Average Error: 23.5 → 11.4
Time: 3.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.58632046728181 \cdot 10^{+124}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
\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 2.58632046728181 \cdot 10^{+124}:\\
\;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r24719944 = alpha;
        double r24719945 = beta;
        double r24719946 = r24719944 + r24719945;
        double r24719947 = r24719945 - r24719944;
        double r24719948 = r24719946 * r24719947;
        double r24719949 = 2.0;
        double r24719950 = i;
        double r24719951 = r24719949 * r24719950;
        double r24719952 = r24719946 + r24719951;
        double r24719953 = r24719948 / r24719952;
        double r24719954 = 2.0;
        double r24719955 = r24719952 + r24719954;
        double r24719956 = r24719953 / r24719955;
        double r24719957 = 1.0;
        double r24719958 = r24719956 + r24719957;
        double r24719959 = r24719958 / r24719954;
        return r24719959;
}
↓
double f(double alpha, double beta, double i) {
        double r24719960 = alpha;
        double r24719961 = 2.58632046728181e+124;
        bool r24719962 = r24719960 <= r24719961;
        double r24719963 = 1.0;
        double r24719964 = beta;
        double r24719965 = r24719964 + r24719960;
        double r24719966 = 2.0;
        double r24719967 = i;
        double r24719968 = r24719966 * r24719967;
        double r24719969 = r24719968 + r24719965;
        double r24719970 = cbrt(r24719969);
        double r24719971 = r24719965 / r24719970;
        double r24719972 = r24719971 / r24719970;
        double r24719973 = r24719964 - r24719960;
        double r24719974 = r24719973 / r24719970;
        double r24719975 = 2.0;
        double r24719976 = r24719969 + r24719975;
        double r24719977 = r24719974 / r24719976;
        double r24719978 = r24719972 * r24719977;
        double r24719979 = r24719963 + r24719978;
        double r24719980 = r24719979 / r24719975;
        double r24719981 = 8.0;
        double r24719982 = r24719981 / r24719960;
        double r24719983 = 4.0;
        double r24719984 = r24719982 - r24719983;
        double r24719985 = r24719960 * r24719960;
        double r24719986 = r24719984 / r24719985;
        double r24719987 = r24719975 / r24719960;
        double r24719988 = r24719986 + r24719987;
        double r24719989 = r24719988 / r24719975;
        double r24719990 = r24719962 ? r24719980 : r24719989;
        return r24719990;
}

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 < 2.58632046728181e+124

    1. Initial program 14.7

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

    3. Applied *-un-lft-identity14.7

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

    4. Applied *-un-lft-identity14.7

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

    5. Applied times-frac4.1

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

    6. Applied times-frac4.1

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

    7. Simplified4.1

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

    9. Applied *-un-lft-identity4.1

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

    10. Applied *-un-lft-identity4.1

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

    11. Applied distribute-lft-out4.1

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

    12. Applied add-cube-cbrt4.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    13. Applied *-un-lft-identity4.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{1 \cdot \left(\beta - \alpha\right)}}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    14. Applied times-frac4.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    15. Applied times-frac4.2

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{1} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)} + 1.0}{2.0}\]

    16. Applied associate-*r*4.2

      \[\leadsto \frac{\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{1}\right) \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    17. Simplified4.2

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

    if 2.58632046728181e+124 < alpha

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 42.2

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

    3. Simplified42.2

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.58632046728181 \cdot 10^{+124}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))