Average Error: 43.9 → 0.7
Time: 30.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\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 r48059 = x;
        double r48060 = exp(r48059);
        double r48061 = -r48059;
        double r48062 = exp(r48061);
        double r48063 = r48060 + r48062;
        double r48064 = 2.0;
        double r48065 = r48063 / r48064;
        double r48066 = y;
        double r48067 = cos(r48066);
        double r48068 = r48065 * r48067;
        double r48069 = r48060 - r48062;
        double r48070 = r48069 / r48064;
        double r48071 = sin(r48066);
        double r48072 = r48070 * r48071;
        double r48073 = /* ERROR: no complex support in C */;
        double r48074 = /* ERROR: no complex support in C */;
        return r48074;
}
↓
double f(double x, double y) {
        double r48075 = 0.3333333333333333;
        double r48076 = x;
        double r48077 = 3.0;
        double r48078 = pow(r48076, r48077);
        double r48079 = r48075 * r48078;
        double r48080 = 0.016666666666666666;
        double r48081 = 5.0;
        double r48082 = pow(r48076, r48081);
        double r48083 = r48080 * r48082;
        double r48084 = 2.0;
        double r48085 = r48084 * r48076;
        double r48086 = r48083 + r48085;
        double r48087 = r48079 + r48086;
        double r48088 = 2.0;
        double r48089 = r48087 / r48088;
        double r48090 = y;
        double r48091 = sin(r48090);
        double r48092 = r48089 * r48091;
        return r48092;
}

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 2019235 
(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)))))