Average Error: 44.2 → 0.7
Time: 1.1m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2.0} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2.0} \cdot \sin y i\right))
double f(double x, double y) {
        double r2163702 = x;
        double r2163703 = exp(r2163702);
        double r2163704 = -r2163702;
        double r2163705 = exp(r2163704);
        double r2163706 = r2163703 + r2163705;
        double r2163707 = 2.0;
        double r2163708 = r2163706 / r2163707;
        double r2163709 = y;
        double r2163710 = cos(r2163709);
        double r2163711 = r2163708 * r2163710;
        double r2163712 = r2163703 - r2163705;
        double r2163713 = r2163712 / r2163707;
        double r2163714 = sin(r2163709);
        double r2163715 = r2163713 * r2163714;
        double r2163716 = /* ERROR: no complex support in C */;
        double r2163717 = /* ERROR: no complex support in C */;
        return r2163717;
}
↓
double f(double x, double y) {
        double r2163718 = x;
        double r2163719 = exp(r2163718);
        double r2163720 = -r2163718;
        double r2163721 = exp(r2163720);
        double r2163722 = r2163719 + r2163721;
        double r2163723 = 2.0;
        double r2163724 = r2163722 / r2163723;
        double r2163725 = y;
        double r2163726 = cos(r2163725);
        double r2163727 = r2163724 * r2163726;
        double r2163728 = 0.016666666666666666;
        double r2163729 = 5.0;
        double r2163730 = pow(r2163718, r2163729);
        double r2163731 = r2163718 * r2163718;
        double r2163732 = 0.3333333333333333;
        double r2163733 = r2163731 * r2163732;
        double r2163734 = 2.0;
        double r2163735 = r2163733 + r2163734;
        double r2163736 = r2163718 * r2163735;
        double r2163737 = fma(r2163728, r2163730, r2163736);
        double r2163738 = r2163737 / r2163723;
        double r2163739 = sin(r2163725);
        double r2163740 = r2163738 * r2163739;
        double r2163741 = /* ERROR: no complex support in C */;
        double r2163742 = /* ERROR: no complex support in C */;
        return r2163742;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.2

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{e^{x} - e^{-x}}{2.0} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2.0} \cdot \sin y i\right))\]

  3. Simplified0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)\right)}}{2.0} \cdot \sin y i\right))\]

  4. Final simplification0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2.0} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2.0} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))