Average Error: 0.0 → 0.0
Time: 3.7s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r25512 = x;
        double r25513 = exp(r25512);
        double r25514 = -r25512;
        double r25515 = exp(r25514);
        double r25516 = r25513 + r25515;
        double r25517 = 2.0;
        double r25518 = r25516 / r25517;
        double r25519 = y;
        double r25520 = cos(r25519);
        double r25521 = r25518 * r25520;
        double r25522 = r25513 - r25515;
        double r25523 = r25522 / r25517;
        double r25524 = sin(r25519);
        double r25525 = r25523 * r25524;
        double r25526 = /* ERROR: no complex support in C */;
        double r25527 = /* ERROR: no complex support in C */;
        return r25527;
}
↓
double f(double x, double y) {
        double r25528 = x;
        double r25529 = exp(r25528);
        double r25530 = -r25528;
        double r25531 = exp(r25530);
        double r25532 = r25529 + r25531;
        double r25533 = 2.0;
        double r25534 = r25532 / r25533;
        double r25535 = y;
        double r25536 = cos(r25535);
        double r25537 = r25534 * r25536;
        double r25538 = r25529 - r25531;
        double r25539 = r25538 / r25533;
        double r25540 = sin(r25535);
        double r25541 = r25539 * r25540;
        double r25542 = /* ERROR: no complex support in C */;
        double r25543 = /* ERROR: no complex support in C */;
        return r25543;
}

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2020046 
(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)))))