Average Error: 0.0 → 0.0
Time: 17.2s
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 r1253415 = x;
        double r1253416 = exp(r1253415);
        double r1253417 = -r1253415;
        double r1253418 = exp(r1253417);
        double r1253419 = r1253416 + r1253418;
        double r1253420 = 2.0;
        double r1253421 = r1253419 / r1253420;
        double r1253422 = y;
        double r1253423 = cos(r1253422);
        double r1253424 = r1253421 * r1253423;
        double r1253425 = r1253416 - r1253418;
        double r1253426 = r1253425 / r1253420;
        double r1253427 = sin(r1253422);
        double r1253428 = r1253426 * r1253427;
        double r1253429 = /* ERROR: no complex support in C */;
        double r1253430 = /* ERROR: no complex support in C */;
        return r1253430;
}
↓
double f(double x, double y) {
        double r1253431 = y;
        double r1253432 = cos(r1253431);
        double r1253433 = x;
        double r1253434 = exp(r1253433);
        double r1253435 = r1253432 / r1253434;
        double r1253436 = r1253432 * r1253434;
        double r1253437 = r1253435 + r1253436;
        double r1253438 = 2.0;
        double r1253439 = r1253437 / r1253438;
        return r1253439;
}

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) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))