Average Error: 43.5 → 0.8
Time: 33.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{3} \cdot x, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot 2\right)\right)}{2} \cdot \sin y i\right))\]
\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(x \cdot x, \frac{1}{3} \cdot x, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2133390 = x;
        double r2133391 = exp(r2133390);
        double r2133392 = -r2133390;
        double r2133393 = exp(r2133392);
        double r2133394 = r2133391 + r2133393;
        double r2133395 = 2.0;
        double r2133396 = r2133394 / r2133395;
        double r2133397 = y;
        double r2133398 = cos(r2133397);
        double r2133399 = r2133396 * r2133398;
        double r2133400 = r2133391 - r2133393;
        double r2133401 = r2133400 / r2133395;
        double r2133402 = sin(r2133397);
        double r2133403 = r2133401 * r2133402;
        double r2133404 = /* ERROR: no complex support in C */;
        double r2133405 = /* ERROR: no complex support in C */;
        return r2133405;
}
↓
double f(double x, double y) {
        double r2133406 = x;
        double r2133407 = exp(r2133406);
        double r2133408 = -r2133406;
        double r2133409 = exp(r2133408);
        double r2133410 = r2133407 + r2133409;
        double r2133411 = 2.0;
        double r2133412 = r2133410 / r2133411;
        double r2133413 = y;
        double r2133414 = cos(r2133413);
        double r2133415 = r2133412 * r2133414;
        double r2133416 = r2133406 * r2133406;
        double r2133417 = 0.3333333333333333;
        double r2133418 = r2133417 * r2133406;
        double r2133419 = 0.016666666666666666;
        double r2133420 = 5.0;
        double r2133421 = pow(r2133406, r2133420);
        double r2133422 = 2.0;
        double r2133423 = r2133406 * r2133422;
        double r2133424 = fma(r2133419, r2133421, r2133423);
        double r2133425 = fma(r2133416, r2133418, r2133424);
        double r2133426 = r2133425 / r2133411;
        double r2133427 = sin(r2133413);
        double r2133428 = r2133426 * r2133427;
        double r2133429 = /* ERROR: no complex support in C */;
        double r2133430 = /* ERROR: no complex support in C */;
        return r2133430;
}

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019171 +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)))))