Average Error: 43.9 → 0.8
Time: 58.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}{60}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1969828 = x;
        double r1969829 = exp(r1969828);
        double r1969830 = -r1969828;
        double r1969831 = exp(r1969830);
        double r1969832 = r1969829 + r1969831;
        double r1969833 = 2.0;
        double r1969834 = r1969832 / r1969833;
        double r1969835 = y;
        double r1969836 = cos(r1969835);
        double r1969837 = r1969834 * r1969836;
        double r1969838 = r1969829 - r1969831;
        double r1969839 = r1969838 / r1969833;
        double r1969840 = sin(r1969835);
        double r1969841 = r1969839 * r1969840;
        double r1969842 = /* ERROR: no complex support in C */;
        double r1969843 = /* ERROR: no complex support in C */;
        return r1969843;
}
↓
double f(double x, double y) {
        double r1969844 = x;
        double r1969845 = exp(r1969844);
        double r1969846 = -r1969844;
        double r1969847 = exp(r1969846);
        double r1969848 = r1969845 + r1969847;
        double r1969849 = 2.0;
        double r1969850 = r1969848 / r1969849;
        double r1969851 = y;
        double r1969852 = cos(r1969851);
        double r1969853 = r1969850 * r1969852;
        double r1969854 = 0.016666666666666666;
        double r1969855 = 5.0;
        double r1969856 = pow(r1969844, r1969855);
        double r1969857 = 2.0;
        double r1969858 = r1969844 * r1969857;
        double r1969859 = r1969844 * r1969844;
        double r1969860 = 0.3333333333333333;
        double r1969861 = r1969859 * r1969860;
        double r1969862 = r1969861 * r1969844;
        double r1969863 = r1969858 + r1969862;
        double r1969864 = fma(r1969854, r1969856, r1969863);
        double r1969865 = r1969864 / r1969849;
        double r1969866 = sin(r1969851);
        double r1969867 = r1969865 * r1969866;
        double r1969868 = /* ERROR: no complex support in C */;
        double r1969869 = /* ERROR: no complex support in C */;
        return r1969869;
}

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. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied distribute-lft-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 +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)))))