Average Error: 43.9 → 0.8
Time: 34.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{60} \cdot {x}^{5} + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2393273 = x;
        double r2393274 = exp(r2393273);
        double r2393275 = -r2393273;
        double r2393276 = exp(r2393275);
        double r2393277 = r2393274 + r2393276;
        double r2393278 = 2.0;
        double r2393279 = r2393277 / r2393278;
        double r2393280 = y;
        double r2393281 = cos(r2393280);
        double r2393282 = r2393279 * r2393281;
        double r2393283 = r2393274 - r2393276;
        double r2393284 = r2393283 / r2393278;
        double r2393285 = sin(r2393280);
        double r2393286 = r2393284 * r2393285;
        double r2393287 = /* ERROR: no complex support in C */;
        double r2393288 = /* ERROR: no complex support in C */;
        return r2393288;
}
↓
double f(double x, double y) {
        double r2393289 = x;
        double r2393290 = exp(r2393289);
        double r2393291 = -r2393289;
        double r2393292 = exp(r2393291);
        double r2393293 = r2393290 + r2393292;
        double r2393294 = 2.0;
        double r2393295 = r2393293 / r2393294;
        double r2393296 = y;
        double r2393297 = cos(r2393296);
        double r2393298 = r2393295 * r2393297;
        double r2393299 = 0.016666666666666666;
        double r2393300 = 5.0;
        double r2393301 = pow(r2393289, r2393300);
        double r2393302 = r2393299 * r2393301;
        double r2393303 = 0.3333333333333333;
        double r2393304 = r2393289 * r2393289;
        double r2393305 = r2393303 * r2393304;
        double r2393306 = r2393305 * r2393289;
        double r2393307 = r2393294 * r2393289;
        double r2393308 = r2393306 + r2393307;
        double r2393309 = r2393302 + r2393308;
        double r2393310 = r2393309 / r2393294;
        double r2393311 = sin(r2393296);
        double r2393312 = r2393310 * r2393311;
        double r2393313 = /* ERROR: no complex support in C */;
        double r2393314 = /* ERROR: no complex support in C */;
        return r2393314;
}

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

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

  5. Applied distribute-rgt-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 
(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)))))