Average Error: 43.5 → 0.8
Time: 32.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r895230 = x;
        double r895231 = exp(r895230);
        double r895232 = -r895230;
        double r895233 = exp(r895232);
        double r895234 = r895231 + r895233;
        double r895235 = 2.0;
        double r895236 = r895234 / r895235;
        double r895237 = y;
        double r895238 = cos(r895237);
        double r895239 = r895236 * r895238;
        double r895240 = r895231 - r895233;
        double r895241 = r895240 / r895235;
        double r895242 = sin(r895237);
        double r895243 = r895241 * r895242;
        double r895244 = /* ERROR: no complex support in C */;
        double r895245 = /* ERROR: no complex support in C */;
        return r895245;
}
↓
double f(double x, double y) {
        double r895246 = x;
        double r895247 = exp(r895246);
        double r895248 = -r895246;
        double r895249 = exp(r895248);
        double r895250 = r895247 + r895249;
        double r895251 = 2.0;
        double r895252 = r895250 / r895251;
        double r895253 = y;
        double r895254 = cos(r895253);
        double r895255 = r895252 * r895254;
        double r895256 = 5.0;
        double r895257 = pow(r895246, r895256);
        double r895258 = 0.016666666666666666;
        double r895259 = 0.3333333333333333;
        double r895260 = r895246 * r895246;
        double r895261 = r895260 * r895246;
        double r895262 = r895259 * r895261;
        double r895263 = fma(r895257, r895258, r895262);
        double r895264 = fma(r895246, r895251, r895263);
        double r895265 = r895264 / r895251;
        double r895266 = sin(r895253);
        double r895267 = r895265 * r895266;
        double r895268 = /* ERROR: no complex support in C */;
        double r895269 = /* ERROR: no complex support in C */;
        return r895269;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right)\right)}}{2} \cdot \sin y i\right))\]

  4. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))