Average Error: 43.6 → 0.6
Time: 12.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r39397 = x;
        double r39398 = exp(r39397);
        double r39399 = -r39397;
        double r39400 = exp(r39399);
        double r39401 = r39398 + r39400;
        double r39402 = 2.0;
        double r39403 = r39401 / r39402;
        double r39404 = y;
        double r39405 = cos(r39404);
        double r39406 = r39403 * r39405;
        double r39407 = r39398 - r39400;
        double r39408 = r39407 / r39402;
        double r39409 = sin(r39404);
        double r39410 = r39408 * r39409;
        double r39411 = /* ERROR: no complex support in C */;
        double r39412 = /* ERROR: no complex support in C */;
        return r39412;
}
↓
double f(double x, double y) {
        double r39413 = x;
        double r39414 = exp(r39413);
        double r39415 = -r39413;
        double r39416 = exp(r39415);
        double r39417 = r39414 + r39416;
        double r39418 = 2.0;
        double r39419 = r39417 / r39418;
        double r39420 = y;
        double r39421 = cos(r39420);
        double r39422 = r39419 * r39421;
        double r39423 = 0.3333333333333333;
        double r39424 = 3.0;
        double r39425 = pow(r39413, r39424);
        double r39426 = r39423 * r39425;
        double r39427 = 0.016666666666666666;
        double r39428 = 5.0;
        double r39429 = pow(r39413, r39428);
        double r39430 = r39427 * r39429;
        double r39431 = 2.0;
        double r39432 = r39431 * r39413;
        double r39433 = r39430 + r39432;
        double r39434 = r39426 + r39433;
        double r39435 = r39434 / r39418;
        double r39436 = sin(r39420);
        double r39437 = r39435 * r39436;
        double r39438 = /* ERROR: no complex support in C */;
        double r39439 = /* ERROR: no complex support in C */;
        return r39439;
}

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. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019362 
(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)))))