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 r25557 = x;
        double r25558 = exp(r25557);
        double r25559 = -r25557;
        double r25560 = exp(r25559);
        double r25561 = r25558 + r25560;
        double r25562 = 2.0;
        double r25563 = r25561 / r25562;
        double r25564 = y;
        double r25565 = cos(r25564);
        double r25566 = r25563 * r25565;
        double r25567 = r25558 - r25560;
        double r25568 = r25567 / r25562;
        double r25569 = sin(r25564);
        double r25570 = r25568 * r25569;
        double r25571 = /* ERROR: no complex support in C */;
        double r25572 = /* ERROR: no complex support in C */;
        return r25572;
}
↓
double f(double x, double y) {
        double r25573 = x;
        double r25574 = exp(r25573);
        double r25575 = -r25573;
        double r25576 = exp(r25575);
        double r25577 = r25574 + r25576;
        double r25578 = 2.0;
        double r25579 = r25577 / r25578;
        double r25580 = y;
        double r25581 = cos(r25580);
        double r25582 = r25579 * r25581;
        return r25582;
}

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 2020001 
(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)))))