Average Error: 44.1 → 0.7
Time: 15.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r61052 = x;
        double r61053 = exp(r61052);
        double r61054 = -r61052;
        double r61055 = exp(r61054);
        double r61056 = r61053 + r61055;
        double r61057 = 2.0;
        double r61058 = r61056 / r61057;
        double r61059 = y;
        double r61060 = cos(r61059);
        double r61061 = r61058 * r61060;
        double r61062 = r61053 - r61055;
        double r61063 = r61062 / r61057;
        double r61064 = sin(r61059);
        double r61065 = r61063 * r61064;
        double r61066 = /* ERROR: no complex support in C */;
        double r61067 = /* ERROR: no complex support in C */;
        return r61067;
}
↓
double f(double x, double y) {
        double r61068 = 0.3333333333333333;
        double r61069 = x;
        double r61070 = 3.0;
        double r61071 = pow(r61069, r61070);
        double r61072 = r61068 * r61071;
        double r61073 = 0.016666666666666666;
        double r61074 = 5.0;
        double r61075 = pow(r61069, r61074);
        double r61076 = r61073 * r61075;
        double r61077 = 2.0;
        double r61078 = r61077 * r61069;
        double r61079 = r61076 + r61078;
        double r61080 = r61072 + r61079;
        double r61081 = 2.0;
        double r61082 = r61080 / r61081;
        double r61083 = y;
        double r61084 = sin(r61083);
        double r61085 = r61082 * r61084;
        return r61085;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.1

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

  2. Simplified44.1

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

  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 2019322 
(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)))))