Average Error: 44.3 → 0.8
Time: 12.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r46965 = x;
        double r46966 = exp(r46965);
        double r46967 = -r46965;
        double r46968 = exp(r46967);
        double r46969 = r46966 + r46968;
        double r46970 = 2.0;
        double r46971 = r46969 / r46970;
        double r46972 = y;
        double r46973 = cos(r46972);
        double r46974 = r46971 * r46973;
        double r46975 = r46966 - r46968;
        double r46976 = r46975 / r46970;
        double r46977 = sin(r46972);
        double r46978 = r46976 * r46977;
        double r46979 = /* ERROR: no complex support in C */;
        double r46980 = /* ERROR: no complex support in C */;
        return r46980;
}
↓
double f(double x, double y) {
        double r46981 = x;
        double r46982 = exp(r46981);
        double r46983 = -r46981;
        double r46984 = exp(r46983);
        double r46985 = r46982 + r46984;
        double r46986 = 2.0;
        double r46987 = r46985 / r46986;
        double r46988 = y;
        double r46989 = cos(r46988);
        double r46990 = r46987 * r46989;
        double r46991 = 0.3333333333333333;
        double r46992 = 3.0;
        double r46993 = pow(r46981, r46992);
        double r46994 = r46991 * r46993;
        double r46995 = 0.016666666666666666;
        double r46996 = 5.0;
        double r46997 = pow(r46981, r46996);
        double r46998 = r46995 * r46997;
        double r46999 = 2.0;
        double r47000 = r46999 * r46981;
        double r47001 = r46998 + r47000;
        double r47002 = r46994 + r47001;
        double r47003 = r47002 / r46986;
        double r47004 = sin(r46988);
        double r47005 = r47003 * r47004;
        double r47006 = /* ERROR: no complex support in C */;
        double r47007 = /* ERROR: no complex support in C */;
        return r47007;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.3

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020034 
(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)))))