Average Error: 43.5 → 0.6
Time: 20.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49889 = x;
        double r49890 = exp(r49889);
        double r49891 = -r49889;
        double r49892 = exp(r49891);
        double r49893 = r49890 + r49892;
        double r49894 = 2.0;
        double r49895 = r49893 / r49894;
        double r49896 = y;
        double r49897 = cos(r49896);
        double r49898 = r49895 * r49897;
        double r49899 = r49890 - r49892;
        double r49900 = r49899 / r49894;
        double r49901 = sin(r49896);
        double r49902 = r49900 * r49901;
        double r49903 = /* ERROR: no complex support in C */;
        double r49904 = /* ERROR: no complex support in C */;
        return r49904;
}
↓
double f(double x, double y) {
        double r49905 = x;
        double r49906 = exp(r49905);
        double r49907 = -r49905;
        double r49908 = exp(r49907);
        double r49909 = r49906 + r49908;
        double r49910 = 2.0;
        double r49911 = r49909 / r49910;
        double r49912 = y;
        double r49913 = cos(r49912);
        double r49914 = r49911 * r49913;
        double r49915 = 0.3333333333333333;
        double r49916 = 3.0;
        double r49917 = pow(r49905, r49916);
        double r49918 = r49915 * r49917;
        double r49919 = 0.016666666666666666;
        double r49920 = 5.0;
        double r49921 = pow(r49905, r49920);
        double r49922 = r49919 * r49921;
        double r49923 = 2.0;
        double r49924 = r49923 * r49905;
        double r49925 = r49922 + r49924;
        double r49926 = r49918 + r49925;
        double r49927 = r49926 / r49910;
        double r49928 = sin(r49912);
        double r49929 = r49927 * r49928;
        double r49930 = /* ERROR: no complex support in C */;
        double r49931 = /* ERROR: no complex support in C */;
        return r49931;
}

Error

Bits error versus x

Bits error versus y

Derivation

  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 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))