Average Error: 43.4 → 0.8
Time: 12.2s
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 r44024 = x;
        double r44025 = exp(r44024);
        double r44026 = -r44024;
        double r44027 = exp(r44026);
        double r44028 = r44025 + r44027;
        double r44029 = 2.0;
        double r44030 = r44028 / r44029;
        double r44031 = y;
        double r44032 = cos(r44031);
        double r44033 = r44030 * r44032;
        double r44034 = r44025 - r44027;
        double r44035 = r44034 / r44029;
        double r44036 = sin(r44031);
        double r44037 = r44035 * r44036;
        double r44038 = /* ERROR: no complex support in C */;
        double r44039 = /* ERROR: no complex support in C */;
        return r44039;
}
↓
double f(double x, double y) {
        double r44040 = x;
        double r44041 = exp(r44040);
        double r44042 = -r44040;
        double r44043 = exp(r44042);
        double r44044 = r44041 + r44043;
        double r44045 = 2.0;
        double r44046 = r44044 / r44045;
        double r44047 = y;
        double r44048 = cos(r44047);
        double r44049 = r44046 * r44048;
        double r44050 = 0.3333333333333333;
        double r44051 = 3.0;
        double r44052 = pow(r44040, r44051);
        double r44053 = r44050 * r44052;
        double r44054 = 0.016666666666666666;
        double r44055 = 5.0;
        double r44056 = pow(r44040, r44055);
        double r44057 = r44054 * r44056;
        double r44058 = 2.0;
        double r44059 = r44058 * r44040;
        double r44060 = r44057 + r44059;
        double r44061 = r44053 + r44060;
        double r44062 = r44061 / r44045;
        double r44063 = sin(r44047);
        double r44064 = r44062 * r44063;
        double r44065 = /* ERROR: no complex support in C */;
        double r44066 = /* ERROR: no complex support in C */;
        return r44066;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  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 2020060 
(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)))))