Average Error: 43.4 → 0.8
Time: 12.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{\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 i\right))
double f(double x, double y) {
        double r49730 = x;
        double r49731 = exp(r49730);
        double r49732 = -r49730;
        double r49733 = exp(r49732);
        double r49734 = r49731 + r49733;
        double r49735 = 2.0;
        double r49736 = r49734 / r49735;
        double r49737 = y;
        double r49738 = cos(r49737);
        double r49739 = r49736 * r49738;
        double r49740 = r49731 - r49733;
        double r49741 = r49740 / r49735;
        double r49742 = sin(r49737);
        double r49743 = r49741 * r49742;
        double r49744 = /* ERROR: no complex support in C */;
        double r49745 = /* ERROR: no complex support in C */;
        return r49745;
}
↓
double f(double x, double y) {
        double r49746 = x;
        double r49747 = exp(r49746);
        double r49748 = -r49746;
        double r49749 = exp(r49748);
        double r49750 = r49747 + r49749;
        double r49751 = 2.0;
        double r49752 = r49750 / r49751;
        double r49753 = y;
        double r49754 = cos(r49753);
        double r49755 = r49752 * r49754;
        double r49756 = 0.3333333333333333;
        double r49757 = 3.0;
        double r49758 = pow(r49746, r49757);
        double r49759 = 0.016666666666666666;
        double r49760 = 5.0;
        double r49761 = pow(r49746, r49760);
        double r49762 = 2.0;
        double r49763 = r49762 * r49746;
        double r49764 = fma(r49759, r49761, r49763);
        double r49765 = fma(r49756, r49758, r49764);
        double r49766 = r49765 / r49751;
        double r49767 = sin(r49753);
        double r49768 = r49766 * r49767;
        double r49769 = /* ERROR: no complex support in C */;
        double r49770 = /* ERROR: no complex support in C */;
        return r49770;
}

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. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020062 +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)))))