Average Error: 43.0 → 0.5
Time: 1.5m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.014008388915516524:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{{x}^{5} \cdot \frac{1}{60} + \left(\left(\frac{1}{3} \cdot x\right) \cdot x + 2\right) \cdot x}{2} i\right))\\ \mathbf{else}:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{e^{x} \cdot e^{x} - e^{-x} \cdot e^{-x}}{e^{x} + e^{-x}}}{2} i\right))\\ \end{array}\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.014008388915516524:\\
\;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{{x}^{5} \cdot \frac{1}{60} + \left(\left(\frac{1}{3} \cdot x\right) \cdot x + 2\right) \cdot x}{2} i\right))\\

\mathbf{else}:\\
\;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{e^{x} \cdot e^{x} - e^{-x} \cdot e^{-x}}{e^{x} + e^{-x}}}{2} i\right))\\

\end{array}
double f(double x, double y) {
        double r5001789 = x;
        double r5001790 = exp(r5001789);
        double r5001791 = -r5001789;
        double r5001792 = exp(r5001791);
        double r5001793 = r5001790 + r5001792;
        double r5001794 = 2.0;
        double r5001795 = r5001793 / r5001794;
        double r5001796 = y;
        double r5001797 = cos(r5001796);
        double r5001798 = r5001795 * r5001797;
        double r5001799 = r5001790 - r5001792;
        double r5001800 = r5001799 / r5001794;
        double r5001801 = sin(r5001796);
        double r5001802 = r5001800 * r5001801;
        double r5001803 = /* ERROR: no complex support in C */;
        double r5001804 = /* ERROR: no complex support in C */;
        return r5001804;
}
↓
double f(double x, double y) {
        double r5001805 = x;
        double r5001806 = 0.014008388915516524;
        bool r5001807 = r5001805 <= r5001806;
        double r5001808 = exp(r5001805);
        double r5001809 = -r5001805;
        double r5001810 = exp(r5001809);
        double r5001811 = r5001808 + r5001810;
        double r5001812 = 2.0;
        double r5001813 = r5001811 / r5001812;
        double r5001814 = y;
        double r5001815 = cos(r5001814);
        double r5001816 = r5001813 * r5001815;
        double r5001817 = sin(r5001814);
        double r5001818 = 5.0;
        double r5001819 = pow(r5001805, r5001818);
        double r5001820 = 0.016666666666666666;
        double r5001821 = r5001819 * r5001820;
        double r5001822 = 0.3333333333333333;
        double r5001823 = r5001822 * r5001805;
        double r5001824 = r5001823 * r5001805;
        double r5001825 = r5001824 + r5001812;
        double r5001826 = r5001825 * r5001805;
        double r5001827 = r5001821 + r5001826;
        double r5001828 = r5001827 / r5001812;
        double r5001829 = r5001817 * r5001828;
        double r5001830 = /* ERROR: no complex support in C */;
        double r5001831 = /* ERROR: no complex support in C */;
        double r5001832 = r5001808 * r5001808;
        double r5001833 = r5001810 * r5001810;
        double r5001834 = r5001832 - r5001833;
        double r5001835 = r5001834 / r5001811;
        double r5001836 = r5001835 / r5001812;
        double r5001837 = r5001817 * r5001836;
        double r5001838 = /* ERROR: no complex support in C */;
        double r5001839 = /* ERROR: no complex support in C */;
        double r5001840 = r5001807 ? r5001831 : r5001839;
        return r5001840;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < 0.014008388915516524

    1. Initial program 43.5

      \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]

    3. Simplified0.4

      \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}}{2} \cdot \sin y i\right))\]

    if 0.014008388915516524 < x

    1. Initial program 2.3

      \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
    2. Using strategy rm

    3. Applied flip--5.2

      \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - e^{-x} \cdot e^{-x}}{e^{x} + e^{-x}}}}{2} \cdot \sin y i\right))\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.014008388915516524:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{{x}^{5} \cdot \frac{1}{60} + \left(\left(\frac{1}{3} \cdot x\right) \cdot x + 2\right) \cdot x}{2} i\right))\\ \mathbf{else}:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{e^{x} \cdot e^{x} - e^{-x} \cdot e^{-x}}{e^{x} + e^{-x}}}{2} i\right))\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))