Average Error: 43.0 → 0.8
Time: 1.0m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1846607 = x;
        double r1846608 = exp(r1846607);
        double r1846609 = -r1846607;
        double r1846610 = exp(r1846609);
        double r1846611 = r1846608 + r1846610;
        double r1846612 = 2.0;
        double r1846613 = r1846611 / r1846612;
        double r1846614 = y;
        double r1846615 = cos(r1846614);
        double r1846616 = r1846613 * r1846615;
        double r1846617 = r1846608 - r1846610;
        double r1846618 = r1846617 / r1846612;
        double r1846619 = sin(r1846614);
        double r1846620 = r1846618 * r1846619;
        double r1846621 = /* ERROR: no complex support in C */;
        double r1846622 = /* ERROR: no complex support in C */;
        return r1846622;
}
↓
double f(double x, double y) {
        double r1846623 = x;
        double r1846624 = exp(r1846623);
        double r1846625 = -r1846623;
        double r1846626 = exp(r1846625);
        double r1846627 = r1846624 + r1846626;
        double r1846628 = 2.0;
        double r1846629 = r1846627 / r1846628;
        double r1846630 = y;
        double r1846631 = cos(r1846630);
        double r1846632 = r1846629 * r1846631;
        double r1846633 = 0.016666666666666666;
        double r1846634 = 5.0;
        double r1846635 = pow(r1846623, r1846634);
        double r1846636 = r1846633 * r1846635;
        double r1846637 = r1846628 * r1846623;
        double r1846638 = 0.3333333333333333;
        double r1846639 = r1846623 * r1846623;
        double r1846640 = r1846639 * r1846623;
        double r1846641 = r1846638 * r1846640;
        double r1846642 = r1846637 + r1846641;
        double r1846643 = r1846636 + r1846642;
        double r1846644 = r1846643 / r1846628;
        double r1846645 = sin(r1846630);
        double r1846646 = r1846644 * r1846645;
        double r1846647 = /* ERROR: no complex support in C */;
        double r1846648 = /* ERROR: no complex support in C */;
        return r1846648;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.0

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019130 
(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)))))