Average Error: 23.5 → 11.6
Time: 31.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.158427829609547 \cdot 10^{+106}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.158427829609547 \cdot 10^{+106}:\\
\;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r3443918 = alpha;
        double r3443919 = beta;
        double r3443920 = r3443918 + r3443919;
        double r3443921 = r3443919 - r3443918;
        double r3443922 = r3443920 * r3443921;
        double r3443923 = 2.0;
        double r3443924 = i;
        double r3443925 = r3443923 * r3443924;
        double r3443926 = r3443920 + r3443925;
        double r3443927 = r3443922 / r3443926;
        double r3443928 = 2.0;
        double r3443929 = r3443926 + r3443928;
        double r3443930 = r3443927 / r3443929;
        double r3443931 = 1.0;
        double r3443932 = r3443930 + r3443931;
        double r3443933 = r3443932 / r3443928;
        return r3443933;
}
↓
double f(double alpha, double beta, double i) {
        double r3443934 = alpha;
        double r3443935 = 3.158427829609547e+106;
        bool r3443936 = r3443934 <= r3443935;
        double r3443937 = 1.0;
        double r3443938 = beta;
        double r3443939 = r3443938 + r3443934;
        double r3443940 = r3443938 - r3443934;
        double r3443941 = 2.0;
        double r3443942 = i;
        double r3443943 = r3443941 * r3443942;
        double r3443944 = r3443939 + r3443943;
        double r3443945 = r3443940 / r3443944;
        double r3443946 = 2.0;
        double r3443947 = r3443946 + r3443944;
        double r3443948 = cbrt(r3443947);
        double r3443949 = r3443948 * r3443948;
        double r3443950 = r3443945 / r3443949;
        double r3443951 = r3443950 / r3443948;
        double r3443952 = r3443939 * r3443951;
        double r3443953 = r3443937 + r3443952;
        double r3443954 = r3443953 / r3443946;
        double r3443955 = r3443946 / r3443934;
        double r3443956 = 4.0;
        double r3443957 = r3443934 * r3443934;
        double r3443958 = r3443956 / r3443957;
        double r3443959 = r3443955 - r3443958;
        double r3443960 = 8.0;
        double r3443961 = r3443957 * r3443934;
        double r3443962 = r3443960 / r3443961;
        double r3443963 = r3443959 + r3443962;
        double r3443964 = r3443963 / r3443946;
        double r3443965 = r3443936 ? r3443954 : r3443964;
        return r3443965;
}

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 < 3.158427829609547e+106

    1. Initial program 13.8

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

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

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

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

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac3.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac3.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified3.4

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt3.5

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    10. Applied associate-/r*3.5

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    if 3.158427829609547e+106 < alpha

    1. Initial program 58.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 40.8

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified40.8

      \[\leadsto \frac{\color{blue}{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.158427829609547 \cdot 10^{+106}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))