Average Error: 43.9 → 0.8
Time: 35.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x}{2} \cdot \sin y i\right))\]
\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}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r878763 = x;
        double r878764 = exp(r878763);
        double r878765 = -r878763;
        double r878766 = exp(r878765);
        double r878767 = r878764 + r878766;
        double r878768 = 2.0;
        double r878769 = r878767 / r878768;
        double r878770 = y;
        double r878771 = cos(r878770);
        double r878772 = r878769 * r878771;
        double r878773 = r878764 - r878766;
        double r878774 = r878773 / r878768;
        double r878775 = sin(r878770);
        double r878776 = r878774 * r878775;
        double r878777 = /* ERROR: no complex support in C */;
        double r878778 = /* ERROR: no complex support in C */;
        return r878778;
}
↓
double f(double x, double y) {
        double r878779 = x;
        double r878780 = exp(r878779);
        double r878781 = -r878779;
        double r878782 = exp(r878781);
        double r878783 = r878780 + r878782;
        double r878784 = 2.0;
        double r878785 = r878783 / r878784;
        double r878786 = y;
        double r878787 = cos(r878786);
        double r878788 = r878785 * r878787;
        double r878789 = 5.0;
        double r878790 = pow(r878779, r878789);
        double r878791 = 0.016666666666666666;
        double r878792 = r878790 * r878791;
        double r878793 = r878779 * r878779;
        double r878794 = 0.3333333333333333;
        double r878795 = r878793 * r878794;
        double r878796 = r878784 + r878795;
        double r878797 = r878796 * r878779;
        double r878798 = r878792 + r878797;
        double r878799 = r878798 / r878784;
        double r878800 = sin(r878786);
        double r878801 = r878799 * r878800;
        double r878802 = /* ERROR: no complex support in C */;
        double r878803 = /* ERROR: no complex support in C */;
        return r878803;
}

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

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

  5. Applied *-un-lft-identity0.8

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

  6. Applied associate-*r*0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))