smooth-math.md

Smoothed Mathematical Functions

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

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

$$ \text{smooth_abs}(x,c) = \left{\begin{array}{lll} |x| & & \text{if }x < c\text{ or } x > c\[5pt] x^2\left(\frac{2}{c}-\frac{1}{c^2} x\right) & & \text{if }0 \leq x \leq c\[5pt] x^2\left(\frac{2}{c}+\frac{1}{c^2} x\right) & & \text{if }-c \leq x < 0 \end{array}\right. $$

• 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

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

$$ \text{smooth_max}(x,y,c) = 0.5\left(x+y+\text{smooth_abs}(x-y,c)\right) $$

• 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

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

$$ \text{smooth_min}(x,y,c) = 0.5\left(x+y-\text{smooth_abs}(x-y,c)\right) $$

• 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