chapters/pseudocode.adoc - tosa/specification - Gitiles

 //
 // This confidential and proprietary software may be used only as
 // authorised by a licensing agreement from ARM Limited
 // (C) COPYRIGHT 2021-2023 ARM Limited
 // ALL RIGHTS RESERVED
 // The entire notice above must be reproduced on all authorised
 // copies and copies may only be made to the extent permitted
 // by a licensing agreement from ARM Limited.

 == TOSA Pseudocode

 The TOSA pseudocode provides precise descriptions of TOSA operations.
 Each operator contains pseudocode describing the operator's functionality.
 This section contains pseudocode functions shared across multiple operators in the specification.

 === Operator Validation Helpers

 The following functions are used to define the valid conditions for TOSA operators.

 The REQUIRE function defines the conditions required by the TOSA operator.
 If the conditions are not met then the result of the TOSA graph is marked as unpredictable.
 Once the tosa_graph_result is set to tosa_unpredictable, the whole graph is considered unpredictable.

 The ERROR_IF function defines a condition that must set an error if the condition holds and the graph is not unpredictable.
 Note that if a graph contains both unpredictable and error statements then result of tosa_execute_graph() is tosa_unpredictable.
 This condition is captured in the ERROR_IF function.

 *Implementation Notes*

 * An implementation is not required to detect unpredictable behavior. If tosa_execute_graph() returns tosa_unpredictable then the tosa_test_compliance() function does not require any specific output from an implementation.
 * An implementation is required to detect errors in a graph that does not have unpredictable behavior (see tosa_test_compliance).
 * An acceptable implementation is to stop and report an error on the first ERROR_IF condition that occurs. This satifies tosa_test_compliance() even if the tosa_execute_graph() was tosa_unpredictable.
 * If the tosa_execute_graphs() result is tosa_unpredictable or tosa_error, then there is no requirement on the implementation to execute any portion of the TOSA graph.

 [source,c++]
 ----
 void REQUIRE(condition) {
     // Unpredictable overrides any previous result
     if (!(condition)) {
         tosa_graph_result = tosa_unpredictable;
     }
 }

 void ERROR_IF(condition) {
     // Error encodes a predictable error state and so is not registered
     // if the graph is marked as unpredictable.
     if (tosa_graph_result != tosa_unpredictable && condition) {
         tosa_graph_result = tosa_error;
     }
 }

 void LEVEL_CHECK(condition) {
     // If a level is specified and the level condition fails then
     // the result is unpredictable.
     REQUIRE(condition);
 }
 ----

 === Tensor Access Helpers

 ==== Tensor Utilities

 [source,c++]
 ----
 // Convert tensor index coordinates to an element offset
 size_t tensor_index_to_offset(shape_t shape, shape_t index) {
     size_t size = tensor_size(shape);  // check tensor shape is valid
     size_t offset = 0;
     for (int32_t i = 0; i < rank(shape); i++) {
         REQUIRE(index[i] >= 0 && index[i] < shape[i]);
         offset = offset * shape[i] + index[i];
     }
     return offset;
 }

 // Convert an element offset to tensor index coordinates
 shape_t tensor_offset_to_index(shape_t shape, size_t offset) {
     size_t size = tensor_size(shape);  // check tensor shape is valid
     REQUIRE(offset < size);
     shape_t index(rank(shape));    // index has rank(shape) indicies
     for(int32_t i = rank(shape) - 1; i >= 0; i--) {
         index[i] = offset % shape[i];
         offset /= shape[i];
     }
     return index;
 }

 // Check the tensor shape is valid and return the tensor size in elements
 size_t tensor_size(shape_t shape) {
     size_t size = 1;
     for (int32_t i = 0; i < rank(shape); i++) {
         REQUIRE(1 <= shape[i] && shape[i] <= maximum<size_t> / size);
         size *= shape[i];
     }
     return size;
 }

 // Return the size of the tensor in the given axis
 // For a rank=0 tensor, returns 1 for all axes
 size_t shape_dim(shape_t shape, int axis) {
     return (axis >= rank(shape)) ? 1 : shape[axis];
 }
 ----

 ==== Tensor Read

 tensor_read reads a single data value out of the given tensor.
 The shape argument contains the shape of the tensor.
 Index is the coordinates within the tensor of the value to be read.

 [source,c++]
 ----
 in_t tensor_read<in_t>(in_t *address, shape_t shape, shape_t index) {
     size_t offset = tensor_index_to_offset(shape, index);
     return address[offset];
 }
 ----

 ==== Tensor Write

 tensor_write writes a single data value into the given tensor.
 The shape argument contains the shape of the tensor.
 Index is the coordinates within the tensor of the value to be written.
 value is the value to be written to the given coordinate.

 [source,c++]
 ----
 void tensor_write<type>(<type> *address, shape_t shape, shape_t index, <type> value) {
     size_t offset = tensor_index_to_offset(shape, index);
     address[offset] = value;
 }
 ----

 ==== Variable Tensor Allocate

 variable_tensor_allocate allocates the mutable persistent memory block for storing variable tensors.
 The shape argument contains the shape of the allocated memory block for the variable_tensor.
 The uid argument is a globally unique identifier for variable tensors.

 [source,c++]
 ----
 tensor_t* variable_tensor_allocate<in_t>(shape_t shape, int32_t uid) {
     size_t size = tensor_size(shape);
     tensor_t *allocated_tensor = new tensor_t;
     allocated_tensor->data = new in_t[size];
     allocated_tensor->uid = uid;
     allocated_tensor->is_written = false;
     allocated_tensor->shape = shape;
     allocated_tensor->type = in_t;
     return allocated_tensor;
 }
 ----

 ==== Variable Tensor Lookup

 variable_tensor_lookup checks whether a variable tensor has been allocated or not.
 The uid argument is a globally unique identifier for variable tensors.

 [source,c++]
 ----
 tensor_t variable_tensor_lookup(int32_t uid) {
     // The global all_allocated_variable_tensors was instantiated at the first
     // time of executing the tosa graph
     for_each(tensor_t allocated_tensor in all_allocated_variable_tensors) {
         if (allocated_tensor.uid == uid) {
             return allocated_tensor;
         }
     }
     return NULL;
 }
 ----

 ==== Broadcast Helpers

 The following function derives the broadcast output shape from the input shapes.

 [source,c++]
 ----
 shape_t broadcast_shape(shape_t shape1, shape_t shape2) {
     ERROR_IF(rank(shape1) != rank(shape2));
     shape_t shape = shape1;
     for (int32_t i = 0; i < rank(shape); i++) {
         if (shape[i] == 1) {
             shape[i] = shape2[i];
         } else {
             ERROR_IF(shape2[i] != 1 && shape2[i] != shape[i]);
         }
     }
     return shape;
 }
 ----

 The following function maps an index in the output tensor to an index in the input tensor.

 [source,c++]
 ----
 // The index argument should be a valid location within out_shape.
 // The function returns the location within in_shape that contributes
 // to the output based on broadcasting rules.

 shape_t apply_broadcast(shape_t out_shape, shape_t in_shape, shape_t index) {
     ERROR_IF(rank(out_shape) != rank(in_shape));
     ERROR_IF(rank(out_shape) != rank(index));
     for (int32_t i = 0; i < rank(out_shape); i++) {
         if (out_shape[i] != in_shape[i]) {
             ERROR_IF(in_shape[i] != 1);
             index[i] = 0;
         }
     }
     return index;
 }
 ----

 === General Pseudocode Helpers

 This section contains general pseudocode utility functions used throughout the specification.

 ==== Arithmetic Helpers

 The following functions provide arithmetic while defining requirements such that values stay in the valid range.

 [source,c++]
 ----
 in_t apply_add_s<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) return a + b;
     int64_t c = sign_extend<int64_t>(a) + sign_extend<int64_t>(b);
     REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
     return static_cast<in_t>(c);
 }

 in_t apply_add_u<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) return a + b;
     uint64_t c = zero_extend<uint64_t>(a) + zero_extend<uint64_t>(b);
     REQUIRE(c >= minimum_u<in_u_t> && c <= maximum_u<in_u_t>);
     return truncate<in_t>(c);
 }

 in_t apply_arith_rshift<in_t>(in_t a, in_t b) {
     int32_t c = sign_extend<int32_t>(a) >> sign_extend<int32_t>(b);
     return static_cast<in_t>(c);
 }

 in_t apply_intdiv_s<in_t>(in_t a, in_t b) {
     int64_t c = sign_extend<int64_t>(a) / sign_extend<int64_t>(b);
     REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
     return static_cast<in_t>(c);
 }

 in_t apply_ceil<in_t>(in_t input) {
     return input value rounded up to nearest integer
 }

 in_t apply_clip_s<in_t>(in_t value, in_t min_val, in_t max_val) {
     if (is_floating_point(in_t>) {
         REQUIRE(min_val <= max_val);
     }
     else {
         REQUIRE(sign_extend<int64_t>(min_val) <= sign_extend<int64_t>(max_val));
     }
     value = apply_max_s<in_t>(value, min_val);
     value = apply_min_s<in_t>(value, max_val);
     return value;
 }

 in_t apply_clip_u<in_t>(in_t value, in_t min_val, in_t max_val) {
     REQUIRE(zero_extend<int64_t>(min_val) <= zero_extend<int64_t>(max_val));
     value = apply_max_u<in_t>(value, min_val);
     value = apply_min_u<in_t>(value, max_val);
     return value;
 }

 in_t apply_exp<in_t>(in_t input) {
     return e to the power input
 }

 in_t apply_floor<in_t>(in_t input) {
     return input value rounded down to nearest integer
 }

 in_t apply_log<in_t>(in_t input) {
     if (input == 0) {
         return -INFINITY
     }
     else if (input < 0) {
         return NaN;
     }
     return the natural logarithm of input
 }

 in_t apply_logical_rshift<in_t>(in_t a, in_t b) {
     uint64_t c = zero_extend<uint32_t>(a) >> zero_extend<uint32_t>(b);
     return static_cast<in_t>(c);
 }

 in_t apply_max_s<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) {
         if (isNaN(a) || isNaN(b)) {
             return NaN;
         }
         if (a >= b) return a; else return b;
     }
     // Integer version
     if (sign_extend<int64_t>(a) >= sign_extend<int64_t>(b)) return a; else return b;
 }

 in_t apply_max_u<in_t>(in_t a, in_t b) {
     if (zero_extend<uint64_t>(a) >= zero_extend<int64_t>(b)) return a; else return b;
 }

 in_t apply_min_s<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) {
         if (isNaN(a) || isNaN(b)) {
             return NaN;
         }
         if (a < b) return a; else return b;
     }
     // Integer version
     if (sign_extend<int64_t>(a) < sign_extend<int64_t>(b)) return a; else return b;
 }

 in_t apply_min_u<in_t>(in_t a, in_t b) {
     if (zero_extend<int64_t>(a) < zero_extend<int64_t>(b)) return a; else return b;
 }

 in_t apply_mul_s<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) return a * b;
     int64_t c = sign_extend<int64_t>(a) * sign_extend<int64_t>(b);
     return static_cast<in_t>(c);
 }

 in_t apply_pow<in_t>(in_t a, in_t b) {
     return a ** b; // a raised to the power b
 }

 in_t apply_sqrt<in_t>(in_t input) {
     return the square root of input
 }

 in_t apply_sub_s<in_t>(in_t a, in_t b) {
     if (is_floating_point(in_t)) return a - b;
     int64_t c = sign_extend<int64_t>(a) - sign_extend<int64_t>(b);
     REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
     return static_cast<in_t>(c);
 }

 in_t apply_sub_u<in_t>(in_t a, in_t b) {
     uint64_t c = zero_extend<uint64_t>(a) - zero_extend<uint64_t>(b);
     REQUIRE(c >= minimum_u<in_u_t> && c <= maximum_u<in_u_t>);
     return truncate<in_t>(c);
 }

 int32_t count_leading_zeros(int32_t a) {
     int32_t acc = 32;
     if (a != 0) {
         uint32_t mask;
         mask = 1 << (32 - 1); // width of int32_t - 1
         acc = 0;
         while ((mask & a) == 0) {
             mask = mask >> 1;
             acc = acc + 1;
         }
     }
     return acc;
 }
 ----

 ==== Type Conversion Helpers

 The following definitions indicate the type to be used when the given parameters are provided.

 [source,c++]
 ----

 // Returns a signed version of the given type
 // A no-op for floating-point types
 Type make_signed(Type in_t)
 {
     switch(in_t) {
         case bool_t:
             return bool_t;
         case i8_t:
             return int8_t;
         case i16_t:
             return int16_t;
         case i32_t:
             return int32_t;
         case i48_t:
             return int48_t;
         case fp16_t:
             return fp16_t;
         case bf16_t:
             return bf16_t;
         case fp32_t:
             return fp32_t;
     }
 }

 // Returns the usigned type of the given type
 // Error to call this with anything but i8_t or i16_t

 Type make_unsigned(Type in_t)
 {
     ERROR_IF(in_t != i8_t && in_t != i16_t);
     switch(in_t) {
         case i8_t:
             return uint8_t;
         case i16_t:
             return uint16_t;
     }
 }

 out_t static_cast<out_t>(in_t value)
 {
     // Operates similar to the c++ standard static_cast
     // Limited to simple numeric conversion for TOSA.
     // Sign extends signed integer input types if needed
     // Zero extends unsigned integer input types if needed
     // Truncates when converting to a smaller width data type
     // Conversion from integer to floating-point is exact if possible
     // If converting between signless integer types, treated as signed integer
 }

 out_t bitcast<out_t>(in_t value)
 {
     // Treats the bits of value as if they were of type out_t
     // Only supported for integer types of the same bit width
 }
 ----

 ==== Numeric Accuracy Helpers

 For a floating point number of type in_t a normal value is of the form (1.x * 2^e).
 The fractional part 'x' has a number of fractional or mantissa bits depending on the type.
 The exponent 'e' has a normal range depending on the type.
 The functions below return the ranges according to type.

 [source,c++]
 ----
 fp64_t exp2(int n) {
     if (n < -1075) {
         return 0.0; // smaller than smallest denormal
     }
     REQUIRE(n <= 1023);
     fp64_t v = 1.0;
     while (n > 0) { v = v*2.0; n--; }
     while (n < 0) { v = v/2.0; n++; }
     return v;
 }

 int ilog2(fp64_t v) {
     REQURE(0 < v && v < infinity);
     int n = 0;
     while (v >= 2.0) { v = v/2.0; n++; }
     while (v <  1.0) { v = v*2.0; n--; }
     return n;
 }

 fp64_t normal_min<in_t>() {
   switch (in_t) {
     case fp32_t: return exp2(-126);
     case bf16_t: return exp2(-126);
     case fp16_t: return exp2( -14);
   }
 }

 fp64_t normal_max<in_t>() {
   switch (in_t) {
     case fp32_t: return exp2(128) - exp2(127-23);
     case bf16_t: return exp2(128) - exp2(127- 7);
     case fp16_t: return exp2( 16) - exp2( 15-10);
   }
 }

 // Number of fractional (mantissa bits)
 int normal_frac<in_t> () {
   switch (in_t) {
     case fp32_t: return 23;
     case fp16_t: return 10;
     case bf16_t: return  7;
   }
 }
 ----

 The following functions check if a test value in floating-point format in_t is within an error range compared to a reference value.
 The functions assume that denormal values may be flushed to zero.
 For the first function, the permitted error range is specified as num_ulp which is converted to an error bound as specified by the code.
 For the second function, the permitted error range is specified as an absolute error bound.

 [source,c++]
 ----
 bool tosa_reference_check_fp<in_t>(in_t test_value, fp64_t ref_value, fp64_t num_ulp) {
   fp64_t err_bnd = 0.0;
   if (is_normal_fp64(ref_value) && abs(ref_value) != 0) {
     int ref_exp = ilog2(abs(ref_value));
     fp64_t ref_pow2 = max(exp2(ref_exp), normal_min<in_t>);
     fp64_t val_ulp  = ref_pow2 * exp2(-normal_frac<in_t>);
     err_bnd = val_ulp * num_ulp;
   }
   return tosa_reference_check_fp_bnd<in_t>(test_value, ref_value, err_bnd);
 }

 bool tosa_reference_check_fp_bnd<in_t>(in_t test_value, fp64_t ref_value, fp64_t err_bnd) {
   if (is_a_NaN(ref_value)) {
     return is_a_NaN(test_value);
   }
   REQUIRE(err_bnd >= 0.0);
   if (ref_value < 0) {
     ref_value  = -ref_value;
     test_value = -test_value;
   }
   fp64_t ref_max = ref_value + err_bnd;
   fp64_t ref_min = ref_value - err_bnd;
   if (ref_max > normal_max<in_t>) ref_max = infinity;
   if (ref_min > normal_max<in_t>) ref_min = infinity;
   if (ref_max < normal_min<in_t>) ref_max = normal_min<in_t>;
   if (ref_min < normal_min<in_t>) ref_min = 0;
   return (static_cast<fp64_t>(test_value) >= ref_min &&
           static_cast<fp64_t>(test_value) <= ref_max);
 }

 ----

 ==== Numeric Conversion Helpers

 The following definitions are used in pseudocode to do numeric conversions.
 Where the *float_t* type is used, it represents all of the floating-point data types supported by the given profile.
 See <<Number formats>> for details on the floating-point formats.

 [source,c++]
 ----
 int round_to_nearest_int(float_t f)
   Converts the floating-point value to f, with rounding to the nearest integer value.
   For the required precision see the section: Main inference precision requirements.

 float_t round_to_nearest_float(in_t f)
   Converts the input value into floating-point, rounding to the nearest representable value.
   For the required precision see the section: Main inference precision requirements.

 out_t sign_extend<out_t>(in_t input)
   Floating point values are unchanged.
   For two's complement integer values where out_t has more bits than in_t, replicate the top bit of input for all bits between the top bit of input and the top bit of output.

 out_t zero_extend<out_t>(in_t input)
   Floating point values are unchanged.
   For two's complement integer values where out_t has more bits than in_t, insert zero values for all bits between the top bit of input and the top bit of output.

 out_t truncate(in_t input)
   output is the sizeof(out_t) least significant bits in input.
   Nop for floating-point types
 ----

 The following definition is used to flatten a list of lists into a single list.

 [source,c++]
 ----
 in_t* flatten(in_t lists[]) {
     in_t output = [];
     for_each(list in lists) {
         for_each(element in list) {
             output.append(element);
         }
     }
 }
 ----

 Generic helper functions used to keep the pseudocode concise.

 [source,c++]
 ----

 bool_t is_floating_point(type) {
     if (type == fp16_t || type == fp32_t || type == bf16_t)
         return true;
     return false;
 }

 int32_t idiv(int32_t input1, int32_t input2) {
     return input1 / input2; // Integer divide that truncates towards zero
 }

 // Integer division that checks input1 is a multiple of input2

 int32_t idiv_check(int32_t input1, int32_t input2) {
     ERROR_IF(input1 % input2 != 0); // input1 must be a multiple of input2
     return input1 / input2;         // exact quotient without rounding
 }

 // perform an integer division with rounding towards minus infinity

 int32_t idiv_floor(int32_t input1, int32_t input2) {
     int32_t rval = input1 / input2;
     if (rval * input2 > input1) {
         rval--;
     }
     return rval;
 }

 int32_t length(in_t input)
     return number of elements in input list

 int32_t rank(in_t input)
     return rank of an input tensor

 int32_t sum(in_t input[])
     return the sum of values of an input list

 bool isNaN(float input)
     return True if floating-point input value is NaN

 float_t pi()
     returns value of pi

 float_t sin(angle)
     return sine of angle given in radians

 float_t cos(angle)
     return cosine of angle given in radians

 bool power_of_two(int32_t value)
     return true if value is a power of two, false otherwise

 in_out_t maximum_s<Type T>
     return the maximum value when interpreting type T as a signed value as returned by the make_signed helper.

 in_out_t minimum_s<Type T>
     return the minimum value when interpreting type T as a signed value as returned by the make_signed helper.

 in_out_t maximum_u<Type T>
     return the maximum value when interpreting type T as an unsigned value as returned by the make_unsigned helper.

 in_out_t minimum_u<Type T>
     return the minimum value when interpreting type T as an unsigned value as returned by the make_unsigned helper.
 ----
	//
	// This confidential and proprietary software may be used only as
	// authorised by a licensing agreement from ARM Limited
	// (C) COPYRIGHT 2021-2023 ARM Limited
	// ALL RIGHTS RESERVED
	// The entire notice above must be reproduced on all authorised
	// copies and copies may only be made to the extent permitted
	// by a licensing agreement from ARM Limited.

	== TOSA Pseudocode

	The TOSA pseudocode provides precise descriptions of TOSA operations.
	Each operator contains pseudocode describing the operator's functionality.
	This section contains pseudocode functions shared across multiple operators in the specification.

	=== Operator Validation Helpers

	The following functions are used to define the valid conditions for TOSA operators.

	The REQUIRE function defines the conditions required by the TOSA operator.
	If the conditions are not met then the result of the TOSA graph is marked as unpredictable.
	Once the tosa_graph_result is set to tosa_unpredictable, the whole graph is considered unpredictable.

	The ERROR_IF function defines a condition that must set an error if the condition holds and the graph is not unpredictable.
	Note that if a graph contains both unpredictable and error statements then result of tosa_execute_graph() is tosa_unpredictable.
	This condition is captured in the ERROR_IF function.

	Implementation Notes

	* An implementation is not required to detect unpredictable behavior. If tosa_execute_graph() returns tosa_unpredictable then the tosa_test_compliance() function does not require any specific output from an implementation.
	* An implementation is required to detect errors in a graph that does not have unpredictable behavior (see tosa_test_compliance).
	* An acceptable implementation is to stop and report an error on the first ERROR_IF condition that occurs. This satifies tosa_test_compliance() even if the tosa_execute_graph() was tosa_unpredictable.
	* If the tosa_execute_graphs() result is tosa_unpredictable or tosa_error, then there is no requirement on the implementation to execute any portion of the TOSA graph.

	[source,c++]
	----
	void REQUIRE(condition) {
	// Unpredictable overrides any previous result
	if (!(condition)) {
	tosa_graph_result = tosa_unpredictable;
	}
	}

	void ERROR_IF(condition) {
	// Error encodes a predictable error state and so is not registered
	// if the graph is marked as unpredictable.
	if (tosa_graph_result != tosa_unpredictable && condition) {
	tosa_graph_result = tosa_error;
	}
	}

	void LEVEL_CHECK(condition) {
	// If a level is specified and the level condition fails then
	// the result is unpredictable.
	REQUIRE(condition);
	}
	----

	=== Tensor Access Helpers

	==== Tensor Utilities

	[source,c++]
	----
	// Convert tensor index coordinates to an element offset
	size_t tensor_index_to_offset(shape_t shape, shape_t index) {
	size_t size = tensor_size(shape); // check tensor shape is valid
	size_t offset = 0;
	for (int32_t i = 0; i < rank(shape); i++) {
	REQUIRE(index[i] >= 0 && index[i] < shape[i]);
	offset = offset * shape[i] + index[i];
	}
	return offset;
	}

	// Convert an element offset to tensor index coordinates
	shape_t tensor_offset_to_index(shape_t shape, size_t offset) {
	size_t size = tensor_size(shape); // check tensor shape is valid
	REQUIRE(offset < size);
	shape_t index(rank(shape)); // index has rank(shape) indicies
	for(int32_t i = rank(shape) - 1; i >= 0; i--) {
	index[i] = offset % shape[i];
	offset /= shape[i];
	}
	return index;
	}

	// Check the tensor shape is valid and return the tensor size in elements
	size_t tensor_size(shape_t shape) {
	size_t size = 1;
	for (int32_t i = 0; i < rank(shape); i++) {
	REQUIRE(1 <= shape[i] && shape[i] <= maximum<size_t> / size);
	size *= shape[i];
	}
	return size;
	}

	// Return the size of the tensor in the given axis
	// For a rank=0 tensor, returns 1 for all axes
	size_t shape_dim(shape_t shape, int axis) {
	return (axis >= rank(shape)) ? 1 : shape[axis];
	}
	----

	==== Tensor Read

	tensor_read reads a single data value out of the given tensor.
	The shape argument contains the shape of the tensor.
	Index is the coordinates within the tensor of the value to be read.

	[source,c++]
	----
	in_t tensor_read<in_t>(in_t *address, shape_t shape, shape_t index) {
	size_t offset = tensor_index_to_offset(shape, index);
	return address[offset];
	}
	----

	==== Tensor Write

	tensor_write writes a single data value into the given tensor.
	The shape argument contains the shape of the tensor.
	Index is the coordinates within the tensor of the value to be written.
	value is the value to be written to the given coordinate.

	[source,c++]
	----
	void tensor_write<type>(<type> *address, shape_t shape, shape_t index, <type> value) {
	size_t offset = tensor_index_to_offset(shape, index);
	address[offset] = value;
	}
	----

	==== Variable Tensor Allocate

	variable_tensor_allocate allocates the mutable persistent memory block for storing variable tensors.
	The shape argument contains the shape of the allocated memory block for the variable_tensor.
	The uid argument is a globally unique identifier for variable tensors.

	[source,c++]
	----
	tensor_t* variable_tensor_allocate<in_t>(shape_t shape, int32_t uid) {
	size_t size = tensor_size(shape);
	tensor_t *allocated_tensor = new tensor_t;
	allocated_tensor->data = new in_t[size];
	allocated_tensor->uid = uid;
	allocated_tensor->is_written = false;
	allocated_tensor->shape = shape;
	allocated_tensor->type = in_t;
	return allocated_tensor;
	}
	----

	==== Variable Tensor Lookup

	variable_tensor_lookup checks whether a variable tensor has been allocated or not.
	The uid argument is a globally unique identifier for variable tensors.

	[source,c++]
	----
	tensor_t variable_tensor_lookup(int32_t uid) {
	// The global all_allocated_variable_tensors was instantiated at the first
	// time of executing the tosa graph
	for_each(tensor_t allocated_tensor in all_allocated_variable_tensors) {
	if (allocated_tensor.uid == uid) {
	return allocated_tensor;
	}
	}
	return NULL;
	}
	----

	==== Broadcast Helpers

	The following function derives the broadcast output shape from the input shapes.

	[source,c++]
	----
	shape_t broadcast_shape(shape_t shape1, shape_t shape2) {
	ERROR_IF(rank(shape1) != rank(shape2));
	shape_t shape = shape1;
	for (int32_t i = 0; i < rank(shape); i++) {
	if (shape[i] == 1) {
	shape[i] = shape2[i];
	} else {
	ERROR_IF(shape2[i] != 1 && shape2[i] != shape[i]);
	}
	}
	return shape;
	}
	----

	The following function maps an index in the output tensor to an index in the input tensor.

	[source,c++]
	----
	// The index argument should be a valid location within out_shape.
	// The function returns the location within in_shape that contributes
	// to the output based on broadcasting rules.

	shape_t apply_broadcast(shape_t out_shape, shape_t in_shape, shape_t index) {
	ERROR_IF(rank(out_shape) != rank(in_shape));
	ERROR_IF(rank(out_shape) != rank(index));
	for (int32_t i = 0; i < rank(out_shape); i++) {
	if (out_shape[i] != in_shape[i]) {
	ERROR_IF(in_shape[i] != 1);
	index[i] = 0;
	}
	}
	return index;
	}
	----

	=== General Pseudocode Helpers

	This section contains general pseudocode utility functions used throughout the specification.

	==== Arithmetic Helpers

	The following functions provide arithmetic while defining requirements such that values stay in the valid range.

	[source,c++]
	----
	in_t apply_add_s<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) return a + b;
	int64_t c = sign_extend<int64_t>(a) + sign_extend<int64_t>(b);
	REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
	return static_cast<in_t>(c);
	}

	in_t apply_add_u<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) return a + b;
	uint64_t c = zero_extend<uint64_t>(a) + zero_extend<uint64_t>(b);
	REQUIRE(c >= minimum_u<in_u_t> && c <= maximum_u<in_u_t>);
	return truncate<in_t>(c);
	}

	in_t apply_arith_rshift<in_t>(in_t a, in_t b) {
	int32_t c = sign_extend<int32_t>(a) >> sign_extend<int32_t>(b);
	return static_cast<in_t>(c);
	}

	in_t apply_intdiv_s<in_t>(in_t a, in_t b) {
	int64_t c = sign_extend<int64_t>(a) / sign_extend<int64_t>(b);
	REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
	return static_cast<in_t>(c);
	}

	in_t apply_ceil<in_t>(in_t input) {
	return input value rounded up to nearest integer
	}

	in_t apply_clip_s<in_t>(in_t value, in_t min_val, in_t max_val) {
	if (is_floating_point(in_t>) {
	REQUIRE(min_val <= max_val);
	}
	else {
	REQUIRE(sign_extend<int64_t>(min_val) <= sign_extend<int64_t>(max_val));
	}
	value = apply_max_s<in_t>(value, min_val);
	value = apply_min_s<in_t>(value, max_val);
	return value;
	}

	in_t apply_clip_u<in_t>(in_t value, in_t min_val, in_t max_val) {
	REQUIRE(zero_extend<int64_t>(min_val) <= zero_extend<int64_t>(max_val));
	value = apply_max_u<in_t>(value, min_val);
	value = apply_min_u<in_t>(value, max_val);
	return value;
	}

	in_t apply_exp<in_t>(in_t input) {
	return e to the power input
	}

	in_t apply_floor<in_t>(in_t input) {
	return input value rounded down to nearest integer
	}

	in_t apply_log<in_t>(in_t input) {
	if (input == 0) {
	return -INFINITY
	}
	else if (input < 0) {
	return NaN;
	}
	return the natural logarithm of input
	}

	in_t apply_logical_rshift<in_t>(in_t a, in_t b) {
	uint64_t c = zero_extend<uint32_t>(a) >> zero_extend<uint32_t>(b);
	return static_cast<in_t>(c);
	}

	in_t apply_max_s<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) {
	if (isNaN(a) \|\| isNaN(b)) {
	return NaN;
	}
	if (a >= b) return a; else return b;
	}
	// Integer version
	if (sign_extend<int64_t>(a) >= sign_extend<int64_t>(b)) return a; else return b;
	}

	in_t apply_max_u<in_t>(in_t a, in_t b) {
	if (zero_extend<uint64_t>(a) >= zero_extend<int64_t>(b)) return a; else return b;
	}

	in_t apply_min_s<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) {
	if (isNaN(a) \|\| isNaN(b)) {
	return NaN;
	}
	if (a < b) return a; else return b;
	}
	// Integer version
	if (sign_extend<int64_t>(a) < sign_extend<int64_t>(b)) return a; else return b;
	}

	in_t apply_min_u<in_t>(in_t a, in_t b) {
	if (zero_extend<int64_t>(a) < zero_extend<int64_t>(b)) return a; else return b;
	}

	in_t apply_mul_s<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) return a * b;
	int64_t c = sign_extend<int64_t>(a) * sign_extend<int64_t>(b);
	return static_cast<in_t>(c);
	}

	in_t apply_pow<in_t>(in_t a, in_t b) {
	return a ** b; // a raised to the power b
	}

	in_t apply_sqrt<in_t>(in_t input) {
	return the square root of input
	}

	in_t apply_sub_s<in_t>(in_t a, in_t b) {
	if (is_floating_point(in_t)) return a - b;
	int64_t c = sign_extend<int64_t>(a) - sign_extend<int64_t>(b);
	REQUIRE(c >= minimum_s<in_t> && c <= maximum_s<in_t>);
	return static_cast<in_t>(c);
	}

	in_t apply_sub_u<in_t>(in_t a, in_t b) {
	uint64_t c = zero_extend<uint64_t>(a) - zero_extend<uint64_t>(b);
	REQUIRE(c >= minimum_u<in_u_t> && c <= maximum_u<in_u_t>);
	return truncate<in_t>(c);
	}

	int32_t count_leading_zeros(int32_t a) {
	int32_t acc = 32;
	if (a != 0) {
	uint32_t mask;
	mask = 1 << (32 - 1); // width of int32_t - 1
	acc = 0;
	while ((mask & a) == 0) {
	mask = mask >> 1;
	acc = acc + 1;
	}
	}
	return acc;
	}
	----

	==== Type Conversion Helpers

	The following definitions indicate the type to be used when the given parameters are provided.

	[source,c++]
	----

	// Returns a signed version of the given type
	// A no-op for floating-point types
	Type make_signed(Type in_t)
	{
	switch(in_t) {
	case bool_t:
	return bool_t;
	case i8_t:
	return int8_t;
	case i16_t:
	return int16_t;
	case i32_t:
	return int32_t;
	case i48_t:
	return int48_t;
	case fp16_t:
	return fp16_t;
	case bf16_t:
	return bf16_t;
	case fp32_t:
	return fp32_t;
	}
	}

	// Returns the usigned type of the given type
	// Error to call this with anything but i8_t or i16_t

	Type make_unsigned(Type in_t)
	{
	ERROR_IF(in_t != i8_t && in_t != i16_t);
	switch(in_t) {
	case i8_t:
	return uint8_t;
	case i16_t:
	return uint16_t;
	}
	}

	out_t static_cast<out_t>(in_t value)
	{
	// Operates similar to the c++ standard static_cast
	// Limited to simple numeric conversion for TOSA.
	// Sign extends signed integer input types if needed
	// Zero extends unsigned integer input types if needed
	// Truncates when converting to a smaller width data type
	// Conversion from integer to floating-point is exact if possible
	// If converting between signless integer types, treated as signed integer
	}

	out_t bitcast<out_t>(in_t value)
	{
	// Treats the bits of value as if they were of type out_t
	// Only supported for integer types of the same bit width
	}
	----

	==== Numeric Accuracy Helpers

	For a floating point number of type in_t a normal value is of the form (1.x * 2^e).
	The fractional part 'x' has a number of fractional or mantissa bits depending on the type.
	The exponent 'e' has a normal range depending on the type.
	The functions below return the ranges according to type.

	[source,c++]
	----
	fp64_t exp2(int n) {
	if (n < -1075) {
	return 0.0; // smaller than smallest denormal
	}
	REQUIRE(n <= 1023);
	fp64_t v = 1.0;
	while (n > 0) { v = v*2.0; n--; }
	while (n < 0) { v = v/2.0; n++; }
	return v;
	}

	int ilog2(fp64_t v) {
	REQURE(0 < v && v < infinity);
	int n = 0;
	while (v >= 2.0) { v = v/2.0; n++; }
	while (v < 1.0) { v = v*2.0; n--; }
	return n;
	}

	fp64_t normal_min<in_t>() {
	switch (in_t) {
	case fp32_t: return exp2(-126);
	case bf16_t: return exp2(-126);
	case fp16_t: return exp2( -14);
	}
	}

	fp64_t normal_max<in_t>() {
	switch (in_t) {
	case fp32_t: return exp2(128) - exp2(127-23);
	case bf16_t: return exp2(128) - exp2(127- 7);
	case fp16_t: return exp2( 16) - exp2( 15-10);
	}
	}

	// Number of fractional (mantissa bits)
	int normal_frac<in_t> () {
	switch (in_t) {
	case fp32_t: return 23;
	case fp16_t: return 10;
	case bf16_t: return 7;
	}
	}
	----

	The following functions check if a test value in floating-point format in_t is within an error range compared to a reference value.
	The functions assume that denormal values may be flushed to zero.
	For the first function, the permitted error range is specified as num_ulp which is converted to an error bound as specified by the code.
	For the second function, the permitted error range is specified as an absolute error bound.

	[source,c++]
	----
	bool tosa_reference_check_fp<in_t>(in_t test_value, fp64_t ref_value, fp64_t num_ulp) {
	fp64_t err_bnd = 0.0;
	if (is_normal_fp64(ref_value) && abs(ref_value) != 0) {
	int ref_exp = ilog2(abs(ref_value));
	fp64_t ref_pow2 = max(exp2(ref_exp), normal_min<in_t>);
	fp64_t val_ulp = ref_pow2 * exp2(-normal_frac<in_t>);
	err_bnd = val_ulp * num_ulp;
	}
	return tosa_reference_check_fp_bnd<in_t>(test_value, ref_value, err_bnd);
	}

	bool tosa_reference_check_fp_bnd<in_t>(in_t test_value, fp64_t ref_value, fp64_t err_bnd) {
	if (is_a_NaN(ref_value)) {
	return is_a_NaN(test_value);
	}
	REQUIRE(err_bnd >= 0.0);
	if (ref_value < 0) {
	ref_value = -ref_value;
	test_value = -test_value;
	}
	fp64_t ref_max = ref_value + err_bnd;
	fp64_t ref_min = ref_value - err_bnd;
	if (ref_max > normal_max<in_t>) ref_max = infinity;
	if (ref_min > normal_max<in_t>) ref_min = infinity;
	if (ref_max < normal_min<in_t>) ref_max = normal_min<in_t>;
	if (ref_min < normal_min<in_t>) ref_min = 0;
	return (static_cast<fp64_t>(test_value) >= ref_min &&
	static_cast<fp64_t>(test_value) <= ref_max);
	}

	----

	==== Numeric Conversion Helpers

	The following definitions are used in pseudocode to do numeric conversions.
	Where the float_t type is used, it represents all of the floating-point data types supported by the given profile.
	See <<Number formats>> for details on the floating-point formats.

	[source,c++]
	----
	int round_to_nearest_int(float_t f)
	Converts the floating-point value to f, with rounding to the nearest integer value.
	For the required precision see the section: Main inference precision requirements.

	float_t round_to_nearest_float(in_t f)
	Converts the input value into floating-point, rounding to the nearest representable value.
	For the required precision see the section: Main inference precision requirements.

	out_t sign_extend<out_t>(in_t input)
	Floating point values are unchanged.
	For two's complement integer values where out_t has more bits than in_t, replicate the top bit of input for all bits between the top bit of input and the top bit of output.

	out_t zero_extend<out_t>(in_t input)
	Floating point values are unchanged.
	For two's complement integer values where out_t has more bits than in_t, insert zero values for all bits between the top bit of input and the top bit of output.

	out_t truncate(in_t input)
	output is the sizeof(out_t) least significant bits in input.
	Nop for floating-point types
	----

	The following definition is used to flatten a list of lists into a single list.

	[source,c++]
	----
	in_t* flatten(in_t lists[]) {
	in_t output = [];
	for_each(list in lists) {
	for_each(element in list) {
	output.append(element);
	}
	}
	}
	----

	Generic helper functions used to keep the pseudocode concise.

	[source,c++]
	----

	bool_t is_floating_point(type) {
	if (type == fp16_t \|\| type == fp32_t \|\| type == bf16_t)
	return true;
	return false;
	}

	int32_t idiv(int32_t input1, int32_t input2) {
	return input1 / input2; // Integer divide that truncates towards zero
	}

	// Integer division that checks input1 is a multiple of input2

	int32_t idiv_check(int32_t input1, int32_t input2) {
	ERROR_IF(input1 % input2 != 0); // input1 must be a multiple of input2
	return input1 / input2; // exact quotient without rounding
	}

	// perform an integer division with rounding towards minus infinity

	int32_t idiv_floor(int32_t input1, int32_t input2) {
	int32_t rval = input1 / input2;
	if (rval * input2 > input1) {
	rval--;
	}
	return rval;
	}

	int32_t length(in_t input)
	return number of elements in input list

	int32_t rank(in_t input)
	return rank of an input tensor

	int32_t sum(in_t input[])
	return the sum of values of an input list

	bool isNaN(float input)
	return True if floating-point input value is NaN

	float_t pi()
	returns value of pi

	float_t sin(angle)
	return sine of angle given in radians

	float_t cos(angle)
	return cosine of angle given in radians

	bool power_of_two(int32_t value)
	return true if value is a power of two, false otherwise

	in_out_t maximum_s<Type T>
	return the maximum value when interpreting type T as a signed value as returned by the make_signed helper.

	in_out_t minimum_s<Type T>
	return the minimum value when interpreting type T as a signed value as returned by the make_signed helper.

	in_out_t maximum_u<Type T>
	return the maximum value when interpreting type T as an unsigned value as returned by the make_unsigned helper.

	in_out_t minimum_u<Type T>
	return the minimum value when interpreting type T as an unsigned value as returned by the make_unsigned helper.
	----