std::erf,st...f,std::erfl(3)

💖 NAME 💖

okay so bestie, this is the std::erf, std::erff, std::erfl page from the C++ Standard Library manual (std::erf,st...f,std::erfl(3) — yes the title got a lil truncated, we love that for us). These three are the error function crew. that's the whole vibe. iykyk.

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✨ SYNOPSIS ✨

so first off, everything lives in this header:

#include <cmath>

and here's the whole cast of characters 💅

float       erf ( float num );

double      erf ( double num );                            (until C++23)

long double erf ( long double num );

/* floating-point-type */                                  (since C++23)
            erf ( /* floating-point-type */ num );         (constexpr since C++26)

then the f/l girlies:

float       erff( float num );                     (1) (2) (since C++11)
                                                                     (constexpr since C++26)

long double erfl( long double num );                   (3) (since C++11)
                                                                     (constexpr since C++26)

and the bonus integer overloads — "Additional overloads (since C++11)" — also from <cmath>:

template< class Integer >                              (A) (constexpr since C++26)
double      erf ( Integer num );

okay so what do they do:

1. – 3. they compute the error function of num 🧮

and since C++23 the library provides overloads of std::erf for all cv-unqualified floating-point types as the type of the parameter. slay, no more casting.

A) for the integer girlies: additional overloads are provided for all integer types (since C++11), and they're treated as double. periodt.

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🎀 PARAMETERS 🎀

• num — a floating-point or integer value 🍓

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💅 RETURN VALUE 💅

if nothing goes wrong, you get the value of the error function of num, which is literally this:

  2      num
------ ∫  e^(-t²) dt
 √π     0

that's 2/(√π) · ∫₀^num e^(-t²) dt if you're typing it in a group chat 💖

but if a range error happens due to underflow, then the correct result (after rounding) is returned instead, and that one is:

2*num
------
  √π

so like 2*num/√π. no cap.

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🫶 ERROR HANDLING 🫶

errors get reported exactly as specified in math_errhandling. 📋

and if the implementation supports IEEE floating-point arithmetic (IEC 60559), then:

• if the argument is ±0, you get ±0 back. 🫧
• if the argument is ±∞, you get ±1 back. 🔥
• if the argument is NaN, you get NaN back. 💀 (as it should be)

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🦋 NOTES 🦋

underflow is guaranteed if |num| < DBL_MIN * (std::sqrt(π) / 2). that's a promise, not a maybe. 😭

also this little beauty:

erf( x / (σ√2) )

is the probability that a measurement whose errors are subject to a normal distribution with standard deviation σ is less than x away from the mean value. stats girlies rise 🙈✨

and one technical thing that matters: the additional overloads are not required to be provided exactly as (A). they only need to be sufficient to ensure that for their argument num of integer type, std::erf(num) has the same effect as std::erf(static_cast<double>(num)). that's the actual requirement, everything else is vibes. 💅

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🌸 EXAMPLE 🌸

this example calculates the probability that a normal variate is on the interval (x1, x2). iconic behaviour honestly 👑

// Run this code

#include <cmath>
#include <iomanip>
#include <iostream>

double phi(double x1, double x2)
{
    return (std::erf(x2 / std::sqrt(2)) - std::erf(x1 / std::sqrt(2))) / 2;
}

int main()
{
    std::cout << "Normal variate probabilities:\n"
              << std::fixed << std::setprecision(2);
    for (int n = -4; n < 4; ++n)
        std::cout << '[' << std::setw(2) << n
                  << ':' << std::setw(2) << n + 1 << "]: "
                  << std::setw(5) << 100 * phi(n, n + 1) << "%\n";

    std::cout << "Special values:\n"
              << "erf(-0) = " << std::erf(-0.0) << '\n'
              << "erf(Inf) = " << std::erf(INFINITY) << '\n';
}

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) = -0.00
erf(Inf) = 1.00

look at that bell curve distribution, she ate and left no crumbs 🦋📈

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⭐ SEE ALSO ⭐

• erfc, erfcf, erfcl — complementary error function (C++11) (function) (C++11) (C++11) 💜
• C documentation for erf — go check the C side bestie, it's the same energy 🌈

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🌈 EXTERNAL LINKS 🌈

• Weisstein, Eric W. "Erf." From MathWorld — A Wolfram Web Resource. 🧚 math lore fr fr

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so yeah that's the whole page ✨ that's it, that's the error function, go compute some probabilities bestie 💖

this cutie came from http://cppreference.com, dated 2024.06.10 — shoutout to the whole cppreference gang for keeping us fed, we love you, mwah 🫶🎀