Average Error: 43.2 → 0.7
Time: 2.5m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))\]
\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{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r8308582 = x;
        double r8308583 = exp(r8308582);
        double r8308584 = -r8308582;
        double r8308585 = exp(r8308584);
        double r8308586 = r8308583 + r8308585;
        double r8308587 = 2.0;
        double r8308588 = r8308586 / r8308587;
        double r8308589 = y;
        double r8308590 = cos(r8308589);
        double r8308591 = r8308588 * r8308590;
        double r8308592 = r8308583 - r8308585;
        double r8308593 = r8308592 / r8308587;
        double r8308594 = sin(r8308589);
        double r8308595 = r8308593 * r8308594;
        double r8308596 = /* ERROR: no complex support in C */;
        double r8308597 = /* ERROR: no complex support in C */;
        return r8308597;
}
↓
double f(double x, double y) {
        double r8308598 = x;
        double r8308599 = exp(r8308598);
        double r8308600 = -r8308598;
        double r8308601 = exp(r8308600);
        double r8308602 = r8308599 + r8308601;
        double r8308603 = 2.0;
        double r8308604 = r8308602 / r8308603;
        double r8308605 = y;
        double r8308606 = cos(r8308605);
        double r8308607 = r8308604 * r8308606;
        double r8308608 = 5.0;
        double r8308609 = pow(r8308598, r8308608);
        double r8308610 = 0.016666666666666666;
        double r8308611 = r8308609 * r8308610;
        double r8308612 = 0.3333333333333333;
        double r8308613 = r8308598 * r8308612;
        double r8308614 = r8308613 * r8308598;
        double r8308615 = r8308603 + r8308614;
        double r8308616 = r8308615 * r8308598;
        double r8308617 = r8308611 + r8308616;
        double r8308618 = r8308617 / r8308603;
        double r8308619 = sin(r8308605);
        double r8308620 = r8308618 * r8308619;
        double r8308621 = /* ERROR: no complex support in C */;
        double r8308622 = /* ERROR: no complex support in C */;
        return r8308622;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019119 
(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)))))