Error Function (erf) Calculator
Enter a value of x to compute erf(x), erfc(x), and the corresponding standard normal CDF value Φ(x).
What is the erf function?
The error function, written as erf(x), is a special function widely used in probability, statistics, and engineering. It is defined by the integral:
erf(x) = (2 / √π) ∫0x e-t² dt
Intuitively, erf(x) measures area under a bell-shaped curve related to the Gaussian distribution. Because that integral has no elementary closed-form solution, calculators use accurate numerical approximations.
Why people use an erf calculator
- Statistics: Convert between Gaussian probabilities and cumulative values.
- Physics: Solve diffusion and heat transfer equations.
- Signal processing: Analyze noise and detection probabilities.
- Reliability engineering: Model uncertain processes and random variation.
- Quantitative finance: Work with normal-distribution-based risk calculations.
Related functions you get automatically
1) Complementary error function: erfc(x)
The complementary error function is:
erfc(x) = 1 - erf(x)
It is especially convenient when probabilities are very close to 0 or 1.
2) Standard normal CDF: Φ(x)
The standard normal CDF is linked to erf by:
Φ(x) = 0.5 × [1 + erf(x / √2)]
This connection makes erf extremely useful in z-score and probability calculations.
How this calculator computes erf(x)
This page uses a classic high-accuracy approximation (Abramowitz and Stegun style) that is fast and stable for most practical values. For very large |x|, erf(x) approaches ±1 as expected.
Quick reference values
- erf(0) = 0
- erf(0.5) ≈ 0.52049988
- erf(1) ≈ 0.84270079
- erf(1.5) ≈ 0.96610515
- erf(2) ≈ 0.99532227
- erf(-x) = -erf(x) (odd symmetry)
Practical tips
- Use at least 6 decimal places for probability-sensitive work.
- If x is large and positive, erf(x) will be extremely close to 1.
- If x is large and negative, erf(x) will be extremely close to -1.
- For normal probabilities, remember to scale input by √2 where required.
FAQ
Is this suitable for classroom and engineering use?
Yes. The approximation used here is widely accepted and highly accurate for everyday scientific, statistical, and engineering tasks.
Can erf(x) be greater than 1?
No. For real inputs, erf(x) is always in the interval (-1, 1), approaching these limits as |x| grows.
What if I need inverse erf?
This calculator focuses on forward evaluation of erf(x). If you need inverse error function values, use a dedicated inverse erf solver.