Average Error: 0.0 → 0.0
Time: 9.9s
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 r27100 = x;
        double r27101 = exp(r27100);
        double r27102 = -r27100;
        double r27103 = exp(r27102);
        double r27104 = r27101 + r27103;
        double r27105 = 2.0;
        double r27106 = r27104 / r27105;
        double r27107 = y;
        double r27108 = cos(r27107);
        double r27109 = r27106 * r27108;
        double r27110 = r27101 - r27103;
        double r27111 = r27110 / r27105;
        double r27112 = sin(r27107);
        double r27113 = r27111 * r27112;
        double r27114 = /* ERROR: no complex support in C */;
        double r27115 = /* ERROR: no complex support in C */;
        return r27115;
}
↓
double f(double x, double y) {
        double r27116 = x;
        double r27117 = exp(r27116);
        double r27118 = -r27116;
        double r27119 = exp(r27118);
        double r27120 = r27117 + r27119;
        double r27121 = 2.0;
        double r27122 = r27120 / r27121;
        double r27123 = y;
        double r27124 = cos(r27123);
        double r27125 = r27122 * r27124;
        return r27125;
}

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 2019350 
(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)))))