Average Error: 44.2 → 0.8
Time: 32.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 r58947 = x;
        double r58948 = exp(r58947);
        double r58949 = -r58947;
        double r58950 = exp(r58949);
        double r58951 = r58948 + r58950;
        double r58952 = 2.0;
        double r58953 = r58951 / r58952;
        double r58954 = y;
        double r58955 = cos(r58954);
        double r58956 = r58953 * r58955;
        double r58957 = r58948 - r58950;
        double r58958 = r58957 / r58952;
        double r58959 = sin(r58954);
        double r58960 = r58958 * r58959;
        double r58961 = /* ERROR: no complex support in C */;
        double r58962 = /* ERROR: no complex support in C */;
        return r58962;
}
↓
double f(double x, double y) {
        double r58963 = x;
        double r58964 = exp(r58963);
        double r58965 = -r58963;
        double r58966 = exp(r58965);
        double r58967 = r58964 + r58966;
        double r58968 = 2.0;
        double r58969 = r58967 / r58968;
        double r58970 = y;
        double r58971 = cos(r58970);
        double r58972 = r58969 * r58971;
        double r58973 = 0.3333333333333333;
        double r58974 = 3.0;
        double r58975 = pow(r58963, r58974);
        double r58976 = r58973 * r58975;
        double r58977 = 0.016666666666666666;
        double r58978 = 5.0;
        double r58979 = pow(r58963, r58978);
        double r58980 = r58977 * r58979;
        double r58981 = 2.0;
        double r58982 = r58981 * r58963;
        double r58983 = r58980 + r58982;
        double r58984 = r58976 + r58983;
        double r58985 = r58984 / r58968;
        double r58986 = sin(r58970);
        double r58987 = r58985 * r58986;
        double r58988 = /* ERROR: no complex support in C */;
        double r58989 = /* ERROR: no complex support in C */;
        return r58989;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.2

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

  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 2019208 
(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)))))