Average Error: 43.5 → 0.6
Time: 19.4s
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 r45959 = x;
        double r45960 = exp(r45959);
        double r45961 = -r45959;
        double r45962 = exp(r45961);
        double r45963 = r45960 + r45962;
        double r45964 = 2.0;
        double r45965 = r45963 / r45964;
        double r45966 = y;
        double r45967 = cos(r45966);
        double r45968 = r45965 * r45967;
        double r45969 = r45960 - r45962;
        double r45970 = r45969 / r45964;
        double r45971 = sin(r45966);
        double r45972 = r45970 * r45971;
        double r45973 = /* ERROR: no complex support in C */;
        double r45974 = /* ERROR: no complex support in C */;
        return r45974;
}
↓
double f(double x, double y) {
        double r45975 = x;
        double r45976 = exp(r45975);
        double r45977 = -r45975;
        double r45978 = exp(r45977);
        double r45979 = r45976 + r45978;
        double r45980 = 2.0;
        double r45981 = r45979 / r45980;
        double r45982 = y;
        double r45983 = cos(r45982);
        double r45984 = r45981 * r45983;
        double r45985 = 0.3333333333333333;
        double r45986 = 3.0;
        double r45987 = pow(r45975, r45986);
        double r45988 = r45985 * r45987;
        double r45989 = 0.016666666666666666;
        double r45990 = 5.0;
        double r45991 = pow(r45975, r45990);
        double r45992 = r45989 * r45991;
        double r45993 = 2.0;
        double r45994 = r45993 * r45975;
        double r45995 = r45992 + r45994;
        double r45996 = r45988 + r45995;
        double r45997 = r45996 / r45980;
        double r45998 = sin(r45982);
        double r45999 = r45997 * r45998;
        double r46000 = /* ERROR: no complex support in C */;
        double r46001 = /* ERROR: no complex support in C */;
        return r46001;
}

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 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 
(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)))))