Average Error: 0.0 → 0.0
Time: 10.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\left(\frac{1}{e^{x}} + e^{x}\right) \cdot \cos y}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\left(\frac{1}{e^{x}} + e^{x}\right) \cdot \cos y}{2}
double f(double x, double y) {
        double r496938 = x;
        double r496939 = exp(r496938);
        double r496940 = -r496938;
        double r496941 = exp(r496940);
        double r496942 = r496939 + r496941;
        double r496943 = 2.0;
        double r496944 = r496942 / r496943;
        double r496945 = y;
        double r496946 = cos(r496945);
        double r496947 = r496944 * r496946;
        double r496948 = r496939 - r496941;
        double r496949 = r496948 / r496943;
        double r496950 = sin(r496945);
        double r496951 = r496949 * r496950;
        double r496952 = /* ERROR: no complex support in C */;
        double r496953 = /* ERROR: no complex support in C */;
        return r496953;
}
↓
double f(double x, double y) {
        double r496954 = 1.0;
        double r496955 = x;
        double r496956 = exp(r496955);
        double r496957 = r496954 / r496956;
        double r496958 = r496957 + r496956;
        double r496959 = y;
        double r496960 = cos(r496959);
        double r496961 = r496958 * r496960;
        double r496962 = 2.0;
        double r496963 = r496961 / r496962;
        return r496963;
}

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot \frac{1}{e^{x}}} + \cos y \cdot e^{x}}{2}\]

  5. Applied distribute-lft-out0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot \left(\frac{1}{e^{x}} + e^{x}\right)}}{2}\]

  6. Final simplification0.0

    \[\leadsto \frac{\left(\frac{1}{e^{x}} + e^{x}\right) \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))