Average Error: 43.3 → 0.7
Time: 40.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2432950 = x;
        double r2432951 = exp(r2432950);
        double r2432952 = -r2432950;
        double r2432953 = exp(r2432952);
        double r2432954 = r2432951 + r2432953;
        double r2432955 = 2.0;
        double r2432956 = r2432954 / r2432955;
        double r2432957 = y;
        double r2432958 = cos(r2432957);
        double r2432959 = r2432956 * r2432958;
        double r2432960 = r2432951 - r2432953;
        double r2432961 = r2432960 / r2432955;
        double r2432962 = sin(r2432957);
        double r2432963 = r2432961 * r2432962;
        double r2432964 = /* ERROR: no complex support in C */;
        double r2432965 = /* ERROR: no complex support in C */;
        return r2432965;
}
↓
double f(double x, double y) {
        double r2432966 = x;
        double r2432967 = exp(r2432966);
        double r2432968 = -r2432966;
        double r2432969 = exp(r2432968);
        double r2432970 = r2432967 + r2432969;
        double r2432971 = 2.0;
        double r2432972 = r2432970 / r2432971;
        double r2432973 = y;
        double r2432974 = cos(r2432973);
        double r2432975 = r2432972 * r2432974;
        double r2432976 = r2432966 * r2432966;
        double r2432977 = 0.3333333333333333;
        double r2432978 = r2432976 * r2432977;
        double r2432979 = r2432978 + r2432971;
        double r2432980 = r2432966 * r2432979;
        double r2432981 = 5.0;
        double r2432982 = pow(r2432966, r2432981);
        double r2432983 = 0.016666666666666666;
        double r2432984 = r2432982 * r2432983;
        double r2432985 = r2432980 + r2432984;
        double r2432986 = r2432985 / r2432971;
        double r2432987 = sin(r2432973);
        double r2432988 = r2432986 * r2432987;
        double r2432989 = /* ERROR: no complex support in C */;
        double r2432990 = /* ERROR: no complex support in C */;
        return r2432990;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019132 
(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)))))