Average Error: 43.4 → 0.8
Time: 11.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r38841 = x;
        double r38842 = exp(r38841);
        double r38843 = -r38841;
        double r38844 = exp(r38843);
        double r38845 = r38842 + r38844;
        double r38846 = 2.0;
        double r38847 = r38845 / r38846;
        double r38848 = y;
        double r38849 = cos(r38848);
        double r38850 = r38847 * r38849;
        double r38851 = r38842 - r38844;
        double r38852 = r38851 / r38846;
        double r38853 = sin(r38848);
        double r38854 = r38852 * r38853;
        double r38855 = /* ERROR: no complex support in C */;
        double r38856 = /* ERROR: no complex support in C */;
        return r38856;
}
↓
double f(double x, double y) {
        double r38857 = 0.3333333333333333;
        double r38858 = x;
        double r38859 = 3.0;
        double r38860 = pow(r38858, r38859);
        double r38861 = r38857 * r38860;
        double r38862 = 0.016666666666666666;
        double r38863 = 5.0;
        double r38864 = pow(r38858, r38863);
        double r38865 = r38862 * r38864;
        double r38866 = 2.0;
        double r38867 = r38866 * r38858;
        double r38868 = r38865 + r38867;
        double r38869 = r38861 + r38868;
        double r38870 = 2.0;
        double r38871 = r38869 / r38870;
        double r38872 = y;
        double r38873 = sin(r38872);
        double r38874 = r38871 * r38873;
        return r38874;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  2. Simplified43.4

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

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020062 
(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)))))