Cephes Mathematical Functions Library

This project serves solely to generate reference outputs from the original/scipy Cephes Math Library.

Consequently, contributions addressing precision or error corrections may not be accepted. We recommend submitting such improvements to the scipy/xsf project instead.

Tip

For general applications, we recommend scipy/xsf. It provides a modernized (C++17), header-only refactoring of classic special function implementations, including AMOS, SPECFUN, and Cephes.

Build and Test

# Generate Coverage Report
cmake -S . -B build -DCMAKE_BUILD_TYPE=Coverage -DBUILD_TESTS=ON
cmake --build build --parallel 8 && ctest --test-dir build -j8 && cmake --build build --target coverage_html
ctest --test-dir build --rerun-failed --output-on-failure

# Release Build
cmake -B build -DCMAKE_BUILD_TYPE=Release -DCMAKE_INSTALL_PREFIX=./build/install_test
cmake --build build --parallel 8
cmake --install build

Features

• Bug? Features!

Docs

Note

• This project currently focuses on special functions with double precision.
• Only functions listed in public headers are treated as public API.

cephes/*.h Headers

By default, Cephes does not signal any error (domain, singularity, overflow, underflow, precision). It is as non-intrusive as possible and tries to return a value which is hopefully usable: it might be NaN, it might be inf.

List of All Cephes functions

Exponential integral

• ei, Exponential Integral
• expn, Exponential integral En
• sici, Sine and cosine integrals
• shichi, Hyperbolic sine and cosine integrals

Gamma Functions

• fac, Factorial function
• gamma, Gamma function
• lgam, Natural logarithm of gamma function
• igam, Incomplete gamma integral
• igamc, Complemented incomplete gamma integral
• igami, Inverse of complemented incomplete gamma integral
• psi, Psi (digamma) function
• rgamma, Reciprocal gamma function

Beta Functions

• beta, Beta function
• incbet, Incomplete beta integral
• incbi, Inverse of incomplete beta integral

Error Functions

• erf, Error function
• erfc, Complementary error function
• dawsn, Dawson's Integral
• fresnl, Fresnel integral

Bessel functions

Airy Functions

• airy, Airy functions

Bessel function

• j0, Bessel function of order zero
• j1, Bessel function of order one
• jn, Bessel function of integer order
• jv, Bessel function of noninteger order

Modified Bessel functions - second kind

• y0, Bessel function of the second kind, order zero
• y1, Bessel function of the second kind, order one
• yn, Bessel function of second kind of integer order
• yv, Bessel function of noninteger order

Modified Bessel functions

• i0, Modified Bessel function of order zero
• i0e, Exponentially scaled modified Bessel function of order zero
• i1, Modified Bessel function of order one
• i1e, Exponentially scaled modified Bessel function of order one
• iv, Modified Bessel function of noninteger order

Modified Bessel functions - third kind

• k0, Modified Bessel function, third kind, order zero
• k0e, Modified Bessel function, third kind, order zero, exponentially scaled
• k1, Modified Bessel function, third kind, order one
• k1e, Modified Bessel function, third kind, order one, exponentially scaled
• kn, Modified Bessel function, third kind, integer order

Hypergeometric Functions

• hyperg, Confluent hypergeometric function
• hyp2f1, Gauss hypergeometric function

Elliptic Functions

• ellpk, Complete elliptic integral of the first kind
• ellik, Incomplete elliptic integral of the first kind
• ellpe, Complete elliptic integral of the second kind
• ellie, Incomplete elliptic integral of the second kind
• ellpj, Jacobian elliptic functions

Probability Functions

Binomial distribution

• bdtr, Binomial distribution
• bdtrc, Complemented binomial distribution
• bdtri, Inverse binomial distribution

Beta distribution

• btdtr, Beta distribution

Chi-square distribution

• chdtr, Chi-square distribution
• chdtrc, Complemented Chi-square distribution
• chdtri, Inverse of complemented Chi-square distribution

F distribution

• fdtr, F distribution
• fdtrc, Complemented F distribution
• fdtri, Inverse of complemented F distribution

Gamma distribution

• gdtr, Gamma distribution function
• gdtrc, Complemented gamma distribution function

Negative binomial distribution

• nbdtr, Negative binomial distribution
• nbdtrc, Complemented negative binomial distribution
• nbdtri, Functional inverse of negative binomial distribution

Normal distribution

• ndtr, Normal distribution function
• ndtri, Inverse of normal distribution function

Poisson distribution

• pdtr, Poisson distribution function
• pdtrc, Complemented Poisson distribution function
• pdtri, Inverse of Poisson distribution function

Student's t distribution

• stdtr, Student's t distribution
• stdtri, Functional inverse of Student's t distribution

Miscellaneous functions

• spence, Dilogarithm
• zeta, Zeta function of two arguments
• zetac, Riemann zeta function of two arguments
• struve, Struve function
• polylog, Polylogarithms
• plancki, Integral of Planck's black body radiation formula

License

This project is licensed under a BSD-3-Clause style license. See LICENSE for details.

Cephes mathematical library, developed by Stephen L. Moshier.

For historical context, see the email communication with the original author: License request email.