RFC: add `special.erf` to the specification

[kgryte]

This RFC proposes adding a new API for computing the error function to the Array API specification in a special functions extension.

Overview

Based on array library comparison data, the API is available in a majority of array libraries in the PyData ecosystem. The error function is particularly important in statistics (e.g., CDF of a normal distribution).

erf was previously discussed as a candidate for inclusion within the context of a special functions extension. That RFC addressed several topics and design discussions. This RFC seeks to cut through the noise and propose a single API candidate for inclusion in the standard.

Prior Art

• NumPy: not implemented (ref: Adding the erf special function to numpy? numpy/numpy#12515).
• SciPy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.erf.html#scipy.special.erf
• CuPy: https://docs.cupy.dev/en/stable/reference/generated/cupyx.scipy.special.erf.html
• Dask: not implemented.
• JAX: https://docs.jax.dev/en/latest/_autosummary/jax.scipy.special.erf.html
• Torch: https://docs.pytorch.org/docs/stable/special.html#torch.special.erf
• TensorFlow: https://www.tensorflow.org/api_docs/python/tf/math/erf
• ndonnx: not implemented.
• MLX: https://ml-explore.github.io/mlx/build/html/python/_autosummary/mlx.core.erf.html#mlx.core.erf
• C99: https://en.cppreference.com/w/c/numeric/math/erf.html

Proposal

def erf(x: array, /) -> array

within a special functions extension in a manner similar to linalg and fft.

Comments

• SciPy supports complex dtypes. JAX does not. As a first pass, I suggest standardizing support for real-valued floating-point arrays. In a follow-up revision, we can add support for complex dtypes.
• Given erf's presence in C99, special cases should follow suit as we have done for other special functions in the main namespace.

[ev-br]

Just to double-check: is the proposal to add xp.erf or xp.special.erf? If the latter, then it sounds better to consider the whole extension at once, in #725.
If the former, then we should IMO draw the line on what is in scope. You mentioned C99, is that the guiding principle (so that "i0 is not in the main namespace but erf and erfc are, because we follow C99 math.h")?

[kgryte]

@ev-br xp.special.erf.

[cbourjau]

ndonnx: not implemented.

The ONNX standard sports an Erf function operator so we could implement this without a problem .