Average Error: 24.2 → 12.0
Time: 13.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.820006962648860775833447636342447093329 \cdot 10^{95}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\log \left(e^{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 2.820006962648860775833447636342447093329 \cdot 10^{95}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\log \left(e^{\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} + 1}\right)\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r139098 = alpha;
        double r139099 = beta;
        double r139100 = r139098 + r139099;
        double r139101 = r139099 - r139098;
        double r139102 = r139100 * r139101;
        double r139103 = 2.0;
        double r139104 = i;
        double r139105 = r139103 * r139104;
        double r139106 = r139100 + r139105;
        double r139107 = r139102 / r139106;
        double r139108 = r139106 + r139103;
        double r139109 = r139107 / r139108;
        double r139110 = 1.0;
        double r139111 = r139109 + r139110;
        double r139112 = r139111 / r139103;
        return r139112;
}
↓
double f(double alpha, double beta, double i) {
        double r139113 = alpha;
        double r139114 = 2.820006962648861e+95;
        bool r139115 = r139113 <= r139114;
        double r139116 = beta;
        double r139117 = r139113 + r139116;
        double r139118 = r139116 - r139113;
        double r139119 = 2.0;
        double r139120 = i;
        double r139121 = r139119 * r139120;
        double r139122 = r139117 + r139121;
        double r139123 = r139118 / r139122;
        double r139124 = r139122 + r139119;
        double r139125 = r139123 / r139124;
        double r139126 = r139117 * r139125;
        double r139127 = 1.0;
        double r139128 = r139126 + r139127;
        double r139129 = exp(r139128);
        double r139130 = log(r139129);
        double r139131 = 3.0;
        double r139132 = pow(r139130, r139131);
        double r139133 = cbrt(r139132);
        double r139134 = r139133 / r139119;
        double r139135 = 1.0;
        double r139136 = r139135 / r139113;
        double r139137 = r139119 * r139136;
        double r139138 = 8.0;
        double r139139 = pow(r139113, r139131);
        double r139140 = r139135 / r139139;
        double r139141 = r139138 * r139140;
        double r139142 = r139137 + r139141;
        double r139143 = 4.0;
        double r139144 = 2.0;
        double r139145 = pow(r139113, r139144);
        double r139146 = r139135 / r139145;
        double r139147 = r139143 * r139146;
        double r139148 = r139142 - r139147;
        double r139149 = r139148 / r139119;
        double r139150 = r139115 ? r139134 : r139149;
        return r139150;
}

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 < 2.820006962648861e+95

    1. Initial program 13.9

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

    3. Applied *-un-lft-identity13.9

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

    4. Applied *-un-lft-identity13.9

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

    5. Applied times-frac3.1

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

    6. Applied times-frac3.1

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

    7. Simplified3.1

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

    9. Applied add-cbrt-cube3.2

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

    10. Simplified3.2

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

    12. Applied add-log-exp3.2

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

    13. Applied add-log-exp3.2

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

    14. Applied sum-log3.2

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

    15. Simplified3.2

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

    if 2.820006962648861e+95 < alpha

    1. Initial program 58.2

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

    2. Taylor expanded around inf 41.1

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.820006962648860775833447636342447093329 \cdot 10^{95}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\log \left(e^{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))