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/C++

Standard library header <cmath>

This header was originally in the C standard library as <math.h>.

This header is part of the numeric library.

Types

float_t
(C++11)
most efficient floating-point type at least as wide as float
(typedef)
double_t
(C++11)
most efficient floating-point type at least as wide as double
(typedef)

Macros

(C++11)(C++11)
indicates the overflow value for float, double and long double respectively
(macro constant)
(C++11)
evaluates to positive infinity or the value guaranteed to overflow a float
(macro constant)
(C++11)
evaluates to a quiet NaN of type float
(macro constant)
(C++11)(C++11)(C++11)
defines the error handling mechanism used by the common mathematical functions
(macro constant)
Classification
(C++11)(C++11)(C++11)(C++11)(C++11)
indicates a floating-point category
(macro constant)

Functions

Basic operations
(C++11)(C++11)
absolute value of a floating point value (\(\small{|x|}\)|x|)
(function)
(C++11)(C++11)
remainder of the floating point division operation
(function)
(C++11)(C++11)(C++11)
signed remainder of the division operation
(function)
(C++11)(C++11)(C++11)
signed remainder as well as the three last bits of the division operation
(function)
(C++11)(C++11)(C++11)
fused multiply-add operation
(function)
(C++11)(C++11)(C++11)
larger of two floating-point values
(function)
(C++11)(C++11)(C++11)
smaller of two floating point values
(function)
(C++11)(C++11)(C++11)
positive difference of two floating point values (\({\small\max{(0, x-y)} }\)max(0, x-y))
(function)
(C++11)(C++11)(C++11)
not-a-number (NaN)
(function)
Linear interpolation
(C++20)
linear interpolation function
(function)
Exponential functions
(C++11)(C++11)
returns e raised to the given power (\({\small e^x}\)ex)
(function)
(C++11)(C++11)(C++11)
returns 2 raised to the given power (\({\small 2^x}\)2x)
(function)
(C++11)(C++11)(C++11)
returns e raised to the given power, minus one (\({\small e^x-1}\)ex-1)
(function)
(C++11)(C++11)
computes natural (base e) logarithm (\({\small \ln{x} }\)ln(x))
(function)
(C++11)(C++11)
computes common (base 10) logarithm (\({\small \log_{10}{x} }\)log10(x))
(function)
(C++11)(C++11)(C++11)
base 2 logarithm of the given number (\({\small \log_{2}{x} }\)log2(x))
(function)
(C++11)(C++11)(C++11)
natural logarithm (to base e) of 1 plus the given number (\({\small \ln{(1+x)} }\)ln(1+x))
(function)
Power functions
(C++11)(C++11)
raises a number to the given power (\(\small{x^y}\)xy)
(function)
(C++11)(C++11)
computes square root (\(\small{\sqrt{x} }\)x)
(function)
(C++11)(C++11)(C++11)
computes cube root (\(\small{\sqrt[3]{x} }\)3x)
(function)
(C++11)(C++11)(C++11)
computes square root of the sum of the squares of two or three (since C++17) given numbers (\(\scriptsize{\sqrt{x^2+y^2} }\)x2
+y2
), (\(\scriptsize{\sqrt{x^2+y^2+z^2} }\)x2
+y2
+z2
)
(since C++17)

(function)
Trigonometric functions
(C++11)(C++11)
computes sine (\({\small\sin{x} }\)sin(x))
(function)
(C++11)(C++11)
computes cosine (\({\small\cos{x} }\)cos(x))
(function)
(C++11)(C++11)
computes tangent (\({\small\tan{x} }\)tan(x))
(function)
(C++11)(C++11)
computes arc sine (\({\small\arcsin{x} }\)arcsin(x))
(function)
(C++11)(C++11)
computes arc cosine (\({\small\arccos{x} }\)arccos(x))
(function)
(C++11)(C++11)
computes arc tangent (\({\small\arctan{x} }\)arctan(x))
(function)
(C++11)(C++11)
arc tangent, using signs to determine quadrants
(function)
Hyperbolic functions
(C++11)(C++11)
computes hyperbolic sine (\({\small\sinh{x} }\)sinh(x))
(function)
(C++11)(C++11)
computes hyperbolic cosine (\({\small\cosh{x} }\)cosh(x))
(function)
(C++11)(C++11)
computes hyperbolic tangent (\({\small\tanh{x} }\)tanh(x))
(function)
(C++11)(C++11)(C++11)
computes the inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x))
(function)
(C++11)(C++11)(C++11)
computes the inverse hyperbolic cosine (\({\small\operatorname{arcosh}{x} }\)arcosh(x))
(function)
(C++11)(C++11)(C++11)
computes the inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x))
(function)
Error and gamma functions
(C++11)(C++11)(C++11)
error function
(function)
(C++11)(C++11)(C++11)
complementary error function
(function)
(C++11)(C++11)(C++11)
gamma function
(function)
(C++11)(C++11)(C++11)
natural logarithm of the gamma function
(function)
Nearest integer floating-point operations
(C++11)(C++11)
nearest integer not less than the given value
(function)
(C++11)(C++11)
nearest integer not greater than the given value
(function)
(C++11)(C++11)(C++11)
nearest integer not greater in magnitude than the given value
(function)
(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)
nearest integer, rounding away from zero in halfway cases
(function)
(C++11)(C++11)(C++11)
nearest integer using current rounding mode
(function)
(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)
nearest integer using current rounding mode with
exception if the result differs
(function)
Floating-point manipulation functions
(C++11)(C++11)
decomposes a number into significand and a power of 2
(function)
(C++11)(C++11)
multiplies a number by 2 raised to a power
(function)
(C++11)(C++11)
decomposes a number into integer and fractional parts
(function)
(C++11)(C++11)(C++11)(C++11)(C++11)(C++11)
multiplies a number by FLT_RADIX raised to a power
(function)
(C++11)(C++11)(C++11)
extracts exponent of the number
(function)
(C++11)(C++11)(C++11)
extracts exponent of the number
(function)
(C++11)(C++11) (C++11)(C++11)(C++11)(C++11)
next representable floating point value towards the given value
(function)
(C++11)(C++11)(C++11)
copies the sign of a floating point value
(function)
Classification and comparison
(C++11)
categorizes the given floating-point value
(function)
(C++11)
checks if the given number has finite value
(function)
(C++11)
checks if the given number is infinite
(function)
(C++11)
checks if the given number is NaN
(function)
(C++11)
checks if the given number is normal
(function)
(C++11)
checks if the given number is negative
(function)
(C++11)
checks if the first floating-point argument is greater than the second
(function)
(C++11)
checks if the first floating-point argument is greater or equal than the second
(function)
(C++11)
checks if the first floating-point argument is less than the second
(function)
(C++11)
checks if the first floating-point argument is less or equal than the second
(function)
(C++11)
checks if the first floating-point argument is less or greater than the second
(function)
(C++11)
checks if two floating-point values are unordered
(function)
Mathematical special functions
(C++17)(C++17)(C++17)
associated Laguerre polynomials
(function)
(C++17)(C++17)(C++17)
associated Legendre polynomials
(function)
(C++17)(C++17)(C++17)
beta function
(function)
(C++17)(C++17)(C++17)
(complete) elliptic integral of the first kind
(function)
(C++17)(C++17)(C++17)
(complete) elliptic integral of the second kind
(function)
(C++17)(C++17)(C++17)
(complete) elliptic integral of the third kind
(function)
(C++17)(C++17)(C++17)
regular modified cylindrical Bessel functions
(function)
(C++17)(C++17)(C++17)
cylindrical Bessel functions (of the first kind)
(function)
(C++17)(C++17)(C++17)
irregular modified cylindrical Bessel functions
(function)
(C++17)(C++17)(C++17)
cylindrical Neumann functions
(function)
(C++17)(C++17)(C++17)
(incomplete) elliptic integral of the first kind
(function)
(C++17)(C++17)(C++17)
(incomplete) elliptic integral of the second kind
(function)
(C++17)(C++17)(C++17)
(incomplete) elliptic integral of the third kind
(function)
(C++17)(C++17)(C++17)
exponential integral
(function)
(C++17)(C++17)(C++17)
Hermite polynomials
(function)
(C++17)(C++17)(C++17)
Legendre polynomials
(function)
(C++17)(C++17)(C++17)
Laguerre polynomials
(function)
(C++17)(C++17)(C++17)
Riemann zeta function
(function)
(C++17)(C++17)(C++17)
spherical Bessel functions (of the first kind)
(function)
(C++17)(C++17)(C++17)
spherical associated Legendre functions
(function)
(C++17)(C++17)(C++17)
spherical Neumann functions
(function)

Synopsis

For each function with at least one parameter of type /* floating-point-type */, an overload for each cv-unqualified floating-point type is provided where all uses of /* floating-point-type */ in the function signature are replaced with that floating-point type.

For each function with at least one parameter of type /* floating-point-type */ other than std::abs, additional overloads are provided to ensure that, if every argument corresponding to a /* floating-point-type */ parameter has arithmetic type, then every such argument is effectively cast to the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank among the types of all such arguments, where arguments of integer type are considered to have the same floating-point conversion rank as double. If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the provided overloads.

namespace std {
  using float_t = /* see description */;
  using double_t = /* see description */;
}
 
#define HUGE_VAL /* see description */
#define HUGE_VALF /* see description */
#define HUGE_VALL /* see description */
#define INFINITY /* see description */
#define NAN /* see description */
#define FP_INFINITE /* see description */
#define FP_NAN /* see description */
#define FP_NORMAL /* see description */
#define FP_SUBNORMAL /* see description */
#define FP_ZERO /* see description */
#define FP_FAST_FMA /* see description */
#define FP_FAST_FMAF /* see description */
#define FP_FAST_FMAL /* see description */
#define FP_ILOGB0 /* see description */
#define FP_ILOGBNAN /* see description */
#define MATH_ERRNO /* see description */
#define MATH_ERREXCEPT /* see description */
 
#define math_errhandling /* see description */
 
namespace std {
  /* floating-point-type */ acos(/* floating-point-type */ x);
  float acosf(float x);
  long double acosl(long double x);
 
  /* floating-point-type */ asin(/* floating-point-type */ x);
  float asinf(float x);
  long double asinl(long double x);
 
  /* floating-point-type */ atan(/* floating-point-type */ x);
  float atanf(float x);
  long double atanl(long double x);
 
  /* floating-point-type */ atan2(/* floating-point-type */ y,
                                  /* floating-point-type */ x);
  float atan2f(float y, float x);
  long double atan2l(long double y, long double x);
 
  /* floating-point-type */ cos(/* floating-point-type */e x);
  float cosf(float x);
  long double cosl(long double x);
 
  /* floating-point-type */ sin(/* floating-point-type */ x);
  float sinf(float x);
  long double sinl(long double x);
 
  /* floating-point-type */ tan(/* floating-point-type */ x);
  float tanf(float x);
  long double tanl(long double x);
 
  /* floating-point-type */ acosh(/* floating-point-type */ x);
  float acoshf(float x);
  long double acoshl(long double x);
 
  /* floating-point-type */ asinh(/* floating-point-type */ x);
  float asinhf(float x);
  long double asinhl(long double x);
 
  /* floating-point-type */ atanh(/* floating-point-type */ x);
  float atanhf(float x);
  long double atanhl(long double x);
 
  /* floating-point-type */ cosh(/* floating-point-type */ x);
  float coshf(float x);
  long double coshl(long double x);
 
  /* floating-point-type */ sinh(/* floating-point-type */ x);
  float sinhf(float x);
  long double sinhl(long double x);
 
  /* floating-point-type */ tanh(/* floating-point-type */ x);
  float tanhf(float x);
  long double tanhl(long double x);
 
  /* floating-point-type */ exp(/* floating-point-type */ x);
  float expf(float x);
  long double expl(long double x);
 
  /* floating-point-type */ exp2(/* floating-point-type */ x);
  float exp2f(float x);
  long double exp2l(long double x);
 
  /* floating-point-type */ expm1(/* floating-point-type */ x);
  float expm1f(float x);
  long double expm1l(long double x);
 
  constexpr /* floating-point-type */ frexp(/* floating-point-type */ value, int* exp);
  constexpr float frexpf(float value, int* exp);
  constexpr long double frexpl(long double value, int* exp);
 
  constexpr int ilogb(/* floating-point-type */ x);
  constexpr int ilogbf(float x);
  constexpr int ilogbl(long double x);
 
  constexpr /* floating-point-type */ ldexp(/* floating-point-type */ x, int exp);
  constexpr float ldexpf(float x, int exp);
  constexpr long double ldexpl(long double x, int exp);
 
  /* floating-point-type */ log(/* floating-point-type */ x);
  float logf(float x);
  long double logl(long double x);
 
  /* floating-point-type */ log10(/* floating-point-type */ x);
  float log10f(float x);
  long double log10l(long double x);
 
  /* floating-point-type */ log1p(/* floating-point-type */ x);
  float log1pf(float x);
  long double log1pl(long double x);
 
  /* floating-point-type */ log2(/* floating-point-type */ x);
  float log2f(float x);
  long double log2l(long double x);
 
  constexpr /* floating-point-type */ logb(/* floating-point-type */ x);
  constexpr float logbf(float x);
  constexpr long double logbl(long double x);
 
  constexpr /* floating-point-type */ modf(/* floating-point-type */ value,
                                           /* floating-point-type */* iptr);
  constexpr float modff(float value, float* iptr);
  constexpr long double modfl(long double value, long double* iptr);
 
  constexpr /* floating-point-type */ scalbn(/* floating-point-type */ x, int n);
  constexpr float scalbnf(float x, int n);
  constexpr long double scalbnl(long double x, int n);
 
  constexpr /* floating-point-type */ scalbln(/* floating-point-type */ x, long int n);
  constexpr float scalblnf(float x, long int n);
  constexpr long double scalblnl(long double x, long int n);
 
  /* floating-point-type */ cbrt(/* floating-point-type */ x);
  float cbrtf(float x);
  long double cbrtl(long double x);
 
  // absolute values
  constexpr int abs(int j);
  constexpr long int abs(long int j);
  constexpr long long int abs(long long int j);
  constexpr /* floating-point-type */ abs(/* floating-point-type */ j);
 
  constexpr /* floating-point-type */ fabs(/* floating-point-type */ x);
  constexpr float fabsf(float x);
  constexpr long double fabsl(long double x);
 
  /* floating-point-type */ hypot(/* floating-point-type */ x,
                                  /* floating-point-type */ y);
  float hypotf(float x, float y);
  long double hypotl(long double x, long double y);
 
  // three-dimensional hypotenuse
  float hypot(/* floating-point-type */ x,
              /* floating-point-type */ y,
              /* floating-point-type */ z);
 
  /* floating-point-type */ pow(/* floating-point-type */ x,
                                /* floating-point-type */ y);
  float powf(float x, float y);
  long double powl(long double x, long double y);
 
  /* floating-point-type */ sqrt(/* floating-point-type */ x);
  float sqrtf(float x);
  long double sqrtl(long double x);
 
  /* floating-point-type */ erf(/* floating-point-type */ x);
  float erff(float x);
  long double erfl(long double x);
 
  /* floating-point-type */ erfc(/* floating-point-type */ x);
  float erfcf(float x);
  long double erfcl(long double x);
 
  /* floating-point-type */ lgamma(/* floating-point-type */ x);
  float lgammaf(float x);
  long double lgammal(long double x);
 
  /* floating-point-type */ tgamma(/* floating-point-type */ x);
  float tgammaf(float x);
  long double tgammal(long double x);
 
  constexpr /* floating-point-type */ ceil(/* floating-point-type */ x);
  constexpr float ceilf(float x);
  constexpr long double ceill(long double x);
 
  constexpr /* floating-point-type */ floor(/* floating-point-type */ x);
  constexpr float floorf(float x);
  constexpr long double floorl(long double x);
 
  /* floating-point-type */ nearbyint(/* floating-point-type */ x);
  float nearbyintf(float x);
  long double nearbyintl(long double x);
 
  /* floating-point-type */ rint(/* floating-point-type */ x);
  float rintf(float x);
  long double rintl(long double x);
 
  long int lrint(/* floating-point-type */ x);
  long int lrintf(float x);
  long int lrintl(long double x);
 
  long long int llrint(/* floating-point-type */ x);
  long long int llrintf(float x);
  long long int llrintl(long double x);
 
  constexpr /* floating-point-type */ round(/* floating-point-type */ x);
  constexpr float roundf(float x);
  constexpr long double roundl(long double x);
 
  constexpr long int lround(/* floating-point-type */ x);
  constexpr long int lroundf(float x);
  constexpr long int lroundl(long double x);
 
  constexpr long long int llround(/* floating-point-type */ x);
  constexpr long long int llroundf(float x);
  constexpr long long int llroundl(long double x);
 
  constexpr /* floating-point-type */ trunc(/* floating-point-type */ x);
  constexpr float truncf(float x);
  constexpr long double truncl(long double x);
 
  constexpr /* floating-point-type */ fmod(/* floating-point-type */ x,
                                           /* floating-point-type */ y);
  constexpr float fmodf(float x, float y);
  constexpr long double fmodl(long double x, long double y);
 
  constexpr /* floating-point-type */ remainder(/* floating-point-type */ x,
                                                /* floating-point-type */ y);
  constexpr float remainderf(float x, float y);
  constexpr long double remainderl(long double x, long double y);
 
  constexpr /* floating-point-type */ remquo(/* floating-point-type */ x,
                                             /* floating-point-type */ y, int* quo);
  constexpr float remquof(float x, float y, int* quo);
  constexpr long double remquol(long double x, long double y, int* quo);
 
  constexpr /* floating-point-type */ copysign(/* floating-point-type */ x,
                                               /* floating-point-type */ y);
  constexpr float copysignf(float x, float y);
  constexpr long double copysignl(long double x, long double y);
 
  double nan(const char* tagp);
  float nanf(const char* tagp);
  long double nanl(const char* tagp);
 
  constexpr /* floating-point-type */ nextafter(/* floating-point-type */ x,
                                                /* floating-point-type */ y);
  constexpr float nextafterf(float x, float y);
  constexpr long double nextafterl(long double x, long double y);
 
  constexpr /* floating-point-type */ nexttoward(/* floating-point-type */ x,
                                                 long double y);
  constexpr float nexttowardf(float x, long double y);
  constexpr long double nexttowardl(long double x, long double y);
 
  constexpr /* floating-point-type */ fdim(/* floating-point-type */ x,
                                           /* floating-point-type */ y);
  constexpr float fdimf(float x, float y);
  constexpr long double fdiml(long double x, long double y);
 
  constexpr /* floating-point-type */ fmax(/* floating-point-type */ x,
                                           /* floating-point-type */ y);
  constexpr float fmaxf(float x, float y);
  constexpr long double fmaxl(long double x, long double y);
 
  constexpr /* floating-point-type */ fmin(/* floating-point-type */ x,
                                           /* floating-point-type */ y);
  constexpr float fminf(float x, float y);
  constexpr long double fminl(long double x, long double y);
 
  constexpr /* floating-point-type */ fma(/* floating-point-type */ x,
                                          /* floating-point-type */ y,
                                          /* floating-point-type */ z);
  constexpr float fmaf(float x, float y, float z);
  constexpr long double fmal(long double x, long double y, long double z);
 
  // linear interpolation
  constexpr /* floating-point-type */ lerp(/* floating-point-type */ a,
                                           /* floating-point-type */ b,
                                           /* floating-point-type */ t) noexcept;
 
  // classification / comparison functions
  constexpr int fpclassify(/* floating-point-type */ x);
 
  constexpr bool isfinite(/* floating-point-type */ x);
 
  constexpr bool isinf(/* floating-point-type */ x);
 
  constexpr bool isnan(/* floating-point-type */ x);
 
  constexpr bool isnormal(/* floating-point-type */ x);
 
  constexpr bool signbit(/* floating-point-type */ x);
 
  constexpr bool isgreater(/* floating-point-type */ x,
                           /* floating-point-type */ y);
 
  constexpr bool isgreaterequal(/* floating-point-type */ x,
                                /* floating-point-type */ y);
 
  constexpr bool isless(/* floating-point-type */ x,
                        /* floating-point-type */ y);
 
  constexpr bool islessequal(/* floating-point-type */ x,
                             /* floating-point-type */ y);
 
  constexpr bool islessgreater(/* floating-point-type */ x,
                               /* floating-point-type */ y);
 
  constexpr bool isunordered(/* floating-point-type */ x,
                             /* floating-point-type */ y);
 
  // mathematical special functions
 
  // associated Laguerre polynomials
  /* floating-point-type */ assoc_laguerre(unsigned n, unsigned m,
                                           /* floating-point-type */ x);
  float assoc_laguerref(unsigned n, unsigned m, float x);
  long double assoc_laguerrel(unsigned n, unsigned m, long double x);
 
  // associated Legendre functions
  /* floating-point-type */ assoc_legendre(unsigned l, unsigned m,
                                           /* floating-point-type */ x);
  float assoc_legendref(unsigned l, unsigned m, float x);
  long double assoc_legendrel(unsigned l, unsigned m, long double x);
 
  // beta function
  /* floating-point-type */ beta(/* floating-point-type */ x,
                                 /* floating-point-type */ y);
  float betaf(float x, float y);
  long double betal(long double x, long double y);
 
  // complete elliptic integral of the first kind
  /* floating-point-type */ comp_ellint_1(/* floating-point-type */ k);
  float comp_ellint_1f(float k);
  long double comp_ellint_1l(long double k);
 
  // complete elliptic integral of the second kind
  /* floating-point-type */ comp_ellint_2(/* floating-point-type */ k);
  float comp_ellint_2f(float k);
  long double comp_ellint_2l(long double k);
 
  // complete elliptic integral of the third kind
  /* floating-point-type */ comp_ellint_3(/* floating-point-type */ k,
                                          /* floating-point-type */ nu);
  float comp_ellint_3f(float k, float nu);
  long double comp_ellint_3l(long double k, long double nu);
 
  // regular modified cylindrical Bessel functions
  /* floating-point-type */ cyl_bessel_i(/* floating-point-type */ nu,
                                         /* floating-point-type */ x);
  float cyl_bessel_if(float nu, float x);
  long double cyl_bessel_il(long double nu, long double x);
 
  // cylindrical Bessel functions of the first kind
  /* floating-point-type */ cyl_bessel_j(/* floating-point-type */ nu,
                                         /* floating-point-type */ x);
  float cyl_bessel_jf(float nu, float x);
  long double cyl_bessel_jl(long double nu, long double x);
 
  // irregular modified cylindrical Bessel functions
  /* floating-point-type */ cyl_bessel_k(/* floating-point-type */ nu,
                                         /* floating-point-type */ x);
  float cyl_bessel_kf(float nu, float x);
  long double cyl_bessel_kl(long double nu, long double x);
 
  // cylindrical Neumann functions;
  // cylindrical Bessel functions of the second kind
  /* floating-point-type */ cyl_neumann(/* floating-point-type */ nu,
                                        /* floating-point-type */ x);
  float cyl_neumannf(float nu, float x);
  long double cyl_neumannl(long double nu, long double x);
 
  // incomplete elliptic integral of the first kind
  /* floating-point-type */ ellint_1(/* floating-point-type */ k,
                                     /* floating-point-type */ phi);
  float ellint_1f(float k, float phi);
  long double ellint_1l(long double k, long double phi);
 
  // incomplete elliptic integral of the second kind
  /* floating-point-type */ ellint_2(/* floating-point-type */ k,
                                     /* floating-point-type */ phi);
  float ellint_2f(float k, float phi);
  long double ellint_2l(long double k, long double phi);
 
  // incomplete elliptic integral of the third kind
  /* floating-point-type */ ellint_3(/* floating-point-type */ k,
                                     /* floating-point-type */ nu,
                                     /* floating-point-type */ phi);
  float ellint_3f(float k, float nu, float phi);
  long double ellint_3l(long double k, long double nu, long double phi);
 
  // exponential integral
  /* floating-point-type */ expint(/* floating-point-type */ x);
  float expintf(float x);
  long double expintl(long double x);
 
  // Hermite polynomials
  /* floating-point-type */ hermite(unsigned n, /* floating-point-type */ x);
  float hermitef(unsigned n, float x);
  long double hermitel(unsigned n, long double x);
 
  // Laguerre polynomials
  /* floating-point-type */ laguerre(unsigned n, /* floating-point-type */ x);
  float laguerref(unsigned n, float x);
  long double laguerrel(unsigned n, long double x);
 
  // Legendre polynomials
  /* floating-point-type */ legendre(unsigned l, /* floating-point-type */ x);
  float legendref(unsigned l, float x);
  long double legendrel(unsigned l, long double x);
 
  // Riemann zeta function
  /* floating-point-type */ riemann_zeta(/* floating-point-type */ x);
  float riemann_zetaf(float x);
  long double riemann_zetal(long double x);
 
  // spherical Bessel functions of the first kind
  /* floating-point-type */ sph_bessel(unsigned n, /* floating-point-type */ x);
  float sph_besself(unsigned n, float x);
  long double sph_bessell(unsigned n, long double x);
 
  // spherical associated Legendre functions
  /* floating-point-type */ sph_legendre(unsigned l, unsigned m,
                                         /* floating-point-type */ theta);
  float sph_legendref(unsigned l, unsigned m, float theta);
  long double  sph_legendrel(unsigned l, unsigned m, long double theta);
 
  // spherical Neumann functions;
  // spherical Bessel functions of the second kind
  /* floating-point-type */ sph_neumann(unsigned n, /* floating-point-type */ x);
  float sph_neumannf(unsigned n, float x);
  long double sph_neumannl(unsigned n, long double x);
}

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