Average Error: 0.0 → 0.0
Time: 16.4s
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 r26279 = x;
        double r26280 = exp(r26279);
        double r26281 = -r26279;
        double r26282 = exp(r26281);
        double r26283 = r26280 + r26282;
        double r26284 = 2.0;
        double r26285 = r26283 / r26284;
        double r26286 = y;
        double r26287 = cos(r26286);
        double r26288 = r26285 * r26287;
        double r26289 = r26280 - r26282;
        double r26290 = r26289 / r26284;
        double r26291 = sin(r26286);
        double r26292 = r26290 * r26291;
        double r26293 = /* ERROR: no complex support in C */;
        double r26294 = /* ERROR: no complex support in C */;
        return r26294;
}
↓
double f(double x, double y) {
        double r26295 = x;
        double r26296 = exp(r26295);
        double r26297 = -r26295;
        double r26298 = exp(r26297);
        double r26299 = r26296 + r26298;
        double r26300 = 2.0;
        double r26301 = r26299 / r26300;
        double r26302 = y;
        double r26303 = cos(r26302);
        double r26304 = r26301 * r26303;
        return r26304;
}

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 2019323 
(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)))))