Average Error: 0.0 → 0.0
Time: 4.6s
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 r25608 = x;
        double r25609 = exp(r25608);
        double r25610 = -r25608;
        double r25611 = exp(r25610);
        double r25612 = r25609 + r25611;
        double r25613 = 2.0;
        double r25614 = r25612 / r25613;
        double r25615 = y;
        double r25616 = cos(r25615);
        double r25617 = r25614 * r25616;
        double r25618 = r25609 - r25611;
        double r25619 = r25618 / r25613;
        double r25620 = sin(r25615);
        double r25621 = r25619 * r25620;
        double r25622 = /* ERROR: no complex support in C */;
        double r25623 = /* ERROR: no complex support in C */;
        return r25623;
}
↓
double f(double x, double y) {
        double r25624 = x;
        double r25625 = exp(r25624);
        double r25626 = -r25624;
        double r25627 = exp(r25626);
        double r25628 = r25625 + r25627;
        double r25629 = 2.0;
        double r25630 = r25628 / r25629;
        double r25631 = y;
        double r25632 = cos(r25631);
        double r25633 = r25630 * r25632;
        return r25633;
}

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 2019346 +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)))))