Average Error: 0.0 → 0.0
Time: 18.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + \left(\sqrt{e^{x}} \cdot \cos y\right) \cdot \sqrt{e^{x}}\right) \cdot \frac{1}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\frac{\cos y}{e^{x}} + \left(\sqrt{e^{x}} \cdot \cos y\right) \cdot \sqrt{e^{x}}\right) \cdot \frac{1}{2}
double f(double x, double y) {
        double r1015065 = x;
        double r1015066 = exp(r1015065);
        double r1015067 = -r1015065;
        double r1015068 = exp(r1015067);
        double r1015069 = r1015066 + r1015068;
        double r1015070 = 2.0;
        double r1015071 = r1015069 / r1015070;
        double r1015072 = y;
        double r1015073 = cos(r1015072);
        double r1015074 = r1015071 * r1015073;
        double r1015075 = r1015066 - r1015068;
        double r1015076 = r1015075 / r1015070;
        double r1015077 = sin(r1015072);
        double r1015078 = r1015076 * r1015077;
        double r1015079 = /* ERROR: no complex support in C */;
        double r1015080 = /* ERROR: no complex support in C */;
        return r1015080;
}
↓
double f(double x, double y) {
        double r1015081 = y;
        double r1015082 = cos(r1015081);
        double r1015083 = x;
        double r1015084 = exp(r1015083);
        double r1015085 = r1015082 / r1015084;
        double r1015086 = sqrt(r1015084);
        double r1015087 = r1015086 * r1015082;
        double r1015088 = r1015087 * r1015086;
        double r1015089 = r1015085 + r1015088;
        double r1015090 = 0.5;
        double r1015091 = r1015089 * r1015090;
        return r1015091;
}

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 
(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)))))