.0;
        double r103386 = pow(r103361, r103385);
        double r103387 = r103379 / r103386;
        double r103388 = r103384 * r103387;
        double r103389 = fma(r103366, r103383, r103388);
        double r103390 = -r103389;
        double r103391 = fma(r103378, r103382, r103390);
        double r103392 = r103368 - r103391;
        double r103393 = r103392 / r103366;
        double r103394 = r103363 ? r103377 : r103393;
        return r103394;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.566215071237633e+18

    1. Initial program 0.5

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

    3. Applied div-sub0.5

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

    4. Applied associate-+l-0.5

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

    6. Applied add-log-exp0.5

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.6

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

    9. Simplified0.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \color{blue}{\left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}}{2}\]
    10. Using strategy rm

    11. Applied add-log-exp0.5

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

    if 6.566215071237633e+18 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.1

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

    5. Taylor expanded around inf 17.9

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

    6. Simplified17.9

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6566215071237633024:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))