Average Error: 43.1 → 0.9
Time: 12.3s
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 r46712 = x;
        double r46713 = exp(r46712);
        double r46714 = -r46712;
        double r46715 = exp(r46714);
        double r46716 = r46713 + r46715;
        double r46717 = 2.0;
        double r46718 = r46716 / r46717;
        double r46719 = y;
        double r46720 = cos(r46719);
        double r46721 = r46718 * r46720;
        double r46722 = r46713 - r46715;
        double r46723 = r46722 / r46717;
        double r46724 = sin(r46719);
        double r46725 = r46723 * r46724;
        double r46726 = /* ERROR: no complex support in C */;
        double r46727 = /* ERROR: no complex support in C */;
        return r46727;
}
↓
double f(double x, double y) {
        double r46728 = 0.3333333333333333;
        double r46729 = x;
        double r46730 = 3.0;
        double r46731 = pow(r46729, r46730);
        double r46732 = 0.016666666666666666;
        double r46733 = 5.0;
        double r46734 = pow(r46729, r46733);
        double r46735 = 2.0;
        double r46736 = r46735 * r46729;
        double r46737 = fma(r46732, r46734, r46736);
        double r46738 = fma(r46728, r46731, r46737);
        double r46739 = 2.0;
        double r46740 = r46738 / r46739;
        double r46741 = y;
        double r46742 = sin(r46741);
        double r46743 = r46740 * r46742;
        return r46743;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  2. Simplified43.1

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

  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 2020039 +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)))))