Average Error: 43.7 → 0.7
Time: 2.5m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))\]
\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 \frac{1}{3}\right) \cdot x\right) \cdot x}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r8069194 = x;
        double r8069195 = exp(r8069194);
        double r8069196 = -r8069194;
        double r8069197 = exp(r8069196);
        double r8069198 = r8069195 + r8069197;
        double r8069199 = 2.0;
        double r8069200 = r8069198 / r8069199;
        double r8069201 = y;
        double r8069202 = cos(r8069201);
        double r8069203 = r8069200 * r8069202;
        double r8069204 = r8069195 - r8069197;
        double r8069205 = r8069204 / r8069199;
        double r8069206 = sin(r8069201);
        double r8069207 = r8069205 * r8069206;
        double r8069208 = /* ERROR: no complex support in C */;
        double r8069209 = /* ERROR: no complex support in C */;
        return r8069209;
}
↓
double f(double x, double y) {
        double r8069210 = x;
        double r8069211 = exp(r8069210);
        double r8069212 = -r8069210;
        double r8069213 = exp(r8069212);
        double r8069214 = r8069211 + r8069213;
        double r8069215 = 2.0;
        double r8069216 = r8069214 / r8069215;
        double r8069217 = y;
        double r8069218 = cos(r8069217);
        double r8069219 = r8069216 * r8069218;
        double r8069220 = 5.0;
        double r8069221 = pow(r8069210, r8069220);
        double r8069222 = 0.016666666666666666;
        double r8069223 = r8069221 * r8069222;
        double r8069224 = 0.3333333333333333;
        double r8069225 = r8069210 * r8069224;
        double r8069226 = r8069225 * r8069210;
        double r8069227 = r8069215 + r8069226;
        double r8069228 = r8069227 * r8069210;
        double r8069229 = r8069223 + r8069228;
        double r8069230 = r8069229 / r8069215;
        double r8069231 = sin(r8069217);
        double r8069232 = r8069230 * r8069231;
        double r8069233 = /* ERROR: no complex support in C */;
        double r8069234 = /* ERROR: no complex support in C */;
        return r8069234;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019104 
(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)))))