Average Error: 0.0 → 0.0
Time: 18.7s
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 r60103 = x;
        double r60104 = exp(r60103);
        double r60105 = -r60103;
        double r60106 = exp(r60105);
        double r60107 = r60104 + r60106;
        double r60108 = 2.0;
        double r60109 = r60107 / r60108;
        double r60110 = y;
        double r60111 = cos(r60110);
        double r60112 = r60109 * r60111;
        double r60113 = r60104 - r60106;
        double r60114 = r60113 / r60108;
        double r60115 = sin(r60110);
        double r60116 = r60114 * r60115;
        double r60117 = /* ERROR: no complex support in C */;
        double r60118 = /* ERROR: no complex support in C */;
        return r60118;
}
↓
double f(double x, double y) {
        double r60119 = x;
        double r60120 = exp(r60119);
        double r60121 = -r60119;
        double r60122 = exp(r60121);
        double r60123 = r60120 + r60122;
        double r60124 = 2.0;
        double r60125 = r60123 / r60124;
        double r60126 = y;
        double r60127 = cos(r60126);
        double r60128 = r60125 * r60127;
        return r60128;
}

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 2019308 
(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)))))