# Mathematical Expressions#

DALI allows you to use regular Python arithmetic operations and other mathematical functions in the pipeline definition (via `@pipeline_def` or within `define_graph()`) on the values that are returned from invoking other operators.

The expressions that are used will be incorporated into the pipeline without needing to explicitly instantiate operators and will describe the element-wise operations on tensors:

```@pipeline_def
def my_pipeline():
"""Create a pipeline which reads and decodes the images, scales channels by
broadcasting and clamps the result to [128, 255) range."""
images = fn.decoders.image(img_files, device="mixed")
red_highlight = images * nvidia.dali.types.Constant(np.float32([1.25, 0.75, 0.75]))
result = nvidia.dali.math.clamp(red_highlight, 128, 255)
return result
```

At least one of the inputs to the arithmetic expression must be returned by other DALI operator - that is a value of `nvidia.dali.pipeline.DataNode` representing a batch of tensors. The other input can be `nvidia.dali.types.Constant()` or regular Python value of type `bool`, `int`, or `float`. As the operations performed are element-wise, the shapes of all operands must be compatible - either match exactly or be broadcastable.

For details and examples see expressions tutorials.

Note

Keep in mind to wrap the tensor constants used in mathematical expressions (like NumPy array) with the `nvidia.dali.types.Constant()`. If used directly, the operator implementation from that tensor’s library may be picked up and the result might be undefined. Using the line from the previous example, the first two variants are equivalent, while the third will be wrong:

```# Correct approach:
red_highlight_0 = images *
nvidia.dali.types.Constant(np.float32([1.25, 0.75, 0.75]))
red_highlight_1 = nvidia.dali.types.Constant(np.float32([1.25, 0.75, 0.75])) *
images
# Wrong approach:
# red_highlight_2 = np.float32([1.25, 0.75, 0.75]) * images
```

## Type Promotion Rules#

For operations that accept two (or more) arguments, type promotions apply. The resulting type is calculated in accordance to the table below.

Operand Type

Operand Type

Result Type

T

T

T

floatX

T

floatX

where T is not a float

floatX

floatY

floatZ

where Z = max(X, Y)

intX

intY

intZ

where Z = max(X, Y)

uintX

uintY

uintZ

where Z = max(X, Y)

intX

uintY

int2Y

if X <= Y

intX

uintY

intX

if X > Y

`T` stands for any one of the supported numerical types: `bool`, `int8`, `int16`, `int32`, `int64`, `uint8`, `uint16`, `uint32`, `uint64`, `float32`, and `float64`.

`bool` type is considered the smallest unsigned integer type and is treated as `uint1` with respect to the table above.

Note

Type promotion is commutative.

For more than two arguments, the resulting type is calculated as a reduction from left to right - first calculating the result of operating on first two arguments, next between that intermediate result and the third argument and so on, until we have only the result type left.

## Supported Arithmetic Operations#

Currently, DALI supports the following operations:

Unary arithmetic operators: +, -

Unary operators that implement `__pos__(self)` and `__neg__(self)`. The result of a unary arithmetic operation always preserves the input type. Unary operators accept only TensorList inputs from other operators.

Return type:

TensorList of the same type

Binary arithmetic operations: +, -, *, /, //, **

Binary operators that implement `__add__`, `__sub__`, `__mul__`, `__truediv__`, `__floordiv__` and `__pow__` respectively.

The result of an arithmetic operation between two operands is described above, with the exception of `/`, the `__truediv__` operation, which always returns `float32` or `float64` type.

Note

The only allowed arithmetic operation between two `bool` values is multiplication `(*)`.

Return type:

TensorList of the type that is calculated based on the type promotion rules.

Comparison operations: ==, !=, <, <=, >, >=

Comparison operations.

Return type:

TensorList of `bool` type.

Bitwise binary operations: &, |, ^

The bitwise binary operations follow the same type promotion rules as arithmetic binary operations, but their inputs are restricted to integral types (including `bool`).

Note

A bitwise operation can be applied to two boolean inputs. Those operations can be used to emulate element-wise logical operations on Tensors.

Return type:

TensorList of the type that is calculated based on the type promotion rules.

The term “broadcasting” refers to how tensors with different shapes are treated in mathematical expressions. A value from a smaller tensor is “broadcast” so it contributes to multiple output values. At its simplest, a scalar value is broadcast to all output values. In more complex cases, the values can be broadcast along some dimensions if one of the operands has size 1 and the other is larger:

```            [[D, E],       [[ A+D,  B+E ],
[[A, B]] +   [F, G],   ==   [ A+F,  B+G ],
[H, J]]        [ A+H,  B+J ]]
```

In the example above, the operands have shapes of (1, 2) and (3, 2). The values from the array [[A, B]] are broadcast along axis 0. It’s possible that both operands are subject to broadcasting along different dimensions:

```            [[D],      [[ A+D,  B+D ],
[[A, B ]] +  [E],  ==   [ A+E,  B+E ],
[F]]       [ A+F,  B+F ]]
```

In this example, the shapes are (1, 2) and (3, 1) - the first operand is broadcast along axis 0 and the second is broadcast along axis 1.

### Shape extension#

For convenience, if the arrays have different number of dimensions, the shapes are padded with outer unit dimensions:

```shape of A == (480, 640, 3)
shape of B == (3)
shape of A + B == (480, 640, 3)   # b is treated as if it was shaped (1, 1, 3)
```

### Limitations#

The broadcasting operations in DALI can have only limited complexity. When broadcasting, the adjacent axes that need or do not need broadcasting are grouped. There can be up to six alternating broadcast/non-broadcast groups. Example of grouping:

```shape of A == a, b, 1, c, d
shape of B == a, b, e, 1, 1
grouping dimensions (0..1) and (3..4)
grouped shapes:
a*b, 1, c*d
a*b, e, 1
```

## Mathematical Functions#

Similarly to arithmetic expressions, one can use selected mathematical functions in the Pipeline graph definition. They also accept `nvidia.dali.pipeline.DataNode`, `nvidia.dali.types.Constant()` or regular Python value of type `bool`, `int`, or `float` as arguments. At least one of the inputs must be the output of other DALI Operator.

nvidia.dali.math.abs(input)#

Computes absolute value of values in `input`.

Return type:

TensorList of abs(input). The type is preserved.

nvidia.dali.math.fabs(input)#

Computes float absolute value of values in `input`.

Return type:

TensorList of fabs(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.floor(input)#

Computes floor of values in `input`.

Return type:

TensorList of floor(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.ceil(input)#

Computes ceil of values in `input`.

Return type:

TensorList of ceil(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.pow(base, exponent)#

Computes base to the power of exponents, that is base ** exponent.

Return type:

TensorList of pow(base, exponent). Type is calculated based on the type promotion rules.

nvidia.dali.math.fpow(base, exponent)#

Computes base to the power of exponents as floating point numbers.

Return type:

TensorList of pow(base, exponent). If all inputs are integers, the result will be float, otherwise the type is preserved.

nvidia.dali.math.min(left, right)#

Computes minima of corresponding values in `left` and `right`.

Return type:

TensorList of the type that is calculated based on the type promotion rules.

nvidia.dali.math.max(left, right)#

Computes maxima of corresponding values in `left` and `right`.

Return type:

TensorList of the type that is calculated based on the type promotion rules.

nvidia.dali.math.clamp(value, lo, hi)#

Produces a tensor of values from `value` clamped to the range `[lo, hi]`.

Return type:

TensorList of the type that is calculated based on the type promotion rules.

### Exponents and logarithms#

nvidia.dali.math.sqrt(input)#

Computes square root of values in `input`.

Return type:

TensorList of sqrt(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.rsqrt(input)#

Computes reciprocal of the square root of values in `input`.

Return type:

TensorList of rsqrt(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.cbrt(input)#

Computes cube root of values in `input`.

Return type:

TensorList of cbrt(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.exp(input)#

Computes exponential of values in `input`.

Return type:

TensorList of exp(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.log(input)#

Computes natural logarithm (base-e) of values in `input`.

Return type:

TensorList of log(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.log2(input)#

Computes logarithm (base-2) of values in `input`.

Return type:

TensorList of log2(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.log10(input)#

Computes logarithm (base-10) of values in `input`.

Return type:

TensorList of log10(input). If input is an integer, the result will be float, otherwise the type is preserved.

### Trigonometric Functions#

nvidia.dali.math.sin(input)#

Computes sine of values in `input`.

Return type:

TensorList of sin(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.cos(input)#

Computes cosine of values in `input`.

Return type:

TensorList of cos(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.tan(input)#

Computes tangent of values in `input`.

Return type:

TensorList of tan(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.asin(input)#

Computes arcus sine of values in `input`.

Return type:

TensorList of asin(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.acos(input)#

Computes arcus cosine of values in `input`.

Return type:

TensorList of acos(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.atan(input)#

Computes arcus tangent of values in `input`.

Return type:

TensorList of atan(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.atan2(x, y)#

Computes arcus tangent of corresponding values in x / y.

Return type:

TensorList of atan2(x, y). If all inputs are integers, the result will be float, otherwise the type is preserved.

### Hyperbolic Functions#

nvidia.dali.math.sinh(input)#

Computes hyperbolic sine of values in `input`.

Return type:

TensorList of sinh(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.cosh(input)#

Computes hyperbolic cosine of values in `input`.

Return type:

TensorList of cosh(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.tanh(input)#

Computes hyperbolic tangent of values in `input`.

Return type:

TensorList of tanh(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.asinh(input)#

Computes inverse hyperbolic sine of values in `input`.

Return type:

TensorList of asinh(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.acosh(input)#

Computes inverse hyperbolic cosine of values in `input`.

Return type:

TensorList of acosh(input). If input is an integer, the result will be float, otherwise the type is preserved.

nvidia.dali.math.atanh(input)#

Computes inverse hyperbolic tangent of values in `input`.

Return type:

TensorList of atanh(input). If input is an integer, the result will be float, otherwise the type is preserved.