Average Error: 43.4 → 0.8
Time: 29.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r47634 = x;
        double r47635 = exp(r47634);
        double r47636 = -r47634;
        double r47637 = exp(r47636);
        double r47638 = r47635 + r47637;
        double r47639 = 2.0;
        double r47640 = r47638 / r47639;
        double r47641 = y;
        double r47642 = cos(r47641);
        double r47643 = r47640 * r47642;
        double r47644 = r47635 - r47637;
        double r47645 = r47644 / r47639;
        double r47646 = sin(r47641);
        double r47647 = r47645 * r47646;
        double r47648 = /* ERROR: no complex support in C */;
        double r47649 = /* ERROR: no complex support in C */;
        return r47649;
}
↓
double f(double x, double y) {
        double r47650 = 0.3333333333333333;
        double r47651 = x;
        double r47652 = 3.0;
        double r47653 = pow(r47651, r47652);
        double r47654 = r47650 * r47653;
        double r47655 = 0.016666666666666666;
        double r47656 = 5.0;
        double r47657 = pow(r47651, r47656);
        double r47658 = r47655 * r47657;
        double r47659 = 2.0;
        double r47660 = r47659 * r47651;
        double r47661 = r47658 + r47660;
        double r47662 = r47654 + r47661;
        double r47663 = 2.0;
        double r47664 = r47662 / r47663;
        double r47665 = y;
        double r47666 = sin(r47665);
        double r47667 = r47664 * r47666;
        return r47667;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Simplified43.4

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019198 
(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)))))