Average Error: 43.6 → 0.7
Time: 17.5s
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 r42199 = x;
        double r42200 = exp(r42199);
        double r42201 = -r42199;
        double r42202 = exp(r42201);
        double r42203 = r42200 + r42202;
        double r42204 = 2.0;
        double r42205 = r42203 / r42204;
        double r42206 = y;
        double r42207 = cos(r42206);
        double r42208 = r42205 * r42207;
        double r42209 = r42200 - r42202;
        double r42210 = r42209 / r42204;
        double r42211 = sin(r42206);
        double r42212 = r42210 * r42211;
        double r42213 = /* ERROR: no complex support in C */;
        double r42214 = /* ERROR: no complex support in C */;
        return r42214;
}
↓
double f(double x, double y) {
        double r42215 = 0.3333333333333333;
        double r42216 = x;
        double r42217 = 3.0;
        double r42218 = pow(r42216, r42217);
        double r42219 = r42215 * r42218;
        double r42220 = 0.016666666666666666;
        double r42221 = 5.0;
        double r42222 = pow(r42216, r42221);
        double r42223 = r42220 * r42222;
        double r42224 = 2.0;
        double r42225 = r42224 * r42216;
        double r42226 = r42223 + r42225;
        double r42227 = r42219 + r42226;
        double r42228 = 2.0;
        double r42229 = r42227 / r42228;
        double r42230 = y;
        double r42231 = sin(r42230);
        double r42232 = r42229 * r42231;
        return r42232;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Simplified43.6

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

  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 2020045 
(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)))))