Average Error: 43.9 → 0.7
Time: 11.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r33864 = x;
        double r33865 = exp(r33864);
        double r33866 = -r33864;
        double r33867 = exp(r33866);
        double r33868 = r33865 + r33867;
        double r33869 = 2.0;
        double r33870 = r33868 / r33869;
        double r33871 = y;
        double r33872 = cos(r33871);
        double r33873 = r33870 * r33872;
        double r33874 = r33865 - r33867;
        double r33875 = r33874 / r33869;
        double r33876 = sin(r33871);
        double r33877 = r33875 * r33876;
        double r33878 = /* ERROR: no complex support in C */;
        double r33879 = /* ERROR: no complex support in C */;
        return r33879;
}
↓
double f(double x, double y) {
        double r33880 = 0.3333333333333333;
        double r33881 = x;
        double r33882 = 3.0;
        double r33883 = pow(r33881, r33882);
        double r33884 = 0.016666666666666666;
        double r33885 = 5.0;
        double r33886 = pow(r33881, r33885);
        double r33887 = 2.0;
        double r33888 = r33887 * r33881;
        double r33889 = fma(r33884, r33886, r33888);
        double r33890 = fma(r33880, r33883, r33889);
        double r33891 = 2.0;
        double r33892 = r33890 / r33891;
        double r33893 = y;
        double r33894 = sin(r33893);
        double r33895 = r33892 * r33894;
        return r33895;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020001 +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)))))