Average Error: 54.3 → 36.9
Time: 1.2m
Precision: 64
\[\alpha \gt -1.0 \land \beta \gt -1.0 \land i \gt 1.0\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\
\;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta, double i) {
        double r3761751 = i;
        double r3761752 = alpha;
        double r3761753 = beta;
        double r3761754 = r3761752 + r3761753;
        double r3761755 = r3761754 + r3761751;
        double r3761756 = r3761751 * r3761755;
        double r3761757 = r3761753 * r3761752;
        double r3761758 = r3761757 + r3761756;
        double r3761759 = r3761756 * r3761758;
        double r3761760 = 2.0;
        double r3761761 = r3761760 * r3761751;
        double r3761762 = r3761754 + r3761761;
        double r3761763 = r3761762 * r3761762;
        double r3761764 = r3761759 / r3761763;
        double r3761765 = 1.0;
        double r3761766 = r3761763 - r3761765;
        double r3761767 = r3761764 / r3761766;
        return r3761767;
}
↓
double f(double alpha, double beta, double i) {
        double r3761768 = beta;
        double r3761769 = 1.605441052792236e+216;
        bool r3761770 = r3761768 <= r3761769;
        double r3761771 = i;
        double r3761772 = alpha;
        double r3761773 = r3761768 + r3761772;
        double r3761774 = r3761771 + r3761773;
        double r3761775 = r3761771 * r3761774;
        double r3761776 = r3761772 * r3761768;
        double r3761777 = r3761775 + r3761776;
        double r3761778 = 2.0;
        double r3761779 = r3761778 * r3761771;
        double r3761780 = r3761773 + r3761779;
        double r3761781 = r3761777 / r3761780;
        double r3761782 = r3761775 / r3761780;
        double r3761783 = 1.0;
        double r3761784 = sqrt(r3761783);
        double r3761785 = r3761784 + r3761780;
        double r3761786 = r3761782 / r3761785;
        double r3761787 = r3761781 * r3761786;
        double r3761788 = 1.0;
        double r3761789 = r3761780 - r3761784;
        double r3761790 = r3761788 / r3761789;
        double r3761791 = r3761787 * r3761790;
        double r3761792 = 0.0;
        double r3761793 = r3761770 ? r3761791 : r3761792;
        return r3761793;
}

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 beta < 1.605441052792236e+216

    1. Initial program 53.3

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

    3. Applied add-sqr-sqrt53.3

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

    4. Applied difference-of-squares53.3

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

    5. Applied times-frac38.7

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

    6. Applied times-frac36.4

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

    8. Applied div-inv36.4

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

    9. Applied associate-*r*36.4

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

    if 1.605441052792236e+216 < beta

    1. Initial program 64.0

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

    2. Taylor expanded around inf 41.0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification36.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))