SciPy error function: erf, erfc (complementary) and erfinv (inverse), tested

The error function lives in scipy.special: erf(x), the complementary erfc(x) (use it instead of 1 - erf(x) for large x) and the inverse erfinv(y) for y in [-1, 1]. np.erf does not exist and math.erf rejects arrays.

1. erf, erfc, erfinv, erfcinv and the normal distribution

import math
import numpy as np
from scipy import special, stats

x = np.array([0.0, 0.5, 1.0, 2.0, 5.0])
print("erf      ", special.erf(x))
print("math.erf ", [math.erf(v) for v in x])
print("erfc     ", special.erfc(x))
print("1 - erf  ", 1 - special.erf(x), "<- x=5: wrong from the 5th digit")
print("erfc(10) ", special.erfc(10.0), " 1-erf(10):", 1 - special.erf(10.0))
print("erfinv   ", special.erfinv([0.0, 0.5, 0.95, 1.0, 1.5]))
print("erfcinv  ", special.erfcinv([1.0, 0.5, 1e-20]))

# link to the normal distribution
z = 1.96
print("norm.cdf ", stats.norm.cdf(z), " 0.5*(1+erf(z/sqrt2)):", 0.5 * (1 + special.erf(z / np.sqrt(2))))
print("norm.ppf ", stats.norm.ppf(0.975), " sqrt2*erfinv(0.95):", np.sqrt(2) * special.erfinv(0.95))
print("complex  ", special.erf(1 + 1j))

Output (SciPy 1.18.1, NumPy 2.5.4, Python 3.13.5):

erf       [0.         0.52049988 0.84270079 0.99532227 1.        ]
math.erf  [0.0, 0.5204998778130465, 0.8427007929497149, 0.9953222650189527, 0.9999999999984626]
erfc      [1.00000000e+00 4.79500122e-01 1.57299207e-01 4.67773498e-03
 1.53745979e-12]
1 - erf   [1.00000000e+00 4.79500122e-01 1.57299207e-01 4.67773498e-03
 1.53743684e-12] <- x=5: wrong from the 5th digit
erfc(10)  2.0884875837625446e-45  1-erf(10): 0.0
erfinv    [0.         0.47693628 1.38590382        inf        nan]
erfcinv   [-0.          0.47693628  6.60158062]
norm.cdf  0.9750021048517795  0.5*(1+erf(z/sqrt2)): 0.9750021048517796
norm.ppf  1.959963984540054  sqrt2*erfinv(0.95): 1.9599639845400538
complex   (1.3161512816979477+0.19045346923783463j)

• scipy.special.erf is vectorized and matches math.erf value by value (NumPy prints 1. for 0.9999999999984626 at x=5 because of display rounding).
• Complementary error function: use scipy.special.erfc(x), not 1 - erf(x). At x=5 the subtraction is already wrong from the 5th significant digit, and at x=10 it returns 0.0 while erfc(10) gives 2.09e-45.
• Inverse error function: erfinv(y) is defined on [-1, 1]; erfinv(1) is inf and outside the range you get nan (no exception, no warning in this run). erfcinv is the inverse of erfc and stays accurate for tiny arguments like 1e-20.
• Normal distribution: norm.cdf(z) = 0.5*(1 + erf(z/sqrt(2))) and norm.ppf(p) = sqrt(2)*erfinv(2p - 1); both pairs agree to the last digit or two above. If you only need probabilities, scipy.stats.norm is clearer.
• special.erf also accepts complex input.

2. "has no attribute erf" and "only 0-dimensional arrays"

import math
import numpy as np

def show(label, fn):
    try:
        print(label, "->", fn())
    except Exception as e:
        print(label, "->", type(e).__name__ + ":", e)

show("np.erf          ", lambda: np.erf(0.5))
show("math.erf(array) ", lambda: math.erf(np.array([0.5, 1.0])))
show("np.vectorize    ", lambda: np.vectorize(math.erf)(np.array([0.5, 1.0])))
show("scipy.erf       ", lambda: __import__("scipy").erf(0.5))
from scipy.special import erf
show("special.erf list", lambda: erf([0, 1, 2]))

Output:

np.erf           -> AttributeError: module 'numpy' has no attribute 'erf'
math.erf(array)  -> TypeError: only 0-dimensional arrays can be converted to Python scalars
np.vectorize     -> [0.52049988 0.84270079]
scipy.erf        -> AttributeError: Module 'scipy' has no attribute 'erf'
special.erf list -> [0.         0.84270079 0.99532227]

• NumPy has no erf, and top-level scipy does not export it: import it from scipy.special.
• math.erf works on single floats only; on an array it raises TypeError. Use scipy.special.erf (fast, vectorized) rather than np.vectorize(math.erf), which is a Python loop.

Related: SciPy standard error (stats.sem), scipy minimize errors.

Written with AI assistance. Measured on 2026-10-23 with SciPy 1.18.1, NumPy 2.5.4 and Python 3.13.5 on Linux. Scripts ran verbatim and the output was pasted unedited. PlainTables is our extension; SciPy is free.