Average Error: 0.0 → 0.0
Time: 4.6s
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 r37084 = x;
        double r37085 = exp(r37084);
        double r37086 = -r37084;
        double r37087 = exp(r37086);
        double r37088 = r37085 + r37087;
        double r37089 = 2.0;
        double r37090 = r37088 / r37089;
        double r37091 = y;
        double r37092 = cos(r37091);
        double r37093 = r37090 * r37092;
        double r37094 = r37085 - r37087;
        double r37095 = r37094 / r37089;
        double r37096 = sin(r37091);
        double r37097 = r37095 * r37096;
        double r37098 = /* ERROR: no complex support in C */;
        double r37099 = /* ERROR: no complex support in C */;
        return r37099;
}
↓
double f(double x, double y) {
        double r37100 = x;
        double r37101 = exp(r37100);
        double r37102 = -r37100;
        double r37103 = exp(r37102);
        double r37104 = r37101 + r37103;
        double r37105 = 2.0;
        double r37106 = r37104 / r37105;
        double r37107 = y;
        double r37108 = cos(r37107);
        double r37109 = r37106 * r37108;
        return r37109;
}

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 2020062 
(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)))))