Average Error: 0.0 → 0.0
Time: 18.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{e^{x} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r38502 = x;
        double r38503 = exp(r38502);
        double r38504 = -r38502;
        double r38505 = exp(r38504);
        double r38506 = r38503 + r38505;
        double r38507 = 2.0;
        double r38508 = r38506 / r38507;
        double r38509 = y;
        double r38510 = cos(r38509);
        double r38511 = r38508 * r38510;
        double r38512 = r38503 - r38505;
        double r38513 = r38512 / r38507;
        double r38514 = sin(r38509);
        double r38515 = r38513 * r38514;
        double r38516 = /* ERROR: no complex support in C */;
        double r38517 = /* ERROR: no complex support in C */;
        return r38517;
}
↓
double f(double x, double y) {
        double r38518 = x;
        double r38519 = exp(r38518);
        double r38520 = -r38518;
        double r38521 = exp(r38520);
        double r38522 = r38519 + r38521;
        double r38523 = 2.0;
        double r38524 = r38522 / r38523;
        double r38525 = y;
        double r38526 = cos(r38525);
        double r38527 = r38524 * r38526;
        return r38527;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} + e^{-x}}{2} \cdot \cos y\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  :precision binary64
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))