Average Error: 24.2 → 11.7
Time: 21.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \frac{1}{{\alpha}^{2}}, \mathsf{fma}\left(8, \frac{1}{{\alpha}^{3}}, \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\
\;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r184883 = alpha;
        double r184884 = beta;
        double r184885 = r184883 + r184884;
        double r184886 = r184884 - r184883;
        double r184887 = r184885 * r184886;
        double r184888 = 2.0;
        double r184889 = i;
        double r184890 = r184888 * r184889;
        double r184891 = r184885 + r184890;
        double r184892 = r184887 / r184891;
        double r184893 = r184891 + r184888;
        double r184894 = r184892 / r184893;
        double r184895 = 1.0;
        double r184896 = r184894 + r184895;
        double r184897 = r184896 / r184888;
        return r184897;
}
↓
double f(double alpha, double beta, double i) {
        double r184898 = alpha;
        double r184899 = 5.849296461317042e+141;
        bool r184900 = r184898 <= r184899;
        double r184901 = beta;
        double r184902 = r184898 + r184901;
        double r184903 = 1.0;
        double r184904 = r184902 / r184903;
        double r184905 = r184904 / r184903;
        double r184906 = r184901 - r184898;
        double r184907 = 2.0;
        double r184908 = i;
        double r184909 = r184907 * r184908;
        double r184910 = r184902 + r184909;
        double r184911 = r184906 / r184910;
        double r184912 = r184910 + r184907;
        double r184913 = r184911 / r184912;
        double r184914 = 1.0;
        double r184915 = fma(r184905, r184913, r184914);
        double r184916 = log(r184915);
        double r184917 = exp(r184916);
        double r184918 = r184917 / r184907;
        double r184919 = 4.0;
        double r184920 = -r184919;
        double r184921 = 2.0;
        double r184922 = pow(r184898, r184921);
        double r184923 = r184903 / r184922;
        double r184924 = 8.0;
        double r184925 = 3.0;
        double r184926 = pow(r184898, r184925);
        double r184927 = r184903 / r184926;
        double r184928 = r184907 / r184898;
        double r184929 = fma(r184924, r184927, r184928);
        double r184930 = fma(r184920, r184923, r184929);
        double r184931 = r184930 / r184907;
        double r184932 = r184900 ? r184918 : r184931;
        return r184932;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.849296461317042e+141

    1. Initial program 15.4

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

    3. Applied *-un-lft-identity15.4

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

    4. Applied *-un-lft-identity15.4

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

    5. Applied times-frac4.8

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

    6. Applied times-frac4.8

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

    7. Applied fma-def4.7

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2}\]
    8. Using strategy rm

    9. Applied add-exp-log4.8

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

    if 5.849296461317042e+141 < alpha

    1. Initial program 62.6

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

    3. Applied *-un-lft-identity62.6

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

    4. Applied *-un-lft-identity62.6

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

    5. Applied times-frac46.8

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

    6. Applied times-frac46.8

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

    7. Applied fma-def47.6

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

    8. Taylor expanded around inf 41.8

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

    9. Simplified41.8

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \frac{1}{{\alpha}^{2}}, \mathsf{fma}\left(8, \frac{1}{{\alpha}^{3}}, \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))