Formulas 5. Mathematical functions
abs (x)
absolute value
round (x)
nearest integer; round (1.5) = 2
floor (x)
round down: highest integer value not greater than x
ceiling (x)
round up: lowest integer value not less than x
sqrt (x)
square root: √x, x ≥ 0
min (x, ...)
the minimum of a series of numbers, e.g. min (7.2, -5, 3) = -5
max (x, ...)
the maximum of a series of numbers, e.g. max (7.2, -5, 3) = 7.2
imin (x, ...)
the location of the minimum, e.g. imin (7.2, -5, 3) = 2
imax (x, ...)
the location of the maximum, e.g. imax (7.2, -5, 3) = 1
sin (x)
sine
cos (x)
cosine
tan (x)
tangent
arcsin (x)
arcsine, -1 ≤ x ≤ 1
arccos (x)
arccosine, -1 ≤ x ≤ 1
arctan (x)
arctangent
arctan2 (y, x)
argument angle
sinc (x)
sinus cardinalis: sin (x) / x
sincpi (x)
sincπ: sin (πx) / (πx)
exp (x)
exponentiation: ex; same as e^x
ln (x)
natural logarithm, base e
log10 (x)
logarithm, base 10
log2 (x)
logarithm, base 2
sinh (x)
hyperbolic sine: (ex - e-x) / 2
cosh (x)
hyperbolic cosine: (ex + e-x) / 2
tanh (x)
hyperbolic tangent: sinh (x) / cosh (x)
arcsinh (x)
inverse hyperbolic sine: ln (x + √(1+x2))
arccosh (x)
inverse hyperbolic cosine: ln (x + √(x2–1))
arctanh (x)
inverse hyperbolic tangent
sigmoid (x)
R → (0,1): 1 / (1 + ex) or 1 – 1 / (1 + ex)
invSigmoid (x)
(0,1) → R: ln (x / (1 – x))
erf (x)
the error function: 2/√π 0x exp(-t2) dt
erfc (x)
the complement of the error function: 1 - erf (x)
randomUniform (min, max)
uniform random real number between min (inclusive) and max (exclusive)
randomInteger (min, max)
uniform random integer number between min and max (inclusive)
randomGauss (μ, σ)
Gaussian random real number with mean μ and standard deviation σ
randomPoisson (mean)
Poisson random real number
lnGamma (x)
logarithm of the Γ function
gaussP (z)
the area under the Gaussian distribution between –∞ and z
gaussQ (z)
the area under the Gaussian distribution between z and +∞: the one-tailed "statistical significance p" of a value that is z standard deviations away from the mean of a Gaussian distribution
invGaussQ (q)
the value of z for which `gaussQ` (z) = q
chiSquareP (chiSquare, df)
the area under the χ2 distribution between 0 and chiSquare, for df degrees of freedom
chiSquareQ (chiSquare, df)
the area under the χ2 distribution between chiSquare and +∞, for df degrees of freedom: the "statistical significance p" of the χ2 difference between two distributions in df+1 dimensions
invChiSquareQ (q, df)
the value of χ2 for which `chiSquareQ` (χ2, df) = q
studentP (t, df)
the area under the student T-distribution from -∞ to t
studentQ (t, df)
the area under the student T-distribution from t to +∞
invStudentQ (q, df)
the value of t for which `studentQ` (t, df) = q
fisherP (f, df1, df2)
the area under Fisher's F-distribution from 0 to f
fisherQ (f, df1, df2)
the area under Fisher's F-distribution from f to +∞
invFisherQ (q, df1, df2)
the value of f for which `fisherQ` (f, df1, df2) = q
binomialP (p, k, n)
the probability that in n experiments, an event with probability p will occur at most k times
binomialQ (p, k, n)
the probability that in n experiments, an event with probability p will occur at least k times; equals 1 - `binomialP` (p, k - 1, n)
invBinomialP (P, k, n)
the value of p for which `binomialP` (p, k, n) = P
invBinomialQ (Q, k, n)
the value of p for which `binomialQ` (p, k, n) = Q
hertzToBark (x)
from acoustic frequency to Bark-rate (perceptual spectral frequency; place on basilar membrane): 7 ln (x/650 + √(1 + (x/650)2))
barkToHertz (x)
650 sinh (x / 7)
hertzToMel (x)
from acoustic frequency to perceptual pitch: 550 ln (1 + x / 550)
melToHertz (x)
550 (exp (x / 550) - 1)
hertzToSemitones (x)
from acoustic frequency to a logarithmic musical scale, relative to 100 Hz: 12 ln (x / 100) / ln 2
semitonesToHertz (x)
100 exp (x ln 2 / 12)
erb (f)
the perceptual equivalent rectangular bandwidth (ERB) in hertz, for a specified acoustic frequency (also in hertz): 6.23·10-6 f2 + 0.09339 f + 28.52
hertzToErb (x)
from acoustic frequency to ERB-rate: 11.17 ln ((x + 312) / (x + 14680)) + 43
erbToHertz (x)
(14680 d - 312) / (1 - d) where d = exp ((x - 43) / 11.17)
phonToDifferenceLimens (x)
from perceptual loudness (intensity sensation) level in phon, to the number of intensity difference limens above threshold: 30 · ((61/60) x – 1).
differenceLimensToPhon (x)
the inverse of the previous: ln (1 + x / 30) / ln (61 / 60).
beta (x, y)
besselI (n, x)
besselK (n, x)

For functions with arrays, see Scripting 5.7. Vectors and matrices.