Average Error: 16.1 → 5.9
Time: 6.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 818395241689.588745:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 818395241689.588745:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\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)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r99247 = beta;
        double r99248 = alpha;
        double r99249 = r99247 - r99248;
        double r99250 = r99248 + r99247;
        double r99251 = 2.0;
        double r99252 = r99250 + r99251;
        double r99253 = r99249 / r99252;
        double r99254 = 1.0;
        double r99255 = r99253 + r99254;
        double r99256 = r99255 / r99251;
        return r99256;
}
↓
double f(double alpha, double beta) {
        double r99257 = alpha;
        double r99258 = 818395241689.5887;
        bool r99259 = r99257 <= r99258;
        double r99260 = beta;
        double r99261 = r99257 + r99260;
        double r99262 = 2.0;
        double r99263 = r99261 + r99262;
        double r99264 = r99260 / r99263;
        double r99265 = r99257 / r99263;
        double r99266 = 1.0;
        double r99267 = r99265 - r99266;
        double r99268 = r99264 - r99267;
        double r99269 = log(r99268);
        double r99270 = exp(r99269);
        double r99271 = r99270 / r99262;
        double r99272 = 1.0;
        double r99273 = sqrt(r99263);
        double r99274 = r99272 / r99273;
        double r99275 = r99260 / r99273;
        double r99276 = r99274 * r99275;
        double r99277 = 4.0;
        double r99278 = 2.0;
        double r99279 = pow(r99257, r99278);
        double r99280 = r99272 / r99279;
        double r99281 = r99272 / r99257;
        double r99282 = 8.0;
        double r99283 = 3.0;
        double r99284 = pow(r99257, r99283);
        double r99285 = r99272 / r99284;
        double r99286 = r99282 * r99285;
        double r99287 = fma(r99262, r99281, r99286);
        double r99288 = -r99287;
        double r99289 = fma(r99277, r99280, r99288);
        double r99290 = r99276 - r99289;
        double r99291 = -r99272;
        double r99292 = r99291 + r99272;
        double r99293 = r99267 * r99292;
        double r99294 = r99290 + r99293;
        double r99295 = r99294 / r99262;
        double r99296 = r99259 ? r99271 : r99295;
        return r99296;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 818395241689.5887

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied add-exp-log0.2

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

    if 818395241689.5887 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Applied associate-+l-48.5

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

    6. Applied add-cube-cbrt48.6

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

    7. Applied add-sqr-sqrt48.6

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

    8. Applied *-un-lft-identity48.6

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

    9. Applied times-frac48.6

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

    10. Applied prod-diff48.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}, \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}}, -\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right)}}{2}\]

    11. Simplified48.6

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right)}{2}\]

    12. Simplified48.6

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

    13. Taylor expanded around inf 18.1

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

    14. Simplified18.1

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 818395241689.588745:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))