Average Error: 43.5 → 0.7
Time: 33.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(\left(x + x\right) + \frac{1}{60} \cdot {x}^{5}\right)}{2} \cdot \sin y i\right))\]
\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 \left(x \cdot \left(x \cdot x\right)\right) + \left(\left(x + x\right) + \frac{1}{60} \cdot {x}^{5}\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1018977 = x;
        double r1018978 = exp(r1018977);
        double r1018979 = -r1018977;
        double r1018980 = exp(r1018979);
        double r1018981 = r1018978 + r1018980;
        double r1018982 = 2.0;
        double r1018983 = r1018981 / r1018982;
        double r1018984 = y;
        double r1018985 = cos(r1018984);
        double r1018986 = r1018983 * r1018985;
        double r1018987 = r1018978 - r1018980;
        double r1018988 = r1018987 / r1018982;
        double r1018989 = sin(r1018984);
        double r1018990 = r1018988 * r1018989;
        double r1018991 = /* ERROR: no complex support in C */;
        double r1018992 = /* ERROR: no complex support in C */;
        return r1018992;
}
↓
double f(double x, double y) {
        double r1018993 = x;
        double r1018994 = exp(r1018993);
        double r1018995 = -r1018993;
        double r1018996 = exp(r1018995);
        double r1018997 = r1018994 + r1018996;
        double r1018998 = 2.0;
        double r1018999 = r1018997 / r1018998;
        double r1019000 = y;
        double r1019001 = cos(r1019000);
        double r1019002 = r1018999 * r1019001;
        double r1019003 = 0.3333333333333333;
        double r1019004 = r1018993 * r1018993;
        double r1019005 = r1018993 * r1019004;
        double r1019006 = r1019003 * r1019005;
        double r1019007 = r1018993 + r1018993;
        double r1019008 = 0.016666666666666666;
        double r1019009 = 5.0;
        double r1019010 = pow(r1018993, r1019009);
        double r1019011 = r1019008 * r1019010;
        double r1019012 = r1019007 + r1019011;
        double r1019013 = r1019006 + r1019012;
        double r1019014 = r1019013 / r1018998;
        double r1019015 = sin(r1019000);
        double r1019016 = r1019014 * r1019015;
        double r1019017 = /* ERROR: no complex support in C */;
        double r1019018 = /* ERROR: no complex support in C */;
        return r1019018;
}

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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))