Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
double f(double x, double y) {
        double r527560 = x;
        double r527561 = exp(r527560);
        double r527562 = -r527560;
        double r527563 = exp(r527562);
        double r527564 = r527561 + r527563;
        double r527565 = 2.0;
        double r527566 = r527564 / r527565;
        double r527567 = y;
        double r527568 = cos(r527567);
        double r527569 = r527566 * r527568;
        double r527570 = r527561 - r527563;
        double r527571 = r527570 / r527565;
        double r527572 = sin(r527567);
        double r527573 = r527571 * r527572;
        double r527574 = /* ERROR: no complex support in C */;
        double r527575 = /* ERROR: no complex support in C */;
        return r527575;
}
↓
double f(double x, double y) {
        double r527576 = x;
        double r527577 = exp(r527576);
        double r527578 = -r527576;
        double r527579 = exp(r527578);
        double r527580 = r527577 + r527579;
        double r527581 = 2.0;
        double r527582 = r527580 / r527581;
        double r527583 = y;
        double r527584 = cos(r527583);
        double r527585 = r527582 * r527584;
        double r527586 = r527577 - r527579;
        double r527587 = r527586 / r527581;
        double r527588 = sin(r527583);
        double r527589 = r527587 * r527588;
        double r527590 = /* ERROR: no complex support in C */;
        double r527591 = /* ERROR: no complex support in C */;
        return r527591;
}


\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))

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 2019102 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))