Average Error: 24.0 → 11.3
Time: 28.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.142935579114712 \cdot 10^{152}:\\ \;\;\;\;\frac{\log \left(e^{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 1.142935579114712 \cdot 10^{152}:\\
\;\;\;\;\frac{\log \left(e^{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}\right)}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r87822 = alpha;
        double r87823 = beta;
        double r87824 = r87822 + r87823;
        double r87825 = r87823 - r87822;
        double r87826 = r87824 * r87825;
        double r87827 = 2.0;
        double r87828 = i;
        double r87829 = r87827 * r87828;
        double r87830 = r87824 + r87829;
        double r87831 = r87826 / r87830;
        double r87832 = r87830 + r87827;
        double r87833 = r87831 / r87832;
        double r87834 = 1.0;
        double r87835 = r87833 + r87834;
        double r87836 = r87835 / r87827;
        return r87836;
}
↓
double f(double alpha, double beta, double i) {
        double r87837 = alpha;
        double r87838 = 1.142935579114712e+152;
        bool r87839 = r87837 <= r87838;
        double r87840 = beta;
        double r87841 = r87840 + r87837;
        double r87842 = r87840 - r87837;
        double r87843 = r87837 + r87840;
        double r87844 = 2.0;
        double r87845 = i;
        double r87846 = r87844 * r87845;
        double r87847 = r87843 + r87846;
        double r87848 = r87842 / r87847;
        double r87849 = r87847 + r87844;
        double r87850 = r87848 / r87849;
        double r87851 = r87841 * r87850;
        double r87852 = 1.0;
        double r87853 = r87851 + r87852;
        double r87854 = exp(r87853);
        double r87855 = log(r87854);
        double r87856 = r87855 / r87844;
        double r87857 = r87844 / r87837;
        double r87858 = 8.0;
        double r87859 = 3.0;
        double r87860 = pow(r87837, r87859);
        double r87861 = r87858 / r87860;
        double r87862 = r87857 + r87861;
        double r87863 = 4.0;
        double r87864 = r87837 * r87837;
        double r87865 = r87863 / r87864;
        double r87866 = r87862 - r87865;
        double r87867 = r87866 / r87844;
        double r87868 = r87839 ? r87856 : r87867;
        return r87868;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.142935579114712e+152

    1. Initial program 15.9

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

    3. Applied *-un-lft-identity15.9

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

    4. Applied *-un-lft-identity15.9

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

    5. Applied times-frac5.5

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

    6. Applied times-frac5.5

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

    7. Simplified5.5

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

    9. Applied add-log-exp5.5

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

    10. Applied add-log-exp5.5

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

    11. Applied sum-log5.5

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

    12. Simplified5.5

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

    if 1.142935579114712e+152 < alpha

    1. Initial program 63.8

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

    2. Taylor expanded around inf 40.2

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

    3. Simplified40.2

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.142935579114712 \cdot 10^{152}:\\ \;\;\;\;\frac{\log \left(e^{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))