Average Error: 43.8 → 0.8
Time: 51.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{x \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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 \frac{1}{3}\right) \cdot x + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1565059 = x;
        double r1565060 = exp(r1565059);
        double r1565061 = -r1565059;
        double r1565062 = exp(r1565061);
        double r1565063 = r1565060 + r1565062;
        double r1565064 = 2.0;
        double r1565065 = r1565063 / r1565064;
        double r1565066 = y;
        double r1565067 = cos(r1565066);
        double r1565068 = r1565065 * r1565067;
        double r1565069 = r1565060 - r1565062;
        double r1565070 = r1565069 / r1565064;
        double r1565071 = sin(r1565066);
        double r1565072 = r1565070 * r1565071;
        double r1565073 = /* ERROR: no complex support in C */;
        double r1565074 = /* ERROR: no complex support in C */;
        return r1565074;
}
↓
double f(double x, double y) {
        double r1565075 = x;
        double r1565076 = exp(r1565075);
        double r1565077 = -r1565075;
        double r1565078 = exp(r1565077);
        double r1565079 = r1565076 + r1565078;
        double r1565080 = 2.0;
        double r1565081 = r1565079 / r1565080;
        double r1565082 = y;
        double r1565083 = cos(r1565082);
        double r1565084 = r1565081 * r1565083;
        double r1565085 = 0.3333333333333333;
        double r1565086 = r1565075 * r1565085;
        double r1565087 = r1565086 * r1565075;
        double r1565088 = r1565087 + r1565080;
        double r1565089 = r1565075 * r1565088;
        double r1565090 = 5.0;
        double r1565091 = pow(r1565075, r1565090);
        double r1565092 = 0.016666666666666666;
        double r1565093 = r1565091 * r1565092;
        double r1565094 = r1565089 + r1565093;
        double r1565095 = r1565094 / r1565080;
        double r1565096 = sin(r1565082);
        double r1565097 = r1565095 * r1565096;
        double r1565098 = /* ERROR: no complex support in C */;
        double r1565099 = /* ERROR: no complex support in C */;
        return r1565099;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019149 
(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)))))