Average Error: 0.0 → 0.0
Time: 20.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]
\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} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}
double f(double x, double y) {
        double r1212199 = x;
        double r1212200 = exp(r1212199);
        double r1212201 = -r1212199;
        double r1212202 = exp(r1212201);
        double r1212203 = r1212200 + r1212202;
        double r1212204 = 2.0;
        double r1212205 = r1212203 / r1212204;
        double r1212206 = y;
        double r1212207 = cos(r1212206);
        double r1212208 = r1212205 * r1212207;
        double r1212209 = r1212200 - r1212202;
        double r1212210 = r1212209 / r1212204;
        double r1212211 = sin(r1212206);
        double r1212212 = r1212210 * r1212211;
        double r1212213 = /* ERROR: no complex support in C */;
        double r1212214 = /* ERROR: no complex support in C */;
        return r1212214;
}
↓
double f(double x, double y) {
        double r1212215 = x;
        double r1212216 = exp(r1212215);
        double r1212217 = y;
        double r1212218 = cos(r1212217);
        double r1212219 = r1212216 * r1212218;
        double r1212220 = r1212218 / r1212216;
        double r1212221 = r1212219 + r1212220;
        double r1212222 = 2.0;
        double r1212223 = r1212221 / r1212222;
        return r1212223;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))