Average Error: 43.5 → 0.8
Time: 10.8s
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 r35795 = x;
        double r35796 = exp(r35795);
        double r35797 = -r35795;
        double r35798 = exp(r35797);
        double r35799 = r35796 + r35798;
        double r35800 = 2.0;
        double r35801 = r35799 / r35800;
        double r35802 = y;
        double r35803 = cos(r35802);
        double r35804 = r35801 * r35803;
        double r35805 = r35796 - r35798;
        double r35806 = r35805 / r35800;
        double r35807 = sin(r35802);
        double r35808 = r35806 * r35807;
        double r35809 = /* ERROR: no complex support in C */;
        double r35810 = /* ERROR: no complex support in C */;
        return r35810;
}
↓
double f(double x, double y) {
        double r35811 = 0.3333333333333333;
        double r35812 = x;
        double r35813 = 3.0;
        double r35814 = pow(r35812, r35813);
        double r35815 = 0.016666666666666666;
        double r35816 = 5.0;
        double r35817 = pow(r35812, r35816);
        double r35818 = 2.0;
        double r35819 = r35818 * r35812;
        double r35820 = fma(r35815, r35817, r35819);
        double r35821 = fma(r35811, r35814, r35820);
        double r35822 = 2.0;
        double r35823 = r35821 / r35822;
        double r35824 = y;
        double r35825 = sin(r35824);
        double r35826 = r35823 * r35825;
        return r35826;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  2. Simplified43.5

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

  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. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2020049 +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)))))