Average Error: 43.7 → 0.7
Time: 29.5s
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 r49969 = x;
        double r49970 = exp(r49969);
        double r49971 = -r49969;
        double r49972 = exp(r49971);
        double r49973 = r49970 + r49972;
        double r49974 = 2.0;
        double r49975 = r49973 / r49974;
        double r49976 = y;
        double r49977 = cos(r49976);
        double r49978 = r49975 * r49977;
        double r49979 = r49970 - r49972;
        double r49980 = r49979 / r49974;
        double r49981 = sin(r49976);
        double r49982 = r49980 * r49981;
        double r49983 = /* ERROR: no complex support in C */;
        double r49984 = /* ERROR: no complex support in C */;
        return r49984;
}
↓
double f(double x, double y) {
        double r49985 = 0.3333333333333333;
        double r49986 = x;
        double r49987 = 3.0;
        double r49988 = pow(r49986, r49987);
        double r49989 = r49985 * r49988;
        double r49990 = 0.016666666666666666;
        double r49991 = 5.0;
        double r49992 = pow(r49986, r49991);
        double r49993 = r49990 * r49992;
        double r49994 = 2.0;
        double r49995 = r49994 * r49986;
        double r49996 = r49993 + r49995;
        double r49997 = r49989 + r49996;
        double r49998 = 2.0;
        double r49999 = r49997 / r49998;
        double r50000 = y;
        double r50001 = sin(r50000);
        double r50002 = r49999 * r50001;
        return r50002;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019212 
(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)))))