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Cephes Mathematical Library

A common lisp CFFI wrapper for the SciPy version of Cephes special functions.

For many years Cephes was considered the gold standard in cross-platform mathematical libraries, superior to the libm distributions of the time (early 1990's). Since then C99 and later ANSI C specifications have closed the gap, but there are still certain special and statistical functions available in Cephes that are not in the standard C libraries.

The SciPy version differs from standard Cephes in that it has some additional functions, improvements in accuracy, and is better documented. Only double-float versions are provided.


The ASDF file will automatically build the shared library as part of the load operation. If you need to build on a system other than MS Windows or UNIX, you will need to modify the make file to account for the linker command to create a shared library.

As delivered, the Makefile is set-up to work on MS Windows or UNIX, and you can build it manually like so:

cd scipy-cephes && make

If you build this on another platform, please drop a note into a Cephes repository issue with the build instructions so we can update the Makefile and system definition.

The init.lisp file, where CFFI loads the library, should work out of the box if libmd on the path somewhere, regardless of platform. If in doubt, place a copy in the same directory as this README.


If you know your way around special functions, the table below should suffice to get started. For a more complete description, see the doc strings in cephes.lisp. Finally, the C source code itself, referenced in cephes.lisp, is thorough and complete from a mathematical perspective.

You can also use the scipy.special online documentation.


There is no overlap between the wrapped Cephes functions and the Common Lisp numerical tower. All functions are in the cephes package.

Exported functions are:

function description
airy Airy function
bdtr Binomial distribution
bdtrc Complement of binomial distribution
bdtri Inverse binomial distribution
besselpoly Weighted integral of the Bessel function of the first kind
beta Beta function
lbeta Natural log of beta
btdtr incomplete beta integral
cbrt cube root
chdtr Chi-square distribution
chdtrc Complemented Chi-square distribution
chdtri Inverse of complemented Chi-square distribution
dawsn Dawson's Integral
ellik Incomplete elliptic integral of the first kind
ellie Incomplete elliptic integral of the second kind
ellpk Complete elliptic integral of the first kind
ellpe Complete elliptic integral of the second kind
jacobian-elliptic jacobian Elliptic Functions
exp2 Base 2 exponential function
exp10 Base 10 exponential function (Common antilogarithm)
expn Exponential integral
fdtr F distribution
fdtrc Complemented F distribution
fdtri Inverse of F distribution
fresnl Fresnel integral
gamma Gamma function
log-gamma Natural logarithm of Gamma function
gdtr Gamma distribution function
gdtrc Complemented Gamma distribution function
gdtri Inverse Gamma distribution function (?) - not documented in src
hyp2f1 Gauss hypergeometric function
hyperg Confluent hypergeometric function
i0 Modified Bessel function of order zero
i0e Modified Bessel function of order zero, exponentially scaled
i1 Modified Bessel function of order one
i1e Modified Bessel function of order one, exponentially scaled
igam Regularized lower incomplete gamma function
igamc Regularized upper incomplete gamma function
igami Inverse of the lower incomplete gamma function
igamci Inverse of the upper incomplete gamma function
incbet Incomplete beta integral
incbi Inverse of incomplete beta integral
iv Modified Bessel function of noninteger order
j0 Bessel function of order zero
y0 Bessel function of the second kind, order zero
j1 Bessel function of order one
y1 Bessel function of second kind of order one
jv Bessel function of noninteger order
k0 Modified Bessel function, third kind, order zero
k0e Modified Bessel function, third kind, order zero, exponentially scaled
k1 Modified Bessel function of the third kind of order one
k1e Modified Bessel function of the third kind of order one, exponentially scaled
kn Modified Bessel function, third kind, integer order
nbdtr Negative binomial distribution
nbdtrc Complemented negative binomial distribution
nbdtri Inverse complemented negative binomial distribution
ndtr Normal distribution function
log-ndtr Log of the normal distribution function
erf Error function
erfc Complementary error function
erfinv Inverse of the error function
erfcinv Inverse of the complementary error function
ndtri Inverse of Normal distribution function
pdtr Poisson distribution
pdtrc Complemented poisson distribution
pdtri Inverse Poisson distribution
poch Pochhammer symbol (a)_m = gamma(a + m) / gamma(a)
psi Psi (digamma) function
rgamma Reciprocal Gamma function
shichi Hyperbolic sine and cosine integrals
sici Sine and cosine integrals
sindg Circular sine of angle in degrees
cosdg Circular cosine of angle in degrees
sinpi Compute sin(pi * x)
cospi Compute cos(pi * x)
spence Dilogarithm
stdtr Student's t distribution
stdtri Functional inverse of Student's t distribution
yv Bessel function of noninteger order
tandg Circular tangent of angle in degrees
cotdg Circular cotangent of argument in degrees
log1p log(1 + x)
log1pmx log(1 + x) - x
expm1 exp(x) - 1
cosm1 cos(x) - 1
lgam1p lgam(x + 1)
yn Bessel function of second kind of integer order
zeta Riemann zeta function of two arguments
zetac Riemann zeta function
owens-t Owen's T-Function


When contributing to this repository, please first discuss major changes to the existing code you wish to make via a github issue. Minor changes and major additions are welcome. Please write good commit messages.


CEPHES.CL is available under the Microsoft Public License.