erf, erff, erfl

 

C

Compiler support

Language

Headers

Type support

Program utilities

Variadic function support

Error handling

Dynamic memory management

Strings library

Algorithms

Numerics

Date and time utilities

Input/output support

Localization support

Concurrency support(C11)

Technical Specifications

Symbol index

 

Numerics

Common mathematical functions

Floating-point environment(C99)

Pseudo-random number generation

Complex number arithmetic(C99)

Type-generic math(C99)

Bit manipulation(C23)

Checked integer arithmetic(C23)

 

Common mathematical functions

Functions Basic operations abslabsllabsimaxabs (C99)(C99) fabs divldivlldivimaxdiv (C99)(C99) fmod remainder (C99) remquo (C99) fma (C99) fdim (C99) nannanfnanlnandN (C99)(C99)(C99)(C23) Maximum/minimum operations fmax (C99) fmin (C99) fmaximum (C23) fminimum (C23) fmaximum_mag (C23)  fmaximum_num (C23) fminimum_mag (C23) fminimum_num (C23) fmaximum_mag_num (C23) fminimum_mag_num (C23)   Exponential functions exp exp10 (C23) exp2 (C99) expm1 (C99) exp10m1 (C23) exp2m1 (C23) log log10 log2 (C99) log1plogp1 (C99)(C23) log10p1 (C23) log2p1 (C23) Power functions sqrt cbrt (C99) rootn (C23) rsqrt (C23) hypot (C99) compound (C23) pow pown (C23) powr (C23) Trigonometric and hyperbolic functions sin cos tan asin acos atan atan2 sinpi (C23) cospi (C23) tanpi (C23) asinpi (C23) acospi (C23) atanpi (C23) atan2pi (C23) sinh cosh tanh asinh (C99) acosh (C99) atanh (C99)

Nearest integer floating-point ceil floor roundlroundllround (C99)(C99)(C99) roundeven (C23) trunc (C99) nearbyint (C99) rintlrintllrint (C99)(C99)(C99) fromfpfromfpxufromfpufromfpx (C23)(C23)(C23)(C23) Floating-point manipulation ldexp frexp scalbnscalbln (C99)(C99) ilogbllogb (C99)(C23) logb (C99) modf nextafternexttoward (C99)(C99) nextupnextdown (C23)(C23) copysign (C99) canonicalize (C23) Narrowing operations fadd (C23) fsub (C23) fmul (C23) fdiv (C23) ffma (C23) fsqrt (C23) Quantum and quantum exponent quantizedN (C23) quantumdN (C23) samequantumdN (C23) llquantexpdN (C23) Decimal re-encoding functions encodedecdN (C23) decodedecdN (C23) encodebindN (C23) decodebindN (C23) Total order and payload functions totalorder (C23) getpayload (C23) setpayload (C23) setpayloadsig (C23) Classification fpclassify (C99) iscanonical (C23)  isfinite (C99) isinf (C99) isnan (C99) isnormal (C99) signbit (C99) issubnormal (C23) iszero (C23) isgreater (C99) isgreaterequal (C99)   isless (C99) islessequal (C99) islessgreater (C99) isunordered (C99) issignaling (C23) iseqsig (C23)

Error and gamma functions erf (C99) erfc (C99) lgamma (C99) tgamma (C99) Types div_tldiv_tlldiv_timaxdiv_t (C99)(C99) float_tdouble_t (C99)(C99) _Decimal32_t_Decimal64_t (C23)(C23) Macro constants Special floating-point values HUGE_VALHUGE_VALFHUGE_VALLHUGE_VALDN (C99)(C99)(C23) INFINITYDEC_INFINITY (C99)(C23) NANDEC_NAN (C99)(C23) Arguments and return values FP_ILOGB0FP_ILOGBNAN (C99)(C99) FP_NORMALFP_SUBNORMALFP_ZEROFP_INFINITEFP_NAN (C99)(C99)(C99)(C99)(C99) FP_LLOGB0FP_LLOGBNAN (C23)(C23) FP_INT_UPWARDFP_INT_DOWNWARDFP_INT_TOWARDZEROFP_INT_TONEARESTFROMZEROFP_INT_TONEAREST (C23)(C23)(C23)(C23)(C23) Error handling MATH_ERRNOMATH_ERRNOEXCEPT (C99)(C99)  math_errhandling (C99) Fast operation indicators FP_FAST_FMAFFP_FAST_FMA (C99)(C99) FP_FAST_FADDFP_FAST_FADDLFP_FAST_DADDLFP_FAST_DMADDDN (C23)(C23)(C23)(C23) FP_FAST_FMULFP_FAST_FMULLFP_FAST_DMULLFP_FAST_DMMULDN (C23)(C23)(C23)(C23) FP_FAST_FFMAFP_FAST_FFMALFP_FAST_DFMALFP_FAST_DMFMADN (C23)(C23)(C23)(C23) FP_FAST_FMALFP_FAST_FMADN (C99)(C23) FP_FAST_FSUBFP_FAST_FSUBLFP_FAST_DSUBLFP_FAST_DMSUBDN (C23)(C23)(C23)(C23) FP_FAST_FDIVFP_FAST_FDIVLFP_FAST_DDIVLFP_FAST_DMDIVDN (C23)(C23)(C23)(C23) FP_FAST_FSQRTFP_FAST_FSQRTLFP_FAST_DSQRTLFP_FAST_DMSQRTDN (C23)(C23)(C23)(C23)

 

Defined in header <math.h>
float       erff(float arg );	(1)	(since C99)
double      erf(double arg );	(2)	(since C99)
longdouble erfl(longdouble arg );	(3)	(since C99)
Defined in header <tgmath.h>
#define erf( arg )	(4)	(since C99)

1-3) Computes the error function of arg.

4) Type-generic macro: If arg has type longdouble, erfl is called. Otherwise, if arg has integer type or the type double, erf is called. Otherwise, erff is called.

Parameters

arg

-

floating-point value

Return value

If no errors occur, value of the error function of arg, that is \(\frac{2}{\sqrt{\pi} }\int_{0}^{arg}{e^{-{t^2} }\mathsf{d}t}\)

2

√π

∫arg
0e^-t2
dt, is returned. If a range error occurs due to underflow, the correct result (after rounding), that is \(\frac{2\cdot arg}{\sqrt{\pi} }\)

2*arg

√π

, is returned.

Error handling

Errors are reported as specified in math_errhandling.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

• If the argument is ±0, ±0 is returned
• If the argument is ±∞, ±1 is returned
• If the argument is NaN, NaN is returned

Notes

Underflow is guaranteed if |arg|<DBL_MIN*(sqrt(π)/2).

\(\operatorname{erf}(\frac{x}{\sigma \sqrt{2} })\)erf(

x

σ√2

) is the probability that a measurement whose errors are subject to a normal distribution with standard deviation \(\sigma\)σ is less than \(x\)x away from the mean value.

Example

Run this code

#include <math.h>#include <stdio.h>   double phi(double x1, double x2){return(erf(x2 /sqrt(2))- erf(x1 /sqrt(2)))/2;}   int main(void){puts("normal variate probabilities:");for(int n =-4; n <4;++n)printf("[%2d:%2d]: %5.2f%%\n", n, n +1, 100* phi(n, n +1));   puts("special values:");printf("erf(-0) = %f\n", erf(-0.0));printf("erf(Inf) = %f\n", erf(INFINITY));}

Output:

normal variate probabilities: [-4:-3]: 0.13% [-3:-2]: 2.14% [-2:-1]: 13.59% [-1: 0]: 34.13% [ 0: 1]: 34.13% [ 1: 2]: 13.59% [ 2: 3]: 2.14% [ 3: 4]: 0.13% special values: erf(-0) = -0.000000 erf(Inf) = 1.000000

References

• C11 standard (ISO/IEC 9899:2011):

• 7.12.8.1 The erf functions (p: 249)

• 7.25 Type-generic math <tgmath.h> (p: 373-375)

• F.10.5.1 The erf functions (p: 525)

• C99 standard (ISO/IEC 9899:1999):

• 7.12.8.1 The erf functions (p: 230)

• 7.22 Type-generic math <tgmath.h> (p: 335-337)

• F.9.5.1 The erf functions (p: 462)

See also

erfcerfcferfcl (C99)(C99)(C99)	computes complementary error function (function)
C++ documentation for erf

External links

Weisstein, Eric W. "Erf." From MathWorld — A Wolfram Web Resource.