Average Error: 43.0 → 0.8
Time: 29.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + {x}^{3} \cdot \frac{1}{3}\right)}{2} i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + {x}^{3} \cdot \frac{1}{3}\right)}{2} i\right))
double f(double x, double y) {
        double r51818 = x;
        double r51819 = exp(r51818);
        double r51820 = -r51818;
        double r51821 = exp(r51820);
        double r51822 = r51819 + r51821;
        double r51823 = 2.0;
        double r51824 = r51822 / r51823;
        double r51825 = y;
        double r51826 = cos(r51825);
        double r51827 = r51824 * r51826;
        double r51828 = r51819 - r51821;
        double r51829 = r51828 / r51823;
        double r51830 = sin(r51825);
        double r51831 = r51829 * r51830;
        double r51832 = /* ERROR: no complex support in C */;
        double r51833 = /* ERROR: no complex support in C */;
        return r51833;
}
↓
double f(double x, double y) {
        double r51834 = x;
        double r51835 = exp(r51834);
        double r51836 = -r51834;
        double r51837 = exp(r51836);
        double r51838 = r51835 + r51837;
        double r51839 = 2.0;
        double r51840 = r51838 / r51839;
        double r51841 = y;
        double r51842 = cos(r51841);
        double r51843 = r51840 * r51842;
        double r51844 = sin(r51841);
        double r51845 = 0.016666666666666666;
        double r51846 = 5.0;
        double r51847 = pow(r51834, r51846);
        double r51848 = r51845 * r51847;
        double r51849 = 2.0;
        double r51850 = r51834 * r51849;
        double r51851 = 3.0;
        double r51852 = pow(r51834, r51851);
        double r51853 = 0.3333333333333333;
        double r51854 = r51852 * r51853;
        double r51855 = r51850 + r51854;
        double r51856 = r51848 + r51855;
        double r51857 = r51856 / r51839;
        double r51858 = r51844 * r51857;
        double r51859 = /* ERROR: no complex support in C */;
        double r51860 = /* ERROR: no complex support in C */;
        return r51860;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))