Average Error: 0.0 → 0.0
Time: 22.9s
Precision: 64
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\cos y, e^{x}, \frac{1}{e^{x}} \cdot \cos y\right)\]
\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(\cos y, e^{x}, \frac{1}{e^{x}} \cdot \cos y\right)
double f(double x, double y) {
        double r729989 = x;
        double r729990 = exp(r729989);
        double r729991 = -r729989;
        double r729992 = exp(r729991);
        double r729993 = r729990 + r729992;
        double r729994 = 2.0;
        double r729995 = r729993 / r729994;
        double r729996 = y;
        double r729997 = cos(r729996);
        double r729998 = r729995 * r729997;
        double r729999 = r729990 - r729992;
        double r730000 = r729999 / r729994;
        double r730001 = sin(r729996);
        double r730002 = r730000 * r730001;
        double r730003 = /* ERROR: no complex support in C */;
        double r730004 = /* ERROR: no complex support in C */;
        return r730004;
}
↓
double f(double x, double y) {
        double r730005 = 0.5;
        double r730006 = y;
        double r730007 = cos(r730006);
        double r730008 = x;
        double r730009 = exp(r730008);
        double r730010 = 1.0;
        double r730011 = r730010 / r730009;
        double r730012 = r730011 * r730007;
        double r730013 = fma(r730007, r730009, r730012);
        double r730014 = r730005 * r730013;
        return r730014;
}

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

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 +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)))))