Average Error: 43.3 → 0.9
Time: 23.9s
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 r54812 = x;
        double r54813 = exp(r54812);
        double r54814 = -r54812;
        double r54815 = exp(r54814);
        double r54816 = r54813 + r54815;
        double r54817 = 2.0;
        double r54818 = r54816 / r54817;
        double r54819 = y;
        double r54820 = cos(r54819);
        double r54821 = r54818 * r54820;
        double r54822 = r54813 - r54815;
        double r54823 = r54822 / r54817;
        double r54824 = sin(r54819);
        double r54825 = r54823 * r54824;
        double r54826 = /* ERROR: no complex support in C */;
        double r54827 = /* ERROR: no complex support in C */;
        return r54827;
}
↓
double f(double x, double y) {
        double r54828 = 0.3333333333333333;
        double r54829 = x;
        double r54830 = 3.0;
        double r54831 = pow(r54829, r54830);
        double r54832 = r54828 * r54831;
        double r54833 = 0.016666666666666666;
        double r54834 = 5.0;
        double r54835 = pow(r54829, r54834);
        double r54836 = r54833 * r54835;
        double r54837 = 2.0;
        double r54838 = r54837 * r54829;
        double r54839 = r54836 + r54838;
        double r54840 = r54832 + r54839;
        double r54841 = 2.0;
        double r54842 = r54840 / r54841;
        double r54843 = y;
        double r54844 = sin(r54843);
        double r54845 = r54842 * r54844;
        return r54845;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Taylor expanded around 0 0.9

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

  3. Final simplification0.9

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

Reproduce

herbie shell --seed 2019308 
(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)))))