Average Error: 0.0 → 0.0
Time: 5.9s
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 r39085 = x;
        double r39086 = exp(r39085);
        double r39087 = -r39085;
        double r39088 = exp(r39087);
        double r39089 = r39086 + r39088;
        double r39090 = 2.0;
        double r39091 = r39089 / r39090;
        double r39092 = y;
        double r39093 = cos(r39092);
        double r39094 = r39091 * r39093;
        double r39095 = r39086 - r39088;
        double r39096 = r39095 / r39090;
        double r39097 = sin(r39092);
        double r39098 = r39096 * r39097;
        double r39099 = /* ERROR: no complex support in C */;
        double r39100 = /* ERROR: no complex support in C */;
        return r39100;
}
↓
double f(double x, double y) {
        double r39101 = x;
        double r39102 = exp(r39101);
        double r39103 = -r39101;
        double r39104 = exp(r39103);
        double r39105 = r39102 + r39104;
        double r39106 = 2.0;
        double r39107 = r39105 / r39106;
        double r39108 = y;
        double r39109 = cos(r39108);
        double r39110 = r39107 * r39109;
        return r39110;
}

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 2019353 
(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)))))