Average Error: 23.7 → 11.3
Time: 53.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.272018169352953388394985640277233981571 \cdot 10^{256}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 8.272018169352953388394985640277233981571 \cdot 10^{256}:\\
\;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}\right)}{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 r145719 = alpha;
        double r145720 = beta;
        double r145721 = r145719 + r145720;
        double r145722 = r145720 - r145719;
        double r145723 = r145721 * r145722;
        double r145724 = 2.0;
        double r145725 = i;
        double r145726 = r145724 * r145725;
        double r145727 = r145721 + r145726;
        double r145728 = r145723 / r145727;
        double r145729 = r145727 + r145724;
        double r145730 = r145728 / r145729;
        double r145731 = 1.0;
        double r145732 = r145730 + r145731;
        double r145733 = r145732 / r145724;
        return r145733;
}
↓
double f(double alpha, double beta, double i) {
        double r145734 = alpha;
        double r145735 = 8.272018169352953e+256;
        bool r145736 = r145734 <= r145735;
        double r145737 = beta;
        double r145738 = r145734 + r145737;
        double r145739 = 2.0;
        double r145740 = i;
        double r145741 = r145739 * r145740;
        double r145742 = r145738 + r145741;
        double r145743 = r145742 + r145739;
        double r145744 = r145737 - r145734;
        double r145745 = r145744 / r145742;
        double r145746 = r145743 / r145745;
        double r145747 = r145738 / r145746;
        double r145748 = 1.0;
        double r145749 = r145747 + r145748;
        double r145750 = exp(r145749);
        double r145751 = log(r145750);
        double r145752 = r145751 / r145739;
        double r145753 = 1.0;
        double r145754 = r145753 / r145734;
        double r145755 = r145739 * r145754;
        double r145756 = 8.0;
        double r145757 = 3.0;
        double r145758 = pow(r145734, r145757);
        double r145759 = r145753 / r145758;
        double r145760 = r145756 * r145759;
        double r145761 = r145755 + r145760;
        double r145762 = 4.0;
        double r145763 = 2.0;
        double r145764 = pow(r145734, r145763);
        double r145765 = r145753 / r145764;
        double r145766 = r145762 * r145765;
        double r145767 = r145761 - r145766;
        double r145768 = r145767 / r145739;
        double r145769 = r145736 ? r145752 : r145768;
        return r145769;
}

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 < 8.272018169352953e+256

    1. Initial program 21.3

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

    3. Applied *-un-lft-identity21.3

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

    4. Applied times-frac9.4

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

    5. Applied associate-/l*9.4

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

    7. Applied add-log-exp9.4

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

    8. Applied add-log-exp9.4

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

    9. Applied sum-log9.4

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

    10. Simplified9.4

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

    if 8.272018169352953e+256 < 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. Using strategy rm

    3. Applied *-un-lft-identity64.0

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

    4. Applied times-frac53.4

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

    5. Applied associate-/l*53.4

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

    6. Taylor expanded around inf 43.2

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.272018169352953388394985640277233981571 \cdot 10^{256}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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))