# expm1#

ivy.expm1(x, /, *, out=None)[source]#

Calculate an implementation-dependent approximation to `exp(x)-1`, having domain `[-infinity, +infinity]` and codomain `[-1, +infinity]`, for each element `x_i` of the input array `x`.

Note

The purpose of this function is to calculate `exp(x)-1.0` more accurately when `x` is close to zero. Accordingly, conforming implementations should avoid implementing this function as simply `exp(x)-1.0`. See FDLIBM, or some other IEEE 754-2019 compliant mathematical library, for a potential reference implementation.

Note

For complex floating-point operands, `expm1(conj(x))` must equal `conj(expm1(x))`.

Note

The exponential function is an entire function in the complex plane and has no branch cuts.

Special cases

For floating-point operands,

• If `x_i` is `NaN`, the result is `NaN`.

• If `x_i` is `+0`, the result is `+0`.

• If `x_i` is `-0`, the result is `-0`.

• If `x_i` is `+infinity`, the result is `+infinity`.

• If `x_i` is `-infinity`, the result is `-1`.

For complex floating-point operands, let `a = real(x_i)`, `b = imag(x_i)`, and

• If `a` is either `+0` or `-0` and `b` is `+0`, the result is `0 + 0j`.

• If `a` is a finite number and `b` is `+infinity`, the result is `NaN + NaN j`.

• If `a` is a finite number and `b` is `NaN`, the result is `NaN + NaN j`.

• If `a` is `+infinity` and `b` is `+0`, the result is `+infinity + 0j`.

• If `a` is `-infinity` and `b` is a finite number, the result is `+0 * cis(b) - 1.0`.

• If `a` is `+infinity` and `b` is a nonzero finite number, the result is `+infinity * cis(b) - 1.0`.

• If `a` is `-infinity` and `b` is `+infinity`, the result is `-1 + 0j` (sign of imaginary component is unspecified).

• If `a` is `+infinity` and `b` is `+infinity`, the result is `infinity + NaN j` (sign of real component is unspecified).

• If `a` is `-infinity` and `b` is `NaN`, the result is `-1 + 0j` (sign of imaginary component is unspecified).

• If `a` is `+infinity` and `b` is `NaN`, the result is `infinity + NaN j` (sign of real component is unspecified).

• If `a` is `NaN` and `b` is `+0`, the result is `NaN + 0j`.

• If `a` is `NaN` and `b` is not equal to `0`, the result is `NaN + NaN j`.

• If `a` is `NaN` and `b` is `NaN`, the result is `NaN + NaN j`.

where `cis(v)` is `cos(v) + sin(v)*1j`.

Parameters:
• x (`Union`[`Array`, `NativeArray`]) – input array. Should have a numeric data type.

• out (`Optional`[`Array`], default: `None`) – optional output array, for writing the result to. It must have a shape that the inputs broadcast to.

Return type:

`Array`

Returns:

ret – an array containing the evaluated result for each element in `x`. The returned array must have a floating-point data type determined by type-promotion.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[0, 5, float('-0'), ivy.nan]])
>>> ivy.expm1(x)
ivy.array([[  0., 147.,  -0.,  nan]])
```
```>>> x = ivy.array([ivy.inf, 1, float('-inf')])
>>> y = ivy.zeros(3)
>>> ivy.expm1(x, out=y)
ivy.array([  inf,  1.72, -1.  ])
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(a=ivy.array([-1, 0,]),
...                   b=ivy.array([10, 1]))
>>> ivy.expm1(x)
{
a: ivy.array([-0.632, 0.]),
b: ivy.array([2.20e+04, 1.72e+00])
}
```
Array.expm1(self, *, out=None)[source]#

ivy.Array instance method variant of ivy.expm1. This method simply wraps the function, and so the docstring for ivy.expm1 also applies to this method with minimal changes.

Parameters:
• self (`Array`) – input array. Should have a numeric data type.

• out (`Optional`[`Array`], default: `None`) – optional output array, for writing the result to. It must have a shape that the inputs broadcast to.

Return type:

`Array`

Returns:

ret – an array containing the evaluated result for each element in `x`. The returned array must have a floating-point data type determined by type-promotion.

Examples

```>>> x = ivy.array([5.5, -2.5, 1.5, -0])
>>> y = x.expm1()
>>> print(y)
ivy.array([244.   ,  -0.918,   3.48 ,   0.   ])
```
```>>> y = ivy.array([0., 0.])
>>> x = ivy.array([5., 0.])
>>> _ = x.expm1(out=y)
>>> print(y)
ivy.array([147.,   0.])
```
Container.expm1(self, *, key_chains=None, to_apply=True, prune_unapplied=False, map_sequences=False, out=None)[source]#

ivy.Container instance method variant of ivy.expm1. This method simply wraps the function, and so the docstring for ivy.expm1 also applies to this method with minimal changes.

Parameters:
• self (`Container`) – input container. Should have a floating-point data type.

• key_chains (`Optional`[`Union`[`List`[`str`], `Dict`[`str`, `str`], `Container`]], default: `None`) – The key-chains to apply or not apply the method to. Default is `None`.

• to_apply (`Union`[`bool`, `Container`], default: `True`) – If True, the method will be applied to key_chains, otherwise key_chains will be skipped. Default is `True`.

• prune_unapplied (`Union`[`bool`, `Container`], default: `False`) – Whether to prune key_chains for which the function was not applied. Default is `False`.

• map_sequences (`Union`[`bool`, `Container`], default: `False`) – Whether to also map method to sequences (lists, tuples). Default is `False`.

• out (`Optional`[`Container`], default: `None`) – optional output container, for writing the result to. It must have a shape that the inputs broadcast to.

Return type:

`Container`

Returns:

ret – a container containing the evaluated result for each element in `self`. The returned array must have a real-valued floating-point data type determined by type-promotion.

Examples

```>>> x = ivy.Container(a=ivy.array([2.5, 0.5]),
...                   b=ivy.array([5.4, -3.2]))
>>> y = x.expm1()
>>> print(y)
{
a: ivy.array([11.2, 0.649]),
b: ivy.array([220., -0.959])
}
```
```>>> y = ivy.Container(a=ivy.array([0., 0.]))
>>> x = ivy.Container(a=ivy.array([4., -2.]))
>>> x.expm1(out=y)
>>> print(y)
{
a: ivy.array([53.6, -0.865])
}
```