Average Error: 43.6 → 0.8
Time: 10.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r40303 = x;
        double r40304 = exp(r40303);
        double r40305 = -r40303;
        double r40306 = exp(r40305);
        double r40307 = r40304 + r40306;
        double r40308 = 2.0;
        double r40309 = r40307 / r40308;
        double r40310 = y;
        double r40311 = cos(r40310);
        double r40312 = r40309 * r40311;
        double r40313 = r40304 - r40306;
        double r40314 = r40313 / r40308;
        double r40315 = sin(r40310);
        double r40316 = r40314 * r40315;
        double r40317 = /* ERROR: no complex support in C */;
        double r40318 = /* ERROR: no complex support in C */;
        return r40318;
}
↓
double f(double x, double y) {
        double r40319 = 0.3333333333333333;
        double r40320 = x;
        double r40321 = 3.0;
        double r40322 = pow(r40320, r40321);
        double r40323 = r40319 * r40322;
        double r40324 = 0.016666666666666666;
        double r40325 = 5.0;
        double r40326 = pow(r40320, r40325);
        double r40327 = r40324 * r40326;
        double r40328 = 2.0;
        double r40329 = r40328 * r40320;
        double r40330 = r40327 + r40329;
        double r40331 = r40323 + r40330;
        double r40332 = 2.0;
        double r40333 = r40331 / r40332;
        double r40334 = y;
        double r40335 = sin(r40334);
        double r40336 = r40333 * r40335;
        return r40336;
}

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

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020064 
(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)))))