Average Error: 24.6 → 11.4
Time: 8.6m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.9946683067318548 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 1.9946683067318548 \cdot 10^{159}:\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{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 r181714 = alpha;
        double r181715 = beta;
        double r181716 = r181714 + r181715;
        double r181717 = r181715 - r181714;
        double r181718 = r181716 * r181717;
        double r181719 = 2.0;
        double r181720 = i;
        double r181721 = r181719 * r181720;
        double r181722 = r181716 + r181721;
        double r181723 = r181718 / r181722;
        double r181724 = r181722 + r181719;
        double r181725 = r181723 / r181724;
        double r181726 = 1.0;
        double r181727 = r181725 + r181726;
        double r181728 = r181727 / r181719;
        return r181728;
}
↓
double f(double alpha, double beta, double i) {
        double r181729 = alpha;
        double r181730 = 1.994668306731855e+159;
        bool r181731 = r181729 <= r181730;
        double r181732 = beta;
        double r181733 = r181729 + r181732;
        double r181734 = 1.0;
        double r181735 = 2.0;
        double r181736 = i;
        double r181737 = r181735 * r181736;
        double r181738 = r181733 + r181737;
        double r181739 = r181732 - r181729;
        double r181740 = r181738 / r181739;
        double r181741 = r181734 / r181740;
        double r181742 = r181738 + r181735;
        double r181743 = r181741 / r181742;
        double r181744 = r181733 * r181743;
        double r181745 = 1.0;
        double r181746 = r181744 + r181745;
        double r181747 = r181746 / r181735;
        double r181748 = r181734 / r181729;
        double r181749 = r181735 * r181748;
        double r181750 = 8.0;
        double r181751 = 3.0;
        double r181752 = pow(r181729, r181751);
        double r181753 = r181734 / r181752;
        double r181754 = r181750 * r181753;
        double r181755 = r181749 + r181754;
        double r181756 = 4.0;
        double r181757 = 2.0;
        double r181758 = pow(r181729, r181757);
        double r181759 = r181734 / r181758;
        double r181760 = r181756 * r181759;
        double r181761 = r181755 - r181760;
        double r181762 = r181761 / r181735;
        double r181763 = r181731 ? r181747 : r181762;
        return r181763;
}

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 < 1.994668306731855e+159

    1. Initial program 16.7

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

    3. Applied associate-/l*5.7

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

    5. Applied *-un-lft-identity5.7

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

    6. Applied div-inv5.7

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

    7. Applied times-frac5.7

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

    8. Simplified5.7

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

    if 1.994668306731855e+159 < 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 40.2

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.9946683067318548 \cdot 10^{159}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))