Average Error: 43.8 → 0.7
Time: 10.6s
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 r37845 = x;
        double r37846 = exp(r37845);
        double r37847 = -r37845;
        double r37848 = exp(r37847);
        double r37849 = r37846 + r37848;
        double r37850 = 2.0;
        double r37851 = r37849 / r37850;
        double r37852 = y;
        double r37853 = cos(r37852);
        double r37854 = r37851 * r37853;
        double r37855 = r37846 - r37848;
        double r37856 = r37855 / r37850;
        double r37857 = sin(r37852);
        double r37858 = r37856 * r37857;
        double r37859 = /* ERROR: no complex support in C */;
        double r37860 = /* ERROR: no complex support in C */;
        return r37860;
}
↓
double f(double x, double y) {
        double r37861 = 0.3333333333333333;
        double r37862 = x;
        double r37863 = 3.0;
        double r37864 = pow(r37862, r37863);
        double r37865 = 0.016666666666666666;
        double r37866 = 5.0;
        double r37867 = pow(r37862, r37866);
        double r37868 = 2.0;
        double r37869 = r37868 * r37862;
        double r37870 = fma(r37865, r37867, r37869);
        double r37871 = fma(r37861, r37864, r37870);
        double r37872 = 2.0;
        double r37873 = r37871 / r37872;
        double r37874 = y;
        double r37875 = sin(r37874);
        double r37876 = r37873 * r37875;
        return r37876;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  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 2020065 +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)))))