Average Error: 43.5 → 0.8
Time: 39.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r45864 = x;
        double r45865 = exp(r45864);
        double r45866 = -r45864;
        double r45867 = exp(r45866);
        double r45868 = r45865 + r45867;
        double r45869 = 2.0;
        double r45870 = r45868 / r45869;
        double r45871 = y;
        double r45872 = cos(r45871);
        double r45873 = r45870 * r45872;
        double r45874 = r45865 - r45867;
        double r45875 = r45874 / r45869;
        double r45876 = sin(r45871);
        double r45877 = r45875 * r45876;
        double r45878 = /* ERROR: no complex support in C */;
        double r45879 = /* ERROR: no complex support in C */;
        return r45879;
}
↓
double f(double x, double y) {
        double r45880 = 0.3333333333333333;
        double r45881 = x;
        double r45882 = 3.0;
        double r45883 = pow(r45881, r45882);
        double r45884 = r45880 * r45883;
        double r45885 = 0.016666666666666666;
        double r45886 = 5.0;
        double r45887 = pow(r45881, r45886);
        double r45888 = r45885 * r45887;
        double r45889 = 2.0;
        double r45890 = r45889 * r45881;
        double r45891 = r45888 + r45890;
        double r45892 = r45884 + r45891;
        double r45893 = 2.0;
        double r45894 = r45892 / r45893;
        double r45895 = y;
        double r45896 = sin(r45895);
        double r45897 = r45894 * r45896;
        return r45897;
}

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. Simplified43.5

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019199 
(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)))))