Average Error: 43.1 → 0.8
Time: 59.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}{60} \cdot {x}^{5} + \left(\left(x + x\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1189414 = x;
        double r1189415 = exp(r1189414);
        double r1189416 = -r1189414;
        double r1189417 = exp(r1189416);
        double r1189418 = r1189415 + r1189417;
        double r1189419 = 2.0;
        double r1189420 = r1189418 / r1189419;
        double r1189421 = y;
        double r1189422 = cos(r1189421);
        double r1189423 = r1189420 * r1189422;
        double r1189424 = r1189415 - r1189417;
        double r1189425 = r1189424 / r1189419;
        double r1189426 = sin(r1189421);
        double r1189427 = r1189425 * r1189426;
        double r1189428 = /* ERROR: no complex support in C */;
        double r1189429 = /* ERROR: no complex support in C */;
        return r1189429;
}
↓
double f(double x, double y) {
        double r1189430 = x;
        double r1189431 = exp(r1189430);
        double r1189432 = -r1189430;
        double r1189433 = exp(r1189432);
        double r1189434 = r1189431 + r1189433;
        double r1189435 = 2.0;
        double r1189436 = r1189434 / r1189435;
        double r1189437 = y;
        double r1189438 = cos(r1189437);
        double r1189439 = r1189436 * r1189438;
        double r1189440 = 0.016666666666666666;
        double r1189441 = 5.0;
        double r1189442 = pow(r1189430, r1189441);
        double r1189443 = r1189440 * r1189442;
        double r1189444 = r1189430 + r1189430;
        double r1189445 = r1189430 * r1189430;
        double r1189446 = 0.3333333333333333;
        double r1189447 = r1189445 * r1189446;
        double r1189448 = r1189447 * r1189430;
        double r1189449 = r1189444 + r1189448;
        double r1189450 = r1189443 + r1189449;
        double r1189451 = r1189450 / r1189435;
        double r1189452 = sin(r1189437);
        double r1189453 = r1189451 * r1189452;
        double r1189454 = /* ERROR: no complex support in C */;
        double r1189455 = /* ERROR: no complex support in C */;
        return r1189455;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))