Average Error: 43.6 → 0.6
Time: 11.6s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r55243 = x;
        double r55244 = exp(r55243);
        double r55245 = -r55243;
        double r55246 = exp(r55245);
        double r55247 = r55244 + r55246;
        double r55248 = 2.0;
        double r55249 = r55247 / r55248;
        double r55250 = y;
        double r55251 = cos(r55250);
        double r55252 = r55249 * r55251;
        double r55253 = r55244 - r55246;
        double r55254 = r55253 / r55248;
        double r55255 = sin(r55250);
        double r55256 = r55254 * r55255;
        double r55257 = /* ERROR: no complex support in C */;
        double r55258 = /* ERROR: no complex support in C */;
        return r55258;
}
↓
double f(double x, double y) {
        double r55259 = 0.3333333333333333;
        double r55260 = x;
        double r55261 = 3.0;
        double r55262 = pow(r55260, r55261);
        double r55263 = r55259 * r55262;
        double r55264 = 0.016666666666666666;
        double r55265 = 5.0;
        double r55266 = pow(r55260, r55265);
        double r55267 = r55264 * r55266;
        double r55268 = 2.0;
        double r55269 = r55268 * r55260;
        double r55270 = r55267 + r55269;
        double r55271 = r55263 + r55270;
        double r55272 = 2.0;
        double r55273 = r55271 / r55272;
        double r55274 = y;
        double r55275 = sin(r55274);
        double r55276 = r55273 * r55275;
        return r55276;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Simplified43.6

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020056 
(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)))))