Average Error: 0.0 → 0.0
Time: 13.0s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\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} + e^{-x}}{2} \cdot \cos y
double f(double x, double y) {
        double r14203 = x;
        double r14204 = exp(r14203);
        double r14205 = -r14203;
        double r14206 = exp(r14205);
        double r14207 = r14204 + r14206;
        double r14208 = 2.0;
        double r14209 = r14207 / r14208;
        double r14210 = y;
        double r14211 = cos(r14210);
        double r14212 = r14209 * r14211;
        double r14213 = r14204 - r14206;
        double r14214 = r14213 / r14208;
        double r14215 = sin(r14210);
        double r14216 = r14214 * r14215;
        double r14217 = /* ERROR: no complex support in C */;
        double r14218 = /* ERROR: no complex support in C */;
        return r14218;
}
↓
double f(double x, double y) {
        double r14219 = x;
        double r14220 = exp(r14219);
        double r14221 = -r14219;
        double r14222 = exp(r14221);
        double r14223 = r14220 + r14222;
        double r14224 = 2.0;
        double r14225 = r14223 / r14224;
        double r14226 = y;
        double r14227 = cos(r14226);
        double r14228 = r14225 * r14227;
        return r14228;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 
(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)))))