Average Error: 0.0 → 0.0
Time: 4.2s
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 r29933 = x;
        double r29934 = exp(r29933);
        double r29935 = -r29933;
        double r29936 = exp(r29935);
        double r29937 = r29934 + r29936;
        double r29938 = 2.0;
        double r29939 = r29937 / r29938;
        double r29940 = y;
        double r29941 = cos(r29940);
        double r29942 = r29939 * r29941;
        double r29943 = r29934 - r29936;
        double r29944 = r29943 / r29938;
        double r29945 = sin(r29940);
        double r29946 = r29944 * r29945;
        double r29947 = /* ERROR: no complex support in C */;
        double r29948 = /* ERROR: no complex support in C */;
        return r29948;
}
↓
double f(double x, double y) {
        double r29949 = x;
        double r29950 = exp(r29949);
        double r29951 = -r29949;
        double r29952 = exp(r29951);
        double r29953 = r29950 + r29952;
        double r29954 = 2.0;
        double r29955 = r29953 / r29954;
        double r29956 = y;
        double r29957 = cos(r29956);
        double r29958 = r29955 * r29957;
        return r29958;
}

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 2020001 +o rules:numerics
(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)))))