Average Error: 43.2 → 0.9
Time: 21.4s
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 r36748 = x;
        double r36749 = exp(r36748);
        double r36750 = -r36748;
        double r36751 = exp(r36750);
        double r36752 = r36749 + r36751;
        double r36753 = 2.0;
        double r36754 = r36752 / r36753;
        double r36755 = y;
        double r36756 = cos(r36755);
        double r36757 = r36754 * r36756;
        double r36758 = r36749 - r36751;
        double r36759 = r36758 / r36753;
        double r36760 = sin(r36755);
        double r36761 = r36759 * r36760;
        double r36762 = /* ERROR: no complex support in C */;
        double r36763 = /* ERROR: no complex support in C */;
        return r36763;
}
↓
double f(double x, double y) {
        double r36764 = 0.3333333333333333;
        double r36765 = x;
        double r36766 = 3.0;
        double r36767 = pow(r36765, r36766);
        double r36768 = 0.016666666666666666;
        double r36769 = 5.0;
        double r36770 = pow(r36765, r36769);
        double r36771 = 2.0;
        double r36772 = r36771 * r36765;
        double r36773 = fma(r36768, r36770, r36772);
        double r36774 = fma(r36764, r36767, r36773);
        double r36775 = 2.0;
        double r36776 = r36774 / r36775;
        double r36777 = y;
        double r36778 = sin(r36777);
        double r36779 = r36776 * r36778;
        return r36779;
}

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. Simplified43.2

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

  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 2019350 +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)))))