Average Error: 43.5 → 0.7
Time: 34.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1315263 = x;
        double r1315264 = exp(r1315263);
        double r1315265 = -r1315263;
        double r1315266 = exp(r1315265);
        double r1315267 = r1315264 + r1315266;
        double r1315268 = 2.0;
        double r1315269 = r1315267 / r1315268;
        double r1315270 = y;
        double r1315271 = cos(r1315270);
        double r1315272 = r1315269 * r1315271;
        double r1315273 = r1315264 - r1315266;
        double r1315274 = r1315273 / r1315268;
        double r1315275 = sin(r1315270);
        double r1315276 = r1315274 * r1315275;
        double r1315277 = /* ERROR: no complex support in C */;
        double r1315278 = /* ERROR: no complex support in C */;
        return r1315278;
}
↓
double f(double x, double y) {
        double r1315279 = x;
        double r1315280 = exp(r1315279);
        double r1315281 = -r1315279;
        double r1315282 = exp(r1315281);
        double r1315283 = r1315280 + r1315282;
        double r1315284 = 2.0;
        double r1315285 = r1315283 / r1315284;
        double r1315286 = y;
        double r1315287 = cos(r1315286);
        double r1315288 = r1315285 * r1315287;
        double r1315289 = 0.016666666666666666;
        double r1315290 = 5.0;
        double r1315291 = pow(r1315279, r1315290);
        double r1315292 = 0.3333333333333333;
        double r1315293 = r1315292 * r1315279;
        double r1315294 = r1315279 * r1315293;
        double r1315295 = r1315294 + r1315284;
        double r1315296 = r1315279 * r1315295;
        double r1315297 = fma(r1315289, r1315291, r1315296);
        double r1315298 = r1315297 / r1315284;
        double r1315299 = sin(r1315286);
        double r1315300 = r1315298 * r1315299;
        double r1315301 = /* ERROR: no complex support in C */;
        double r1315302 = /* ERROR: no complex support in C */;
        return r1315302;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019138 +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)))))