Average Error: 43.4 → 1.7
Time: 10.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin y}}\right)\right) \cdot \sqrt[3]{\sin y}\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin y}}\right)\right) \cdot \sqrt[3]{\sin y}
double f(double x, double y) {
        double r34826 = x;
        double r34827 = exp(r34826);
        double r34828 = -r34826;
        double r34829 = exp(r34828);
        double r34830 = r34827 + r34829;
        double r34831 = 2.0;
        double r34832 = r34830 / r34831;
        double r34833 = y;
        double r34834 = cos(r34833);
        double r34835 = r34832 * r34834;
        double r34836 = r34827 - r34829;
        double r34837 = r34836 / r34831;
        double r34838 = sin(r34833);
        double r34839 = r34837 * r34838;
        double r34840 = /* ERROR: no complex support in C */;
        double r34841 = /* ERROR: no complex support in C */;
        return r34841;
}
↓
double f(double x, double y) {
        double r34842 = 0.3333333333333333;
        double r34843 = x;
        double r34844 = 3.0;
        double r34845 = pow(r34843, r34844);
        double r34846 = r34842 * r34845;
        double r34847 = 0.016666666666666666;
        double r34848 = 5.0;
        double r34849 = pow(r34843, r34848);
        double r34850 = r34847 * r34849;
        double r34851 = r34846 + r34850;
        double r34852 = 2.0;
        double r34853 = r34852 * r34843;
        double r34854 = r34851 + r34853;
        double r34855 = 2.0;
        double r34856 = r34854 / r34855;
        double r34857 = y;
        double r34858 = sin(r34857);
        double r34859 = cbrt(r34858);
        double r34860 = r34859 * r34859;
        double r34861 = cbrt(r34860);
        double r34862 = r34859 * r34861;
        double r34863 = cbrt(r34859);
        double r34864 = r34862 * r34863;
        double r34865 = r34856 * r34864;
        double r34866 = r34865 * r34859;
        return r34866;
}

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 0.7

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

  5. Applied associate-+r+0.7

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

  7. Applied add-cube-cbrt1.6

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

  8. Applied associate-*r*1.6

    \[\leadsto \color{blue}{\left(\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2} \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt1.6

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

  11. Applied cbrt-prod1.7

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

  12. Applied associate-*r*1.7

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

  13. Final simplification1.7

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

Reproduce

herbie shell --seed 2019344 
(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)))))