Average Error: 24.1 → 11.7
Time: 8.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.9832343195288184 \cdot 10^{93}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 5.9832343195288184 \cdot 10^{93}:\\
\;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r87881 = alpha;
        double r87882 = beta;
        double r87883 = r87881 + r87882;
        double r87884 = r87882 - r87881;
        double r87885 = r87883 * r87884;
        double r87886 = 2.0;
        double r87887 = i;
        double r87888 = r87886 * r87887;
        double r87889 = r87883 + r87888;
        double r87890 = r87885 / r87889;
        double r87891 = r87889 + r87886;
        double r87892 = r87890 / r87891;
        double r87893 = 1.0;
        double r87894 = r87892 + r87893;
        double r87895 = r87894 / r87886;
        return r87895;
}
↓
double f(double alpha, double beta, double i) {
        double r87896 = alpha;
        double r87897 = 5.983234319528818e+93;
        bool r87898 = r87896 <= r87897;
        double r87899 = beta;
        double r87900 = r87896 + r87899;
        double r87901 = r87899 - r87896;
        double r87902 = 2.0;
        double r87903 = i;
        double r87904 = r87902 * r87903;
        double r87905 = r87900 + r87904;
        double r87906 = r87901 / r87905;
        double r87907 = r87900 * r87906;
        double r87908 = r87905 + r87902;
        double r87909 = r87907 / r87908;
        double r87910 = 1.0;
        double r87911 = r87909 + r87910;
        double r87912 = log(r87911);
        double r87913 = 3.0;
        double r87914 = pow(r87912, r87913);
        double r87915 = cbrt(r87914);
        double r87916 = exp(r87915);
        double r87917 = r87916 / r87902;
        double r87918 = 8.0;
        double r87919 = 1.0;
        double r87920 = pow(r87896, r87913);
        double r87921 = r87919 / r87920;
        double r87922 = r87918 * r87921;
        double r87923 = 4.0;
        double r87924 = 2.0;
        double r87925 = pow(r87896, r87924);
        double r87926 = r87919 / r87925;
        double r87927 = r87923 * r87926;
        double r87928 = r87922 - r87927;
        double r87929 = r87902 / r87896;
        double r87930 = r87928 + r87929;
        double r87931 = r87930 / r87902;
        double r87932 = r87898 ? r87917 : r87931;
        return r87932;
}

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 < 5.983234319528818e+93

    1. Initial program 13.8

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

    3. Applied *-un-lft-identity13.8

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

    4. Applied times-frac3.1

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

    5. Simplified3.1

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

    7. Applied add-exp-log3.1

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

    9. Applied add-cbrt-cube3.1

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

    10. Simplified3.1

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

    if 5.983234319528818e+93 < alpha

    1. Initial program 59.3

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

    3. Applied *-un-lft-identity59.3

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

    4. Applied times-frac43.5

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

    5. Simplified43.5

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

    6. Taylor expanded around inf 41.2

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

    7. Simplified41.2

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.9832343195288184 \cdot 10^{93}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))