Average Error: 24.0 → 11.4
Time: 4.8m
Precision: 64
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.1178533449480407 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}\right)\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}} \cdot e^{1.0}\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 3.592412999901975 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.1854505227927157 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta + \alpha}{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}}{\frac{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
double f(double alpha, double beta, double i) {
        double r18789997 = alpha;
        double r18789998 = beta;
        double r18789999 = r18789997 + r18789998;
        double r18790000 = r18789998 - r18789997;
        double r18790001 = r18789999 * r18790000;
        double r18790002 = 2.0;
        double r18790003 = i;
        double r18790004 = r18790002 * r18790003;
        double r18790005 = r18789999 + r18790004;
        double r18790006 = r18790001 / r18790005;
        double r18790007 = 2.0;
        double r18790008 = r18790005 + r18790007;
        double r18790009 = r18790006 / r18790008;
        double r18790010 = 1.0;
        double r18790011 = r18790009 + r18790010;
        double r18790012 = r18790011 / r18790007;
        return r18790012;
}
↓
double f(double alpha, double beta, double i) {
        double r18790013 = alpha;
        double r18790014 = 1.1178533449480407e+147;
        bool r18790015 = r18790013 <= r18790014;
        double r18790016 = beta;
        double r18790017 = r18790016 + r18790013;
        double r18790018 = 2.0;
        double r18790019 = i;
        double r18790020 = r18790018 * r18790019;
        double r18790021 = r18790020 + r18790017;
        double r18790022 = r18790016 - r18790013;
        double r18790023 = r18790021 / r18790022;
        double r18790024 = r18790017 / r18790023;
        double r18790025 = 2.0;
        double r18790026 = r18790025 + r18790021;
        double r18790027 = r18790024 / r18790026;
        double r18790028 = 1.0;
        double r18790029 = r18790027 + r18790028;
        double r18790030 = exp(r18790029);
        double r18790031 = log(r18790030);
        double r18790032 = r18790029 * r18790031;
        double r18790033 = exp(r18790027);
        double r18790034 = exp(r18790028);
        double r18790035 = r18790033 * r18790034;
        double r18790036 = log(r18790035);
        double r18790037 = r18790032 * r18790036;
        double r18790038 = cbrt(r18790037);
        double r18790039 = r18790038 / r18790025;
        double r18790040 = 3.592412999901975e+171;
        bool r18790041 = r18790013 <= r18790040;
        double r18790042 = 8.0;
        double r18790043 = r18790042 / r18790013;
        double r18790044 = 4.0;
        double r18790045 = r18790043 - r18790044;
        double r18790046 = r18790045 / r18790013;
        double r18790047 = r18790046 / r18790013;
        double r18790048 = r18790025 / r18790013;
        double r18790049 = r18790047 + r18790048;
        double r18790050 = r18790049 / r18790025;
        double r18790051 = 1.1854505227927157e+225;
        bool r18790052 = r18790013 <= r18790051;
        double r18790053 = sqrt(r18790021);
        double r18790054 = r18790017 / r18790053;
        double r18790055 = r18790053 / r18790022;
        double r18790056 = r18790054 / r18790055;
        double r18790057 = r18790056 / r18790026;
        double r18790058 = r18790057 + r18790028;
        double r18790059 = r18790058 / r18790025;
        double r18790060 = r18790052 ? r18790059 : r18790050;
        double r18790061 = r18790041 ? r18790050 : r18790060;
        double r18790062 = r18790015 ? r18790039 : r18790061;
        return r18790062;
}


\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 1.1178533449480407 \cdot 10^{+147}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}\right)\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}} \cdot e^{1.0}\right)}}{2.0}\\

\mathbf{elif}\;\alpha \le 3.592412999901975 \cdot 10^{+171}:\\
\;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\

\mathbf{elif}\;\alpha \le 1.1854505227927157 \cdot 10^{+225}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\beta + \alpha}{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}}{\frac{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\

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

\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if alpha < 1.1178533449480407e+147

    1. Initial program 16.0

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

    3. Applied associate-/l*5.2

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

    5. Applied add-cbrt-cube5.2

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

    7. Applied add-log-exp5.2

      \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\log \left(e^{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}\right)} \cdot \left(\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}{2.0}\]
    8. Using strategy rm

    9. Applied add-log-exp5.2

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

    10. Applied add-log-exp5.2

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

    11. Applied sum-log5.2

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

    if 1.1178533449480407e+147 < alpha < 3.592412999901975e+171 or 1.1854505227927157e+225 < alpha

    1. Initial program 62.4

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

    3. Applied associate-/l*48.7

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

    4. Taylor expanded around -inf 40.9

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

    5. Simplified40.9

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}}{2.0}\]

    if 3.592412999901975e+171 < alpha < 1.1854505227927157e+225

    1. Initial program 63.2

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

    3. Applied associate-/l*42.3

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

    5. Applied *-un-lft-identity42.3

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

    6. Applied *-un-lft-identity42.3

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

    7. Applied distribute-lft-out--42.3

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

    8. Applied add-sqr-sqrt42.5

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

    9. Applied times-frac42.6

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

    10. Applied associate-/r*42.5

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.1178533449480407 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}\right)\right) \cdot \log \left(e^{\frac{\frac{\beta + \alpha}{\frac{2 \cdot i + \left(\beta + \alpha\right)}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}} \cdot e^{1.0}\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 3.592412999901975 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.1854505227927157 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\beta + \alpha}{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}}{\frac{\sqrt{2 \cdot i + \left(\beta + \alpha\right)}}{\beta - \alpha}}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1.0}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(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))