Average Error: 23.7 → 11.0
Time: 15.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 1.5371929371909745 \cdot 10^{209}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r95702 = alpha;
        double r95703 = beta;
        double r95704 = r95702 + r95703;
        double r95705 = r95703 - r95702;
        double r95706 = r95704 * r95705;
        double r95707 = 2.0;
        double r95708 = i;
        double r95709 = r95707 * r95708;
        double r95710 = r95704 + r95709;
        double r95711 = r95706 / r95710;
        double r95712 = r95710 + r95707;
        double r95713 = r95711 / r95712;
        double r95714 = 1.0;
        double r95715 = r95713 + r95714;
        double r95716 = r95715 / r95707;
        return r95716;
}
↓
double f(double alpha, double beta, double i) {
        double r95717 = alpha;
        double r95718 = 1.5371929371909745e+209;
        bool r95719 = r95717 <= r95718;
        double r95720 = beta;
        double r95721 = r95717 + r95720;
        double r95722 = 2.0;
        double r95723 = i;
        double r95724 = r95722 * r95723;
        double r95725 = r95721 + r95724;
        double r95726 = r95725 + r95722;
        double r95727 = sqrt(r95726);
        double r95728 = r95721 / r95727;
        double r95729 = r95720 - r95717;
        double r95730 = r95729 / r95725;
        double r95731 = r95727 / r95730;
        double r95732 = r95728 / r95731;
        double r95733 = 1.0;
        double r95734 = r95732 + r95733;
        double r95735 = r95734 * r95734;
        double r95736 = r95734 * r95735;
        double r95737 = cbrt(r95736);
        double r95738 = r95737 / r95722;
        double r95739 = 8.0;
        double r95740 = 3.0;
        double r95741 = pow(r95717, r95740);
        double r95742 = r95739 / r95741;
        double r95743 = r95722 / r95717;
        double r95744 = r95742 + r95743;
        double r95745 = 4.0;
        double r95746 = r95717 * r95717;
        double r95747 = r95745 / r95746;
        double r95748 = r95744 - r95747;
        double r95749 = r95748 / r95722;
        double r95750 = r95719 ? r95738 : r95749;
        return r95750;
}

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 < 1.5371929371909745e+209

    1. Initial program 18.9

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

    3. Applied add-sqr-sqrt18.9

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

    4. Applied *-un-lft-identity18.9

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

    5. Applied times-frac7.4

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

    6. Applied times-frac7.4

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

    7. Simplified7.4

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

    9. Applied add-sqr-sqrt7.4

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

    10. Applied sqrt-prod7.5

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

    11. Applied *-un-lft-identity7.5

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

    12. Applied times-frac7.5

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

    14. Applied add-cbrt-cube7.4

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

    15. Simplified7.4

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

    if 1.5371929371909745e+209 < alpha

    1. Initial program 64.0

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

    2. Taylor expanded around inf 40.5

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

    3. Simplified40.5

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))