Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]
\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}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}
double f(double x, double y) {
        double r1901144 = x;
        double r1901145 = exp(r1901144);
        double r1901146 = -r1901144;
        double r1901147 = exp(r1901146);
        double r1901148 = r1901145 + r1901147;
        double r1901149 = 2.0;
        double r1901150 = r1901148 / r1901149;
        double r1901151 = y;
        double r1901152 = cos(r1901151);
        double r1901153 = r1901150 * r1901152;
        double r1901154 = r1901145 - r1901147;
        double r1901155 = r1901154 / r1901149;
        double r1901156 = sin(r1901151);
        double r1901157 = r1901155 * r1901156;
        double r1901158 = /* ERROR: no complex support in C */;
        double r1901159 = /* ERROR: no complex support in C */;
        return r1901159;
}
↓
double f(double x, double y) {
        double r1901160 = y;
        double r1901161 = cos(r1901160);
        double r1901162 = x;
        double r1901163 = exp(r1901162);
        double r1901164 = r1901161 / r1901163;
        double r1901165 = r1901163 * r1901161;
        double r1901166 = r1901164 + r1901165;
        double r1901167 = 0.5;
        double r1901168 = r1901166 * r1901167;
        return r1901168;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 
(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)))))