Average Error: 43.5 → 0.9
Time: 40.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2089886 = x;
        double r2089887 = exp(r2089886);
        double r2089888 = -r2089886;
        double r2089889 = exp(r2089888);
        double r2089890 = r2089887 + r2089889;
        double r2089891 = 2.0;
        double r2089892 = r2089890 / r2089891;
        double r2089893 = y;
        double r2089894 = cos(r2089893);
        double r2089895 = r2089892 * r2089894;
        double r2089896 = r2089887 - r2089889;
        double r2089897 = r2089896 / r2089891;
        double r2089898 = sin(r2089893);
        double r2089899 = r2089897 * r2089898;
        double r2089900 = /* ERROR: no complex support in C */;
        double r2089901 = /* ERROR: no complex support in C */;
        return r2089901;
}
↓
double f(double x, double y) {
        double r2089902 = x;
        double r2089903 = exp(r2089902);
        double r2089904 = -r2089902;
        double r2089905 = exp(r2089904);
        double r2089906 = r2089903 + r2089905;
        double r2089907 = 2.0;
        double r2089908 = r2089906 / r2089907;
        double r2089909 = y;
        double r2089910 = cos(r2089909);
        double r2089911 = r2089908 * r2089910;
        double r2089912 = 0.016666666666666666;
        double r2089913 = 5.0;
        double r2089914 = pow(r2089902, r2089913);
        double r2089915 = 0.3333333333333333;
        double r2089916 = r2089915 * r2089902;
        double r2089917 = r2089902 * r2089916;
        double r2089918 = r2089917 + r2089907;
        double r2089919 = r2089902 * r2089918;
        double r2089920 = fma(r2089912, r2089914, r2089919);
        double r2089921 = r2089920 / r2089907;
        double r2089922 = sin(r2089909);
        double r2089923 = r2089921 * r2089922;
        double r2089924 = /* ERROR: no complex support in C */;
        double r2089925 = /* ERROR: no complex support in C */;
        return r2089925;
}

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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)))))