Smoothed mathematics

The functions in this section are smoothed equivalents of the original math functions and can be used to allow computing derivatives around discontinuities.

`smooth_abs`

T smooth_abs(T x, T c = 0.001) is a smoothed version of abs, defined as:

smooth_abs (x, c) = {\begin{cases} | x | & if x < c or x > c \\ x^{2} (\frac{2}{c} - \frac{1}{c^{2}} x) & if 0 \leq x \leq c \\ x^{2} (\frac{2}{c} + \frac{1}{c^{2}} x) & if - c \leq x < 0 \end{cases}

x is the input value
c is the cut-off point for the spline-approximated area (default: 0.001)
returns: The smoothed absolute value, defined as above.

`smooth_max`

T smooth_max(T x, T y, T c = 0.001) is a smoothed version of max, defined as:

smooth_max (x, y, c) = 0.5 (x + y + smooth_abs (x - y, c))

x First argument to max
y Second argument to max
c Cut-off point for the spline-approximated area (default: 0.001)
returns: The smoothed max function, defined as above

`smooth_min`

T smooth_min(T x, T y, T c = 0.001) is a smoothed version of min, defined as:

smooth_min (x, y, c) = 0.5 (x + y - smooth_abs (x - y, c))

x First argument to min
y Second argument to min
c Cut-off point for the spline-approximated area (default: 0.001)
returns: The smoothed min function, defined as above

Smoothed mathematics

smooth_abs

smooth_max

smooth_min

Related

`smooth_abs`

`smooth_max`

`smooth_min`