Average Error: 43.5 → 0.8
Time: 31.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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{x \cdot \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1784821 = x;
        double r1784822 = exp(r1784821);
        double r1784823 = -r1784821;
        double r1784824 = exp(r1784823);
        double r1784825 = r1784822 + r1784824;
        double r1784826 = 2.0;
        double r1784827 = r1784825 / r1784826;
        double r1784828 = y;
        double r1784829 = cos(r1784828);
        double r1784830 = r1784827 * r1784829;
        double r1784831 = r1784822 - r1784824;
        double r1784832 = r1784831 / r1784826;
        double r1784833 = sin(r1784828);
        double r1784834 = r1784832 * r1784833;
        double r1784835 = /* ERROR: no complex support in C */;
        double r1784836 = /* ERROR: no complex support in C */;
        return r1784836;
}
↓
double f(double x, double y) {
        double r1784837 = x;
        double r1784838 = exp(r1784837);
        double r1784839 = -r1784837;
        double r1784840 = exp(r1784839);
        double r1784841 = r1784838 + r1784840;
        double r1784842 = 2.0;
        double r1784843 = r1784841 / r1784842;
        double r1784844 = y;
        double r1784845 = cos(r1784844);
        double r1784846 = r1784843 * r1784845;
        double r1784847 = 2.0;
        double r1784848 = 0.3333333333333333;
        double r1784849 = r1784837 * r1784837;
        double r1784850 = r1784848 * r1784849;
        double r1784851 = r1784847 + r1784850;
        double r1784852 = r1784837 * r1784851;
        double r1784853 = 5.0;
        double r1784854 = pow(r1784837, r1784853);
        double r1784855 = 0.016666666666666666;
        double r1784856 = r1784854 * r1784855;
        double r1784857 = r1784852 + r1784856;
        double r1784858 = r1784857 / r1784842;
        double r1784859 = sin(r1784844);
        double r1784860 = r1784858 * r1784859;
        double r1784861 = /* ERROR: no complex support in C */;
        double r1784862 = /* ERROR: no complex support in C */;
        return r1784862;
}

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. 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

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

  5. Applied *-commutative0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019171 
(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)))))