ST AN4044 APPLICATION NOTE

AN4044

Application note

Using floating-point unit (FPU) with

STM32F405/07xx and STM32F415/417xx microcontrollers

Introduction

This application note explains how to use floating-point units (FPU) with STM32F405/07xx
and STM32F415/417xx microcontrollers and provides a short overview of:

■ Floating-point arithmetic

■ STM32F405/07xx and STM32F415/417xx family floating-point unit

An application example is given at the end of this application note.

Ta bl e 1 lists the microcontrollers and development tools concerned by this application note.

Table 1. Applicable products and tools

Type Applicable products

Microcontrollers

Development tools STM3240G-EVAL evaluation board

STM32F405/07xx and STM32F415/417xx high-performance MCUs with
DSP and FPU instructions

March 2012 Doc ID 022737 Rev 1 1/21

www.st.com

Contents AN4044

Contents

1 Floating-point arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Fixed-point or floating-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Floating-point unit (FPU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 IEEE standard for floating-point arithmetic (IEEE 754) . . . . . . . . . . . . . 7

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Number formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Normalized numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Denormalized numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.4 Infinites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.5 NaN (Not-a-Number) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Rounding modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Number conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Exception and exception handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 STM32F4 floating-point unit (FPU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Special operating modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Floating-point status and control register (FPSCR) . . . . . . . . . . . . . . . . . 12

3.2.1 Code condition bits: N, Z, C, V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 Mode bits: AHP, DN, FZ, RM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.3 Exception flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Exception management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Programmers model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 FPU instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5.1 FPU arithmetic instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5.2 FPU compare & convert instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5.3 FPU load/store instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Application example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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AN4044 Contents

4.1 Julia set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Implementation on STM32F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Reference documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Doc ID 022737 Rev 1 3/21

List of tables AN4044

List of tables

Table 1. Applicable products and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Table 2. Integer numbers dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Table 3. Floating-point numbers dynamic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Table 4. Normalized numbers range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Table 5. Denormalized numbers range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Table 6. Value range for IEEE.754 number formats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Table 7. FPSCR register. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Table 8. Performance comparison with MDK-ARM version 4.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Table 9. Reference documents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Table 10. Document revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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AN4044 Floating-point arithmetic

1 Floating-point arithmetic

Floating-point numbers are used to represent non-integer numbers. They are composed of
3 fields:

● the sign

● the exponent

● the mantissa

Such a representation allows a very wide range of number coding, making floating-point
numbers the best way to deal with real numbers. Floating-point calculations can be
accelerated using a Floating-point unit (FPU) integrated in the processor.

1.1 Fixed-point or floating-point

One alternative to floating-point is fixed-point, where the exponent field is fixed. But if fixed­point is giving better calculation speed on FPUless processors, the range of numbers and
their dynamic is low. As a consequence, a developer using the fixed-point technique will
have to check carefully any scaling/saturation issues in its algorithm.

Table 2. Integer numbers dynamic

Coding Dynamic

Int8 48 dB
Int16 96 dB
Int32 192 dB
Int64 385 dB

The C language offers the float and the double types for floating-point operations. At a
higher level, modelization tools, such as matlab or scilab, are generating C code mainly
using float or double. No floating-point support means modifying the generated code to
adapt it to fixed-point. And all the fixed-point operations have to be handcoded by the
programmer.

Table 3. Floating-point numbers dynamic

Coding Dynamic

half precision 180 dB
single precision 1529 dB
double precision 12318 dB

When used natively in code, floating-point operations will decrease the development time of
a project. It is the most efficient way to implement any mathematical algorithm.

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Floating-point arithmetic AN4044

1.2 Floating-point unit (FPU)

Floating-point calculations require a lot of resources, as for any operation between two
numbers. For example, we need to:

● align the two numbers (have them with the same exponent)

● perform the operation

● round out the result

● code the result

On an FPUless processor, all these operations are done by software through the C compiler
library and are not visible to the programmer; but the performances are very low.

On a processor having an FPU, all of the operations are entirely done by hardware in a
single cycle, for most of the instructions. The C compiler does not use its own floating-point
library but directly generates FPU native instructions.

When implementing a mathematical algorithm on a microprocessor having an FPU, the
programmer does not have to choose between performance and development time. The
FPU brings reliability allowing to use directly any generated code through a high level tool,
such as matlab or scilab, with the highest level of performance.

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AN4044 IEEE standard for floating-point arithmetic (IEEE 754)

2 IEEE standard for floating-point arithmetic (IEEE 754)

The usage of the floating-point arithmetic has always been a need in computer science
since the early ages. At the end of the 30’s, when Konrad Zuse developed his Z series in
Germany, floating-points were already in. But the complexity of implementing a hardware
support for the floating-point arithmetic has discarded its usage for decades.

In the mid 50’s, IBM, with its 704, introduced the FPU in mainframes; and in the 70’s, various
platforms were supporting floating-point operations but with their own coding techniques.
The unification took place in 1985 when the IEEE published the standard 754 to define a
common approach for floating-point arithmetic support.

2.1 Overview

The various types of floating-point implementations over the years led the IEEE to
standardize the following elements:

● number formats

● arithmetic operations

● number conversions

● special values coding

● 4 rounding modes

● 5 exceptions and their handling

2.2 Number formats

All values are composed of three fields:

● Sign: s

● Biased exponent:

– sum of the exponent = e
– constant value = bias

● Fraction (or mantissa): f

The values can be coded on various lengths:

● 16-bit: half precision format

● 32-bit: single precision format

● 64-bit: double precision format

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