Average Error: 23.7 → 11.1
Time: 22.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}{2}\\ \end{array}\]
\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 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\
\;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r94743 = alpha;
        double r94744 = beta;
        double r94745 = r94743 + r94744;
        double r94746 = r94744 - r94743;
        double r94747 = r94745 * r94746;
        double r94748 = 2.0;
        double r94749 = i;
        double r94750 = r94748 * r94749;
        double r94751 = r94745 + r94750;
        double r94752 = r94747 / r94751;
        double r94753 = r94751 + r94748;
        double r94754 = r94752 / r94753;
        double r94755 = 1.0;
        double r94756 = r94754 + r94755;
        double r94757 = r94756 / r94748;
        return r94757;
}
↓
double f(double alpha, double beta, double i) {
        double r94758 = alpha;
        double r94759 = 1.244272110236543e+212;
        bool r94760 = r94758 <= r94759;
        double r94761 = beta;
        double r94762 = r94761 + r94758;
        double r94763 = r94761 - r94758;
        double r94764 = 2.0;
        double r94765 = i;
        double r94766 = r94764 * r94765;
        double r94767 = r94758 + r94766;
        double r94768 = r94761 + r94767;
        double r94769 = r94764 + r94768;
        double r94770 = r94763 / r94769;
        double r94771 = r94762 * r94770;
        double r94772 = r94766 + r94762;
        double r94773 = r94771 / r94772;
        double r94774 = 1.0;
        double r94775 = r94773 + r94774;
        double r94776 = exp(r94775);
        double r94777 = sqrt(r94776);
        double r94778 = log(r94777);
        double r94779 = r94778 + r94778;
        double r94780 = r94779 / r94764;
        double r94781 = r94764 / r94758;
        double r94782 = 8.0;
        double r94783 = 3.0;
        double r94784 = pow(r94758, r94783);
        double r94785 = r94782 / r94784;
        double r94786 = 4.0;
        double r94787 = r94786 / r94758;
        double r94788 = r94787 / r94758;
        double r94789 = r94785 - r94788;
        double r94790 = r94781 + r94789;
        double r94791 = r94790 / r94764;
        double r94792 = r94760 ? r94780 : r94791;
        return r94792;
}

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 < 1.244272110236543e+212

    1. Initial program 18.9

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

    2. Simplified18.9

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

    4. Applied add-log-exp18.9

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

    5. Applied add-log-exp18.9

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

    6. Applied sum-log18.9

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

    7. Simplified7.3

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

    9. Applied div-inv7.3

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

    10. Applied associate-*l*7.3

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

    11. Simplified7.3

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

    13. Applied add-sqr-sqrt7.3

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

    14. Applied log-prod7.3

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

    15. Simplified7.3

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

    16. Simplified7.3

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

    if 1.244272110236543e+212 < alpha

    1. Initial program 64.0

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

    2. Simplified64.0

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

    4. Applied add-log-exp64.0

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

    5. Applied add-log-exp64.0

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

    6. Applied sum-log64.0

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

    7. Simplified50.1

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

    8. Taylor expanded around inf 42.9

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

    9. Simplified42.9

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))