Average Error: 16.4 → 7.0
Time: 7.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0243503852289514 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.0243503852289514 \cdot 10^{45}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \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 r170004 = beta;
        double r170005 = alpha;
        double r170006 = r170004 - r170005;
        double r170007 = r170005 + r170004;
        double r170008 = 2.0;
        double r170009 = r170007 + r170008;
        double r170010 = r170006 / r170009;
        double r170011 = 1.0;
        double r170012 = r170010 + r170011;
        double r170013 = r170012 / r170008;
        return r170013;
}
↓
double f(double alpha, double beta) {
        double r170014 = alpha;
        double r170015 = 1.0243503852289514e+45;
        bool r170016 = r170014 <= r170015;
        double r170017 = beta;
        double r170018 = r170014 + r170017;
        double r170019 = 2.0;
        double r170020 = r170018 + r170019;
        double r170021 = r170017 / r170020;
        double r170022 = r170014 / r170020;
        double r170023 = 1.0;
        double r170024 = r170022 - r170023;
        double r170025 = exp(r170024);
        double r170026 = log(r170025);
        double r170027 = r170021 - r170026;
        double r170028 = r170027 / r170019;
        double r170029 = 4.0;
        double r170030 = 1.0;
        double r170031 = 2.0;
        double r170032 = pow(r170014, r170031);
        double r170033 = r170030 / r170032;
        double r170034 = r170030 / r170014;
        double r170035 = 8.0;
        double r170036 = 3.0;
        double r170037 = pow(r170014, r170036);
        double r170038 = r170030 / r170037;
        double r170039 = r170035 * r170038;
        double r170040 = fma(r170019, r170034, r170039);
        double r170041 = -r170040;
        double r170042 = fma(r170029, r170033, r170041);
        double r170043 = r170021 - r170042;
        double r170044 = r170043 / r170019;
        double r170045 = r170016 ? r170028 : r170044;
        return r170045;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.0243503852289514e+45

    1. Initial program 2.2

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

    3. Applied div-sub2.2

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

    4. Applied associate-+l-2.2

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

    6. Applied log1p-expm1-u2.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\]
    7. Using strategy rm

    8. Applied add-log-exp2.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}\right)}}{2}\]

    9. Simplified2.2

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

    if 1.0243503852289514e+45 < alpha

    1. Initial program 51.3

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

    3. Applied div-sub51.3

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

    4. Applied associate-+l-49.4

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

    6. Applied log1p-expm1-u49.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\]
    7. Using strategy rm

    8. Applied add-log-exp49.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}\right)}}{2}\]

    9. Simplified49.5

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

    10. Taylor expanded around inf 18.8

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

    11. Simplified18.8

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0243503852289514 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +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))