Average Error: 0.0 → 0.0
Time: 2.6s
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 r18740 = x;
        double r18741 = exp(r18740);
        double r18742 = -r18740;
        double r18743 = exp(r18742);
        double r18744 = r18741 + r18743;
        double r18745 = 2.0;
        double r18746 = r18744 / r18745;
        double r18747 = y;
        double r18748 = cos(r18747);
        double r18749 = r18746 * r18748;
        double r18750 = r18741 - r18743;
        double r18751 = r18750 / r18745;
        double r18752 = sin(r18747);
        double r18753 = r18751 * r18752;
        double r18754 = /* ERROR: no complex support in C */;
        double r18755 = /* ERROR: no complex support in C */;
        return r18755;
}
↓
double f(double x, double y) {
        double r18756 = x;
        double r18757 = exp(r18756);
        double r18758 = -r18756;
        double r18759 = exp(r18758);
        double r18760 = r18757 + r18759;
        double r18761 = 2.0;
        double r18762 = r18760 / r18761;
        double r18763 = y;
        double r18764 = cos(r18763);
        double r18765 = r18762 * r18764;
        return r18765;
}

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 2020025 
(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)))))