Average Error: 43.5 → 0.9
Time: 42.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right)}{2} i\right))\]
\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 + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{3}\right)}{2} i\right))
double f(double x, double y) {
        double r2665450 = x;
        double r2665451 = exp(r2665450);
        double r2665452 = -r2665450;
        double r2665453 = exp(r2665452);
        double r2665454 = r2665451 + r2665453;
        double r2665455 = 2.0;
        double r2665456 = r2665454 / r2665455;
        double r2665457 = y;
        double r2665458 = cos(r2665457);
        double r2665459 = r2665456 * r2665458;
        double r2665460 = r2665451 - r2665453;
        double r2665461 = r2665460 / r2665455;
        double r2665462 = sin(r2665457);
        double r2665463 = r2665461 * r2665462;
        double r2665464 = /* ERROR: no complex support in C */;
        double r2665465 = /* ERROR: no complex support in C */;
        return r2665465;
}
↓
double f(double x, double y) {
        double r2665466 = x;
        double r2665467 = exp(r2665466);
        double r2665468 = -r2665466;
        double r2665469 = exp(r2665468);
        double r2665470 = r2665467 + r2665469;
        double r2665471 = 2.0;
        double r2665472 = r2665470 / r2665471;
        double r2665473 = y;
        double r2665474 = cos(r2665473);
        double r2665475 = r2665472 * r2665474;
        double r2665476 = sin(r2665473);
        double r2665477 = 0.016666666666666666;
        double r2665478 = 5.0;
        double r2665479 = pow(r2665466, r2665478);
        double r2665480 = r2665477 * r2665479;
        double r2665481 = 2.0;
        double r2665482 = r2665466 * r2665481;
        double r2665483 = r2665466 * r2665466;
        double r2665484 = r2665483 * r2665466;
        double r2665485 = 0.3333333333333333;
        double r2665486 = r2665484 * r2665485;
        double r2665487 = r2665482 + r2665486;
        double r2665488 = r2665480 + r2665487;
        double r2665489 = r2665488 / r2665471;
        double r2665490 = r2665476 * r2665489;
        double r2665491 = /* ERROR: no complex support in C */;
        double r2665492 = /* ERROR: no complex support in C */;
        return r2665492;
}

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 
(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)))))