Average Error: 43.9 → 0.7
Time: 12.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r55032 = x;
        double r55033 = exp(r55032);
        double r55034 = -r55032;
        double r55035 = exp(r55034);
        double r55036 = r55033 + r55035;
        double r55037 = 2.0;
        double r55038 = r55036 / r55037;
        double r55039 = y;
        double r55040 = cos(r55039);
        double r55041 = r55038 * r55040;
        double r55042 = r55033 - r55035;
        double r55043 = r55042 / r55037;
        double r55044 = sin(r55039);
        double r55045 = r55043 * r55044;
        double r55046 = /* ERROR: no complex support in C */;
        double r55047 = /* ERROR: no complex support in C */;
        return r55047;
}
↓
double f(double x, double y) {
        double r55048 = 0.3333333333333333;
        double r55049 = x;
        double r55050 = 3.0;
        double r55051 = pow(r55049, r55050);
        double r55052 = r55048 * r55051;
        double r55053 = 0.016666666666666666;
        double r55054 = 5.0;
        double r55055 = pow(r55049, r55054);
        double r55056 = r55053 * r55055;
        double r55057 = 2.0;
        double r55058 = r55057 * r55049;
        double r55059 = r55056 + r55058;
        double r55060 = r55052 + r55059;
        double r55061 = 2.0;
        double r55062 = r55060 / r55061;
        double r55063 = y;
        double r55064 = sin(r55063);
        double r55065 = r55062 * r55064;
        return r55065;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020001 
(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)))))