Average Error: 0.0 → 0.0
Time: 21.1s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r37932 = x;
        double r37933 = exp(r37932);
        double r37934 = -r37932;
        double r37935 = exp(r37934);
        double r37936 = r37933 + r37935;
        double r37937 = 2.0;
        double r37938 = r37936 / r37937;
        double r37939 = y;
        double r37940 = cos(r37939);
        double r37941 = r37938 * r37940;
        double r37942 = r37933 - r37935;
        double r37943 = r37942 / r37937;
        double r37944 = sin(r37939);
        double r37945 = r37943 * r37944;
        double r37946 = /* ERROR: no complex support in C */;
        double r37947 = /* ERROR: no complex support in C */;
        return r37947;
}
↓
double f(double x, double y) {
        double r37948 = x;
        double r37949 = exp(r37948);
        double r37950 = -r37948;
        double r37951 = exp(r37950);
        double r37952 = r37949 + r37951;
        double r37953 = 2.0;
        double r37954 = r37952 / r37953;
        double r37955 = y;
        double r37956 = cos(r37955);
        double r37957 = r37954 * r37956;
        double r37958 = r37949 - r37951;
        double r37959 = r37958 / r37953;
        double r37960 = sin(r37955);
        double r37961 = r37959 * r37960;
        double r37962 = /* ERROR: no complex support in C */;
        double r37963 = /* ERROR: no complex support in C */;
        return r37963;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019208 +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)))))