Average Error: 0.0 → 0.0
Time: 4.1s
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 r20587 = x;
        double r20588 = exp(r20587);
        double r20589 = -r20587;
        double r20590 = exp(r20589);
        double r20591 = r20588 + r20590;
        double r20592 = 2.0;
        double r20593 = r20591 / r20592;
        double r20594 = y;
        double r20595 = cos(r20594);
        double r20596 = r20593 * r20595;
        double r20597 = r20588 - r20590;
        double r20598 = r20597 / r20592;
        double r20599 = sin(r20594);
        double r20600 = r20598 * r20599;
        double r20601 = /* ERROR: no complex support in C */;
        double r20602 = /* ERROR: no complex support in C */;
        return r20602;
}
↓
double f(double x, double y) {
        double r20603 = x;
        double r20604 = exp(r20603);
        double r20605 = -r20603;
        double r20606 = exp(r20605);
        double r20607 = r20604 + r20606;
        double r20608 = 2.0;
        double r20609 = r20607 / r20608;
        double r20610 = y;
        double r20611 = cos(r20610);
        double r20612 = r20609 * r20611;
        return r20612;
}

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 2020027 
(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)))))