Functions stepUp(), stepDown() and stepAny() provide smooth, S-shaped step functions that are useful for "softening" abrupt transitions between two values.

Functions that return the first three derivatives of these step functions are also provided.

y=stepUp(x) for x=0:1 returns a smooth, S-shaped step function y(x), symmetric about the midpoint x=0.5, such that y(0)=0, y(1)=1, y'(0)=y''(0)=y'(1)=y''(1)=0, where the primes indicate differentiation with respect to x. No guarantees are made about the behavior of higher derivatives except to say that y''' does exist.

y=stepDown(x) for x=0:1 is a mirror image S curve that takes y down from 1 to 0 smoothly as x goes from 0 to 1. As with stepUp() y' and y'' are 0 at both ends, but here y(0)=1 and y(1)=0.

The stepAny() function is also available to make a step function that takes y smoothly from y0 to y1 as x goes from x0 to x1.

We also provide functions that calculate the first three derivatives of stepUp(), stepDown(), and stepAny(), namely: dstepUp(), d2stepUp(), d3stepUp(), and similarly dstepDown(), dstepAny() and so on. Note again that the third derivative is not guaranteed to be well behaved.

Costs of the current implementations of these inline functions are:

stepUp() 7 flops
dstepUp() 4 flops
d2stepUp() 6 flops
d3stepUp() 4 flops

The corresponding stepDown() methods cost one extra flop; the stepAny() methods cost 6 extra flops (provided you do some precalculation; see the stepAny() documentation).