Average Error: 43.2 → 0.8
Time: 13.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r54901 = x;
        double r54902 = exp(r54901);
        double r54903 = -r54901;
        double r54904 = exp(r54903);
        double r54905 = r54902 + r54904;
        double r54906 = 2.0;
        double r54907 = r54905 / r54906;
        double r54908 = y;
        double r54909 = cos(r54908);
        double r54910 = r54907 * r54909;
        double r54911 = r54902 - r54904;
        double r54912 = r54911 / r54906;
        double r54913 = sin(r54908);
        double r54914 = r54912 * r54913;
        double r54915 = /* ERROR: no complex support in C */;
        double r54916 = /* ERROR: no complex support in C */;
        return r54916;
}
↓
double f(double x, double y) {
        double r54917 = x;
        double r54918 = exp(r54917);
        double r54919 = -r54917;
        double r54920 = exp(r54919);
        double r54921 = r54918 + r54920;
        double r54922 = 2.0;
        double r54923 = r54921 / r54922;
        double r54924 = y;
        double r54925 = cos(r54924);
        double r54926 = r54923 * r54925;
        double r54927 = 0.3333333333333333;
        double r54928 = 3.0;
        double r54929 = pow(r54917, r54928);
        double r54930 = 0.016666666666666666;
        double r54931 = 5.0;
        double r54932 = pow(r54917, r54931);
        double r54933 = 2.0;
        double r54934 = r54933 * r54917;
        double r54935 = fma(r54930, r54932, r54934);
        double r54936 = fma(r54927, r54929, r54935);
        double r54937 = r54936 / r54922;
        double r54938 = sin(r54924);
        double r54939 = r54937 * r54938;
        double r54940 = /* ERROR: no complex support in C */;
        double r54941 = /* ERROR: no complex support in C */;
        return r54941;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))))