Average Error: 43.4 → 0.9
Time: 12.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r47665 = x;
        double r47666 = exp(r47665);
        double r47667 = -r47665;
        double r47668 = exp(r47667);
        double r47669 = r47666 + r47668;
        double r47670 = 2.0;
        double r47671 = r47669 / r47670;
        double r47672 = y;
        double r47673 = cos(r47672);
        double r47674 = r47671 * r47673;
        double r47675 = r47666 - r47668;
        double r47676 = r47675 / r47670;
        double r47677 = sin(r47672);
        double r47678 = r47676 * r47677;
        double r47679 = /* ERROR: no complex support in C */;
        double r47680 = /* ERROR: no complex support in C */;
        return r47680;
}
↓
double f(double x, double y) {
        double r47681 = 0.3333333333333333;
        double r47682 = x;
        double r47683 = 3.0;
        double r47684 = pow(r47682, r47683);
        double r47685 = 0.016666666666666666;
        double r47686 = 5.0;
        double r47687 = pow(r47682, r47686);
        double r47688 = 2.0;
        double r47689 = r47688 * r47682;
        double r47690 = fma(r47685, r47687, r47689);
        double r47691 = fma(r47681, r47684, r47690);
        double r47692 = 2.0;
        double r47693 = r47691 / r47692;
        double r47694 = y;
        double r47695 = sin(r47694);
        double r47696 = r47693 * r47695;
        return r47696;
}

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 0.9

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

  4. Simplified0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))