Average Error: 0.0 → 0.0
Time: 5.8s
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 r42952 = x;
        double r42953 = exp(r42952);
        double r42954 = -r42952;
        double r42955 = exp(r42954);
        double r42956 = r42953 + r42955;
        double r42957 = 2.0;
        double r42958 = r42956 / r42957;
        double r42959 = y;
        double r42960 = cos(r42959);
        double r42961 = r42958 * r42960;
        double r42962 = r42953 - r42955;
        double r42963 = r42962 / r42957;
        double r42964 = sin(r42959);
        double r42965 = r42963 * r42964;
        double r42966 = /* ERROR: no complex support in C */;
        double r42967 = /* ERROR: no complex support in C */;
        return r42967;
}
↓
double f(double x, double y) {
        double r42968 = x;
        double r42969 = exp(r42968);
        double r42970 = -r42968;
        double r42971 = exp(r42970);
        double r42972 = r42969 + r42971;
        double r42973 = 2.0;
        double r42974 = r42972 / r42973;
        double r42975 = y;
        double r42976 = cos(r42975);
        double r42977 = r42974 * r42976;
        return r42977;
}

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)))))