Average Error: 43.8 → 0.7
Time: 11.5s
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 r43302 = x;
        double r43303 = exp(r43302);
        double r43304 = -r43302;
        double r43305 = exp(r43304);
        double r43306 = r43303 + r43305;
        double r43307 = 2.0;
        double r43308 = r43306 / r43307;
        double r43309 = y;
        double r43310 = cos(r43309);
        double r43311 = r43308 * r43310;
        double r43312 = r43303 - r43305;
        double r43313 = r43312 / r43307;
        double r43314 = sin(r43309);
        double r43315 = r43313 * r43314;
        double r43316 = /* ERROR: no complex support in C */;
        double r43317 = /* ERROR: no complex support in C */;
        return r43317;
}
↓
double f(double x, double y) {
        double r43318 = 0.3333333333333333;
        double r43319 = x;
        double r43320 = 3.0;
        double r43321 = pow(r43319, r43320);
        double r43322 = r43318 * r43321;
        double r43323 = 0.016666666666666666;
        double r43324 = 5.0;
        double r43325 = pow(r43319, r43324);
        double r43326 = r43323 * r43325;
        double r43327 = 2.0;
        double r43328 = r43327 * r43319;
        double r43329 = r43326 + r43328;
        double r43330 = r43322 + r43329;
        double r43331 = 2.0;
        double r43332 = r43330 / r43331;
        double r43333 = y;
        double r43334 = sin(r43333);
        double r43335 = r43332 * r43334;
        return r43335;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020065 
(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)))))