Average Error: 43.5 → 0.9
Time: 39.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3}, x \cdot 2\right)\right)}{2} \cdot \sin y i\right))\]
\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({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3}, x \cdot 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2109906 = x;
        double r2109907 = exp(r2109906);
        double r2109908 = -r2109906;
        double r2109909 = exp(r2109908);
        double r2109910 = r2109907 + r2109909;
        double r2109911 = 2.0;
        double r2109912 = r2109910 / r2109911;
        double r2109913 = y;
        double r2109914 = cos(r2109913);
        double r2109915 = r2109912 * r2109914;
        double r2109916 = r2109907 - r2109909;
        double r2109917 = r2109916 / r2109911;
        double r2109918 = sin(r2109913);
        double r2109919 = r2109917 * r2109918;
        double r2109920 = /* ERROR: no complex support in C */;
        double r2109921 = /* ERROR: no complex support in C */;
        return r2109921;
}
↓
double f(double x, double y) {
        double r2109922 = x;
        double r2109923 = exp(r2109922);
        double r2109924 = -r2109922;
        double r2109925 = exp(r2109924);
        double r2109926 = r2109923 + r2109925;
        double r2109927 = 2.0;
        double r2109928 = r2109926 / r2109927;
        double r2109929 = y;
        double r2109930 = cos(r2109929);
        double r2109931 = r2109928 * r2109930;
        double r2109932 = 5.0;
        double r2109933 = pow(r2109922, r2109932);
        double r2109934 = 0.016666666666666666;
        double r2109935 = r2109922 * r2109922;
        double r2109936 = 0.3333333333333333;
        double r2109937 = r2109922 * r2109936;
        double r2109938 = 2.0;
        double r2109939 = r2109922 * r2109938;
        double r2109940 = fma(r2109935, r2109937, r2109939);
        double r2109941 = fma(r2109933, r2109934, r2109940);
        double r2109942 = r2109941 / r2109927;
        double r2109943 = sin(r2109929);
        double r2109944 = r2109942 * r2109943;
        double r2109945 = /* ERROR: no complex support in C */;
        double r2109946 = /* ERROR: no complex support in C */;
        return r2109946;
}

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

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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)))))