Average Error: 43.7 → 0.8
Time: 33.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + 2 \cdot x\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))\]
\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{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + 2 \cdot x\right) + {x}^{5} \cdot \frac{1}{60}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1996256 = x;
        double r1996257 = exp(r1996256);
        double r1996258 = -r1996256;
        double r1996259 = exp(r1996258);
        double r1996260 = r1996257 + r1996259;
        double r1996261 = 2.0;
        double r1996262 = r1996260 / r1996261;
        double r1996263 = y;
        double r1996264 = cos(r1996263);
        double r1996265 = r1996262 * r1996264;
        double r1996266 = r1996257 - r1996259;
        double r1996267 = r1996266 / r1996261;
        double r1996268 = sin(r1996263);
        double r1996269 = r1996267 * r1996268;
        double r1996270 = /* ERROR: no complex support in C */;
        double r1996271 = /* ERROR: no complex support in C */;
        return r1996271;
}
↓
double f(double x, double y) {
        double r1996272 = x;
        double r1996273 = exp(r1996272);
        double r1996274 = -r1996272;
        double r1996275 = exp(r1996274);
        double r1996276 = r1996273 + r1996275;
        double r1996277 = 2.0;
        double r1996278 = r1996276 / r1996277;
        double r1996279 = y;
        double r1996280 = cos(r1996279);
        double r1996281 = r1996278 * r1996280;
        double r1996282 = 0.3333333333333333;
        double r1996283 = r1996272 * r1996272;
        double r1996284 = r1996282 * r1996283;
        double r1996285 = r1996272 * r1996284;
        double r1996286 = r1996277 * r1996272;
        double r1996287 = r1996285 + r1996286;
        double r1996288 = 5.0;
        double r1996289 = pow(r1996272, r1996288);
        double r1996290 = 0.016666666666666666;
        double r1996291 = r1996289 * r1996290;
        double r1996292 = r1996287 + r1996291;
        double r1996293 = r1996292 / r1996277;
        double r1996294 = sin(r1996279);
        double r1996295 = r1996293 * r1996294;
        double r1996296 = /* ERROR: no complex support in C */;
        double r1996297 = /* ERROR: no complex support in C */;
        return r1996297;
}

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 0.8

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

  3. Simplified0.8

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

  5. Applied distribute-rgt-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 
(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)))))