Average Error: 43.7 → 0.8
Time: 12.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r42281 = x;
        double r42282 = exp(r42281);
        double r42283 = -r42281;
        double r42284 = exp(r42283);
        double r42285 = r42282 + r42284;
        double r42286 = 2.0;
        double r42287 = r42285 / r42286;
        double r42288 = y;
        double r42289 = cos(r42288);
        double r42290 = r42287 * r42289;
        double r42291 = r42282 - r42284;
        double r42292 = r42291 / r42286;
        double r42293 = sin(r42288);
        double r42294 = r42292 * r42293;
        double r42295 = /* ERROR: no complex support in C */;
        double r42296 = /* ERROR: no complex support in C */;
        return r42296;
}
↓
double f(double x, double y) {
        double r42297 = x;
        double r42298 = exp(r42297);
        double r42299 = -r42297;
        double r42300 = exp(r42299);
        double r42301 = r42298 + r42300;
        double r42302 = 2.0;
        double r42303 = r42301 / r42302;
        double r42304 = y;
        double r42305 = cos(r42304);
        double r42306 = r42303 * r42305;
        double r42307 = 0.3333333333333333;
        double r42308 = 3.0;
        double r42309 = pow(r42297, r42308);
        double r42310 = r42307 * r42309;
        double r42311 = 0.016666666666666666;
        double r42312 = 5.0;
        double r42313 = pow(r42297, r42312);
        double r42314 = r42311 * r42313;
        double r42315 = 2.0;
        double r42316 = r42315 * r42297;
        double r42317 = r42314 + r42316;
        double r42318 = r42310 + r42317;
        double r42319 = r42318 / r42302;
        double r42320 = sin(r42304);
        double r42321 = r42319 * r42320;
        double r42322 = /* ERROR: no complex support in C */;
        double r42323 = /* ERROR: no complex support in C */;
        return r42323;
}

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

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020057 
(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)))))