Average Error: 0.0 → 0.0
Time: 15.6s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}
double f(double x, double y) {
        double r1073473 = x;
        double r1073474 = exp(r1073473);
        double r1073475 = -r1073473;
        double r1073476 = exp(r1073475);
        double r1073477 = r1073474 + r1073476;
        double r1073478 = 2.0;
        double r1073479 = r1073477 / r1073478;
        double r1073480 = y;
        double r1073481 = cos(r1073480);
        double r1073482 = r1073479 * r1073481;
        double r1073483 = r1073474 - r1073476;
        double r1073484 = r1073483 / r1073478;
        double r1073485 = sin(r1073480);
        double r1073486 = r1073484 * r1073485;
        double r1073487 = /* ERROR: no complex support in C */;
        double r1073488 = /* ERROR: no complex support in C */;
        return r1073488;
}
↓
double f(double x, double y) {
        double r1073489 = y;
        double r1073490 = cos(r1073489);
        double r1073491 = x;
        double r1073492 = exp(r1073491);
        double r1073493 = r1073490 / r1073492;
        double r1073494 = r1073492 * r1073490;
        double r1073495 = r1073493 + r1073494;
        double r1073496 = 0.5;
        double r1073497 = r1073495 * r1073496;
        return r1073497;
}

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

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}\right)}\]

  3. Final simplification0.0

    \[\leadsto \left(\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019132 
(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)))))