Average Error: 43.5 → 0.9
Time: 39.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)}{2} \cdot \sin y i\right))\]
\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} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2921539 = x;
        double r2921540 = exp(r2921539);
        double r2921541 = -r2921539;
        double r2921542 = exp(r2921541);
        double r2921543 = r2921540 + r2921542;
        double r2921544 = 2.0;
        double r2921545 = r2921543 / r2921544;
        double r2921546 = y;
        double r2921547 = cos(r2921546);
        double r2921548 = r2921545 * r2921547;
        double r2921549 = r2921540 - r2921542;
        double r2921550 = r2921549 / r2921544;
        double r2921551 = sin(r2921546);
        double r2921552 = r2921550 * r2921551;
        double r2921553 = /* ERROR: no complex support in C */;
        double r2921554 = /* ERROR: no complex support in C */;
        return r2921554;
}
↓
double f(double x, double y) {
        double r2921555 = x;
        double r2921556 = exp(r2921555);
        double r2921557 = -r2921555;
        double r2921558 = exp(r2921557);
        double r2921559 = r2921556 + r2921558;
        double r2921560 = 2.0;
        double r2921561 = r2921559 / r2921560;
        double r2921562 = y;
        double r2921563 = cos(r2921562);
        double r2921564 = r2921561 * r2921563;
        double r2921565 = 0.016666666666666666;
        double r2921566 = 5.0;
        double r2921567 = pow(r2921555, r2921566);
        double r2921568 = r2921565 * r2921567;
        double r2921569 = r2921555 * r2921555;
        double r2921570 = 0.3333333333333333;
        double r2921571 = r2921569 * r2921570;
        double r2921572 = 2.0;
        double r2921573 = r2921571 + r2921572;
        double r2921574 = r2921555 * r2921573;
        double r2921575 = r2921568 + r2921574;
        double r2921576 = r2921575 / r2921560;
        double r2921577 = sin(r2921562);
        double r2921578 = r2921576 * r2921577;
        double r2921579 = /* ERROR: no complex support in C */;
        double r2921580 = /* ERROR: no complex support in C */;
        return r2921580;
}

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 0.8

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019169 
(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)))))