Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} + e^{-x}}{2} \cdot \cos y\]
\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 r34278 = x;
        double r34279 = exp(r34278);
        double r34280 = -r34278;
        double r34281 = exp(r34280);
        double r34282 = r34279 + r34281;
        double r34283 = 2.0;
        double r34284 = r34282 / r34283;
        double r34285 = y;
        double r34286 = cos(r34285);
        double r34287 = r34284 * r34286;
        double r34288 = r34279 - r34281;
        double r34289 = r34288 / r34283;
        double r34290 = sin(r34285);
        double r34291 = r34289 * r34290;
        double r34292 = /* ERROR: no complex support in C */;
        double r34293 = /* ERROR: no complex support in C */;
        return r34293;
}
↓
double f(double x, double y) {
        double r34294 = x;
        double r34295 = exp(r34294);
        double r34296 = -r34294;
        double r34297 = exp(r34296);
        double r34298 = r34295 + r34297;
        double r34299 = 2.0;
        double r34300 = r34298 / r34299;
        double r34301 = y;
        double r34302 = cos(r34301);
        double r34303 = r34300 * r34302;
        return r34303;
}

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 2019322 +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)))))