Module fwdk 

Functions

abs

Absolute value. f' = sign(x), f'' = 0 (away from 0). At x = 0 the derivative is not defined; every surface takes the sub-gradient f'(0) = +1 (the x >= 0 branch), matching the winning-branch convention max/min use for payoffs.

acos

Inverse cosine. f' = -1/√(1-x²), f'' = x·f'³.

acosh

Inverse hyperbolic cosine. f' = 1/√(x²-1), f'' = -x·f'³.

asin

Inverse sine. f' = 1/√(1-x²), f'' = x·f'³.

asinh

Inverse hyperbolic sine. f' = 1/√(x²+1), f'' = -x·f'³.

atan

Inverse tangent. f' = 1/(1+x²), f'' = -2x·f'².

atan2

atanh

Inverse hyperbolic tangent. f' = 1/(1-x²), f'' = 2x·f'².

cbrt

Cube root. f' = 1/(3·f²), f'' = -2/(9·x·f²).

cos

Cosine.

cosh

Hyperbolic cosine.

erf

Gaussian error function. f' = (2/√π)·e^{-x²}, f'' = -2x·f'.

erfc

Complementary error function 1 - erf(x). f' = -(2/√π)·e^{-x²}.

exp

Exponential. f = f' = f''.

exp2

Base-2 exponential. f' = f·ln 2, f'' = f·(ln 2)².

exp_m1

exp(x) - 1. f' = f'' = exp(x) = f + 1.

hypot

inv_norm_cdf

Inverse standard normal CDF Φ⁻¹(p). f' = 1/φ(f), f'' = f·f'².

ln

Natural logarithm. f' = 1/x, f'' = -1/x².

ln_1p

ln(1 + x). f' = 1/(1+x), f'' = -f'².

log2

Base-2 logarithm. f' = 1/(x·ln 2), f'' = -f'/x.

log10

Base-10 logarithm. f' = 1/(x·ln 10), f'' = -f'/x.

max

min

norm_cdf

Standard normal CDF Φ(x). f' = φ(x), f'' = -x·φ(x).

norm_pdf

Standard normal density φ(x). f' = -x·φ(x), f'' = (x² − 1)·φ(x).

pow

powf

powi

sin

Sine.

sinh

Hyperbolic sine.

sqrt

Square root. f' = 1/(2√x), f'' = -1/(4·x·√x).

tan

Tangent. f' = 1 + tan², f'' = 2·tan·f'.

tanh

Hyperbolic tangent. f' = 1 - tanh², f'' = -2·tanh·f'.