Average Error: 43.6 → 0.7
Time: 28.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\sin y}{2} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{1}{3}, {x}^{3}, {x}^{5} \cdot \frac{1}{60}\right)\right)
double f(double x, double y) {
        double r51319 = x;
        double r51320 = exp(r51319);
        double r51321 = -r51319;
        double r51322 = exp(r51321);
        double r51323 = r51320 + r51322;
        double r51324 = 2.0;
        double r51325 = r51323 / r51324;
        double r51326 = y;
        double r51327 = cos(r51326);
        double r51328 = r51325 * r51327;
        double r51329 = r51320 - r51322;
        double r51330 = r51329 / r51324;
        double r51331 = sin(r51326);
        double r51332 = r51330 * r51331;
        double r51333 = /* ERROR: no complex support in C */;
        double r51334 = /* ERROR: no complex support in C */;
        return r51334;
}
↓
double f(double x, double y) {
        double r51335 = y;
        double r51336 = sin(r51335);
        double r51337 = 2.0;
        double r51338 = r51336 / r51337;
        double r51339 = 2.0;
        double r51340 = x;
        double r51341 = 0.3333333333333333;
        double r51342 = 3.0;
        double r51343 = pow(r51340, r51342);
        double r51344 = 5.0;
        double r51345 = pow(r51340, r51344);
        double r51346 = 0.016666666666666666;
        double r51347 = r51345 * r51346;
        double r51348 = fma(r51341, r51343, r51347);
        double r51349 = fma(r51339, r51340, r51348);
        double r51350 = r51338 * r51349;
        return r51350;
}

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

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))