Average Error: 43.3 → 0.7
Time: 35.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r52282 = x;
        double r52283 = exp(r52282);
        double r52284 = -r52282;
        double r52285 = exp(r52284);
        double r52286 = r52283 + r52285;
        double r52287 = 2.0;
        double r52288 = r52286 / r52287;
        double r52289 = y;
        double r52290 = cos(r52289);
        double r52291 = r52288 * r52290;
        double r52292 = r52283 - r52285;
        double r52293 = r52292 / r52287;
        double r52294 = sin(r52289);
        double r52295 = r52293 * r52294;
        double r52296 = /* ERROR: no complex support in C */;
        double r52297 = /* ERROR: no complex support in C */;
        return r52297;
}
↓
double f(double x, double y) {
        double r52298 = x;
        double r52299 = exp(r52298);
        double r52300 = -r52298;
        double r52301 = exp(r52300);
        double r52302 = r52299 + r52301;
        double r52303 = 2.0;
        double r52304 = r52302 / r52303;
        double r52305 = y;
        double r52306 = cos(r52305);
        double r52307 = r52304 * r52306;
        double r52308 = 0.3333333333333333;
        double r52309 = 3.0;
        double r52310 = pow(r52298, r52309);
        double r52311 = 0.016666666666666666;
        double r52312 = 5.0;
        double r52313 = pow(r52298, r52312);
        double r52314 = 2.0;
        double r52315 = r52314 * r52298;
        double r52316 = fma(r52311, r52313, r52315);
        double r52317 = fma(r52308, r52310, r52316);
        double r52318 = r52317 / r52303;
        double r52319 = sin(r52305);
        double r52320 = r52318 * r52319;
        double r52321 = /* ERROR: no complex support in C */;
        double r52322 = /* ERROR: no complex support in C */;
        return r52322;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))))