Average Error: 23.2 → 12.0
Time: 32.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.6141649384551372 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.9556047485876425 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 1.9640182087305322 \cdot 10^{+237}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right)\right) + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\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 2.6141649384551372 \cdot 10^{+41}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\

\mathbf{elif}\;\alpha \le 5.9556047485876425 \cdot 10^{+87}:\\
\;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\

\mathbf{elif}\;\alpha \le 1.9640182087305322 \cdot 10^{+237}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right)\right) + 1.0}{2.0}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r6704033 = alpha;
        double r6704034 = beta;
        double r6704035 = r6704033 + r6704034;
        double r6704036 = r6704034 - r6704033;
        double r6704037 = r6704035 * r6704036;
        double r6704038 = 2.0;
        double r6704039 = i;
        double r6704040 = r6704038 * r6704039;
        double r6704041 = r6704035 + r6704040;
        double r6704042 = r6704037 / r6704041;
        double r6704043 = 2.0;
        double r6704044 = r6704041 + r6704043;
        double r6704045 = r6704042 / r6704044;
        double r6704046 = 1.0;
        double r6704047 = r6704045 + r6704046;
        double r6704048 = r6704047 / r6704043;
        return r6704048;
}
↓
double f(double alpha, double beta, double i) {
        double r6704049 = alpha;
        double r6704050 = 2.6141649384551372e+41;
        bool r6704051 = r6704049 <= r6704050;
        double r6704052 = beta;
        double r6704053 = r6704052 + r6704049;
        double r6704054 = r6704052 - r6704049;
        double r6704055 = i;
        double r6704056 = 2.0;
        double r6704057 = r6704055 * r6704056;
        double r6704058 = r6704057 + r6704053;
        double r6704059 = r6704054 / r6704058;
        double r6704060 = 2.0;
        double r6704061 = r6704060 + r6704058;
        double r6704062 = r6704059 / r6704061;
        double r6704063 = r6704053 * r6704062;
        double r6704064 = 1.0;
        double r6704065 = r6704063 + r6704064;
        double r6704066 = r6704065 * r6704065;
        double r6704067 = r6704065 * r6704066;
        double r6704068 = cbrt(r6704067);
        double r6704069 = r6704068 / r6704060;
        double r6704070 = 5.9556047485876425e+87;
        bool r6704071 = r6704049 <= r6704070;
        double r6704072 = r6704060 / r6704049;
        double r6704073 = 8.0;
        double r6704074 = r6704049 * r6704049;
        double r6704075 = r6704049 * r6704074;
        double r6704076 = r6704073 / r6704075;
        double r6704077 = 4.0;
        double r6704078 = r6704077 / r6704074;
        double r6704079 = r6704076 - r6704078;
        double r6704080 = r6704072 + r6704079;
        double r6704081 = r6704080 / r6704060;
        double r6704082 = 1.9640182087305322e+237;
        bool r6704083 = r6704049 <= r6704082;
        double r6704084 = cbrt(r6704062);
        double r6704085 = r6704084 * r6704084;
        double r6704086 = r6704053 * r6704085;
        double r6704087 = r6704084 * r6704086;
        double r6704088 = r6704087 + r6704064;
        double r6704089 = r6704088 / r6704060;
        double r6704090 = r6704083 ? r6704089 : r6704081;
        double r6704091 = r6704071 ? r6704081 : r6704090;
        double r6704092 = r6704051 ? r6704069 : r6704091;
        return r6704092;
}

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 3 regimes

  2. if alpha < 2.6141649384551372e+41

    1. Initial program 11.3

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

    3. Applied *-un-lft-identity11.3

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

    4. Applied *-un-lft-identity11.3

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

    5. Applied times-frac1.2

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

    6. Applied times-frac1.2

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

    7. Simplified1.2

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

    9. Applied add-cbrt-cube1.2

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

    if 2.6141649384551372e+41 < alpha < 5.9556047485876425e+87 or 1.9640182087305322e+237 < alpha

    1. Initial program 53.0

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

    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

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

    if 5.9556047485876425e+87 < alpha < 1.9640182087305322e+237

    1. Initial program 53.0

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

    3. Applied *-un-lft-identity53.0

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

    4. Applied *-un-lft-identity53.0

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

    5. Applied times-frac37.6

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

    6. Applied times-frac37.5

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

    7. Simplified37.5

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

    9. Applied add-cube-cbrt37.5

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

    10. Applied associate-*r*37.5

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.6141649384551372 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1.0\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.9556047485876425 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 1.9640182087305322 \cdot 10^{+237}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \left(\left(\beta + \alpha\right) \cdot \left(\sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right)\right) + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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))