Average Error: 0.0 → 0.0
Time: 3.5s
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 r31990 = x;
        double r31991 = exp(r31990);
        double r31992 = -r31990;
        double r31993 = exp(r31992);
        double r31994 = r31991 + r31993;
        double r31995 = 2.0;
        double r31996 = r31994 / r31995;
        double r31997 = y;
        double r31998 = cos(r31997);
        double r31999 = r31996 * r31998;
        double r32000 = r31991 - r31993;
        double r32001 = r32000 / r31995;
        double r32002 = sin(r31997);
        double r32003 = r32001 * r32002;
        double r32004 = /* ERROR: no complex support in C */;
        double r32005 = /* ERROR: no complex support in C */;
        return r32005;
}
↓
double f(double x, double y) {
        double r32006 = x;
        double r32007 = exp(r32006);
        double r32008 = -r32006;
        double r32009 = exp(r32008);
        double r32010 = r32007 + r32009;
        double r32011 = 2.0;
        double r32012 = r32010 / r32011;
        double r32013 = y;
        double r32014 = cos(r32013);
        double r32015 = r32012 * r32014;
        return r32015;
}

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 2020035 
(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)))))