Average Error: 0.0 → 0.0
Time: 3.0s
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 r33112 = x;
        double r33113 = exp(r33112);
        double r33114 = -r33112;
        double r33115 = exp(r33114);
        double r33116 = r33113 + r33115;
        double r33117 = 2.0;
        double r33118 = r33116 / r33117;
        double r33119 = y;
        double r33120 = cos(r33119);
        double r33121 = r33118 * r33120;
        double r33122 = r33113 - r33115;
        double r33123 = r33122 / r33117;
        double r33124 = sin(r33119);
        double r33125 = r33123 * r33124;
        double r33126 = /* ERROR: no complex support in C */;
        double r33127 = /* ERROR: no complex support in C */;
        return r33127;
}
↓
double f(double x, double y) {
        double r33128 = x;
        double r33129 = exp(r33128);
        double r33130 = -r33128;
        double r33131 = exp(r33130);
        double r33132 = r33129 + r33131;
        double r33133 = 2.0;
        double r33134 = r33132 / r33133;
        double r33135 = y;
        double r33136 = cos(r33135);
        double r33137 = r33134 * r33136;
        return r33137;
}

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 2019354 
(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)))))