Average Error: 43.9 → 0.7
Time: 28.0s
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 r54815 = x;
        double r54816 = exp(r54815);
        double r54817 = -r54815;
        double r54818 = exp(r54817);
        double r54819 = r54816 + r54818;
        double r54820 = 2.0;
        double r54821 = r54819 / r54820;
        double r54822 = y;
        double r54823 = cos(r54822);
        double r54824 = r54821 * r54823;
        double r54825 = r54816 - r54818;
        double r54826 = r54825 / r54820;
        double r54827 = sin(r54822);
        double r54828 = r54826 * r54827;
        double r54829 = /* ERROR: no complex support in C */;
        double r54830 = /* ERROR: no complex support in C */;
        return r54830;
}
↓
double f(double x, double y) {
        double r54831 = 0.3333333333333333;
        double r54832 = x;
        double r54833 = 3.0;
        double r54834 = pow(r54832, r54833);
        double r54835 = 0.016666666666666666;
        double r54836 = 5.0;
        double r54837 = pow(r54832, r54836);
        double r54838 = 2.0;
        double r54839 = r54838 * r54832;
        double r54840 = fma(r54835, r54837, r54839);
        double r54841 = fma(r54831, r54834, r54840);
        double r54842 = 2.0;
        double r54843 = r54841 / r54842;
        double r54844 = y;
        double r54845 = sin(r54844);
        double r54846 = r54843 * r54845;
        return r54846;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019235 +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)))))