Average Error: 0.0 → 0.0
Time: 30.0s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}
double f(double x, double y) {
        double r589563 = x;
        double r589564 = exp(r589563);
        double r589565 = -r589563;
        double r589566 = exp(r589565);
        double r589567 = r589564 + r589566;
        double r589568 = 2.0;
        double r589569 = r589567 / r589568;
        double r589570 = y;
        double r589571 = cos(r589570);
        double r589572 = r589569 * r589571;
        double r589573 = r589564 - r589566;
        double r589574 = r589573 / r589568;
        double r589575 = sin(r589570);
        double r589576 = r589574 * r589575;
        double r589577 = /* ERROR: no complex support in C */;
        double r589578 = /* ERROR: no complex support in C */;
        return r589578;
}
↓
double f(double x, double y) {
        double r589579 = y;
        double r589580 = cos(r589579);
        double r589581 = x;
        double r589582 = exp(r589581);
        double r589583 = r589580 / r589582;
        double r589584 = r589580 * r589582;
        double r589585 = r589583 + r589584;
        double r589586 = 2.0;
        double r589587 = r589585 / r589586;
        return r589587;
}

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

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{x}, \cos y, \frac{\cos y}{e^{x}}\right)}{2}}\]
  3. Using strategy rm

  4. Applied fma-udef0.0

    \[\leadsto \frac{\color{blue}{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}}{2}\]

  5. Final simplification0.0

    \[\leadsto \frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))