Average Error: 0.0 → 0.0
Time: 16.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r31217 = x;
        double r31218 = exp(r31217);
        double r31219 = -r31217;
        double r31220 = exp(r31219);
        double r31221 = r31218 + r31220;
        double r31222 = 2.0;
        double r31223 = r31221 / r31222;
        double r31224 = y;
        double r31225 = cos(r31224);
        double r31226 = r31223 * r31225;
        double r31227 = r31218 - r31220;
        double r31228 = r31227 / r31222;
        double r31229 = sin(r31224);
        double r31230 = r31228 * r31229;
        double r31231 = /* ERROR: no complex support in C */;
        double r31232 = /* ERROR: no complex support in C */;
        return r31232;
}
↓
double f(double x, double y) {
        double r31233 = x;
        double r31234 = exp(r31233);
        double r31235 = -r31233;
        double r31236 = exp(r31235);
        double r31237 = r31234 + r31236;
        double r31238 = 2.0;
        double r31239 = r31237 / r31238;
        double r31240 = y;
        double r31241 = cos(r31240);
        double r31242 = r31239 * r31241;
        double r31243 = r31234 - r31236;
        double r31244 = r31243 / r31238;
        double r31245 = sin(r31240);
        double r31246 = r31244 * r31245;
        double r31247 = /* ERROR: no complex support in C */;
        double r31248 = /* ERROR: no complex support in C */;
        return r31248;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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)))))