Average Error: 23.8 → 12.4
Time: 40.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.021952390523362835514021153198990246696 \cdot 10^{45}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2}\\ \end{array}\]
\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 6.021952390523362835514021153198990246696 \cdot 10^{45}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4706060 = alpha;
        double r4706061 = beta;
        double r4706062 = r4706060 + r4706061;
        double r4706063 = r4706061 - r4706060;
        double r4706064 = r4706062 * r4706063;
        double r4706065 = 2.0;
        double r4706066 = i;
        double r4706067 = r4706065 * r4706066;
        double r4706068 = r4706062 + r4706067;
        double r4706069 = r4706064 / r4706068;
        double r4706070 = r4706068 + r4706065;
        double r4706071 = r4706069 / r4706070;
        double r4706072 = 1.0;
        double r4706073 = r4706071 + r4706072;
        double r4706074 = r4706073 / r4706065;
        return r4706074;
}
↓
double f(double alpha, double beta, double i) {
        double r4706075 = alpha;
        double r4706076 = 6.021952390523363e+45;
        bool r4706077 = r4706075 <= r4706076;
        double r4706078 = 1.0;
        double r4706079 = beta;
        double r4706080 = r4706075 + r4706079;
        double r4706081 = i;
        double r4706082 = 2.0;
        double r4706083 = r4706081 * r4706082;
        double r4706084 = r4706080 + r4706083;
        double r4706085 = r4706079 - r4706075;
        double r4706086 = r4706084 / r4706085;
        double r4706087 = r4706086 / r4706080;
        double r4706088 = r4706078 / r4706087;
        double r4706089 = r4706082 + r4706084;
        double r4706090 = r4706088 / r4706089;
        double r4706091 = 1.0;
        double r4706092 = r4706090 + r4706091;
        double r4706093 = r4706092 * r4706092;
        double r4706094 = r4706093 * r4706092;
        double r4706095 = cbrt(r4706094);
        double r4706096 = r4706095 / r4706082;
        double r4706097 = r4706082 / r4706075;
        double r4706098 = 4.0;
        double r4706099 = r4706075 * r4706075;
        double r4706100 = r4706098 / r4706099;
        double r4706101 = r4706097 - r4706100;
        double r4706102 = 8.0;
        double r4706103 = r4706099 * r4706075;
        double r4706104 = r4706102 / r4706103;
        double r4706105 = r4706101 + r4706104;
        double r4706106 = r4706105 / r4706082;
        double r4706107 = r4706077 ? r4706096 : r4706106;
        return r4706107;
}

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 < 6.021952390523363e+45

    1. Initial program 11.7

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

    3. Applied associate-/l*1.2

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

    5. Applied clear-num1.2

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

    7. Applied add-cbrt-cube1.2

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

    if 6.021952390523363e+45 < alpha

    1. Initial program 54.8

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

    3. Applied associate-/l*40.1

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

    5. Applied *-un-lft-identity40.1

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

    6. Applied associate-/r/40.1

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

    7. Applied times-frac40.1

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

    8. Taylor expanded around inf 41.1

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

    9. Simplified41.1

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.021952390523362835514021153198990246696 \cdot 10^{45}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)\right) \cdot \left(\frac{\frac{1}{\frac{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\beta - \alpha}}{\alpha + \beta}}}{2 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))