Average Error: 0.0 → 0.0
Time: 13.8s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)\]
\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\left(\cos y\right), \left(e^{x}\right), \left(\frac{\cos y}{e^{x}}\right)\right)
double f(double x, double y) {
        double r1022252 = x;
        double r1022253 = exp(r1022252);
        double r1022254 = -r1022252;
        double r1022255 = exp(r1022254);
        double r1022256 = r1022253 + r1022255;
        double r1022257 = 2.0;
        double r1022258 = r1022256 / r1022257;
        double r1022259 = y;
        double r1022260 = cos(r1022259);
        double r1022261 = r1022258 * r1022260;
        double r1022262 = r1022253 - r1022255;
        double r1022263 = r1022262 / r1022257;
        double r1022264 = sin(r1022259);
        double r1022265 = r1022263 * r1022264;
        double r1022266 = /* ERROR: no complex support in C */;
        double r1022267 = /* ERROR: no complex support in C */;
        return r1022267;
}
↓
double f(double x, double y) {
        double r1022268 = 0.5;
        double r1022269 = y;
        double r1022270 = cos(r1022269);
        double r1022271 = x;
        double r1022272 = exp(r1022271);
        double r1022273 = r1022270 / r1022272;
        double r1022274 = fma(r1022270, r1022272, r1022273);
        double r1022275 = r1022268 * r1022274;
        return r1022275;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +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)))))