Average Error: 0.0 → 0.0
Time: 12.0s
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 r21938 = x;
        double r21939 = exp(r21938);
        double r21940 = -r21938;
        double r21941 = exp(r21940);
        double r21942 = r21939 + r21941;
        double r21943 = 2.0;
        double r21944 = r21942 / r21943;
        double r21945 = y;
        double r21946 = cos(r21945);
        double r21947 = r21944 * r21946;
        double r21948 = r21939 - r21941;
        double r21949 = r21948 / r21943;
        double r21950 = sin(r21945);
        double r21951 = r21949 * r21950;
        double r21952 = /* ERROR: no complex support in C */;
        double r21953 = /* ERROR: no complex support in C */;
        return r21953;
}
↓
double f(double x, double y) {
        double r21954 = x;
        double r21955 = exp(r21954);
        double r21956 = -r21954;
        double r21957 = exp(r21956);
        double r21958 = r21955 + r21957;
        double r21959 = 2.0;
        double r21960 = r21958 / r21959;
        double r21961 = y;
        double r21962 = cos(r21961);
        double r21963 = r21960 * r21962;
        return r21963;
}

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 +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)))))