Floating-Point Unit (FPU)

Intro

STM AN4044: One alternative to floating-point is fixed-point, where the exponent field is fixed. But if fixed-point is giving better calculation speed on FPU-less 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 the algorithm.

Coding | Dynamic [dB] |
---|---|

Int32 | 192 |

Int64 | 385 |

Single precision | 1529 |

Double precision | 12318 |

The STM32 ARM Cortex M4F MPUs (e.g. STM32WB, STM32F4, STM32L4) have a single precision floating-point unit. The STM32H7 MPUs have a double precision FPU (not supported yet).

Also from STM AN4044

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 FPU-less processor, all these operations are done by software through the C compiler library (or Forth Words) 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 (or Forth) 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.

Any integer with absolute value less than 2^24 can be exactly represented in the single-precision format, and any integer with absolute value less than 2^53 can be exactly represented in the double-precision format.

Mode | Exponent | Exp. Bias | Exp. Range | Mantissa | Decimal digits | Min. value | Max. Value |
---|---|---|---|---|---|---|---|

Single | 8-bit | 127 | -126,+127 | 23-bit | 7.22 | 1.18E-38 | 3.40E38 |

Double | 11-bit | 1023 | -1022,+1023 | 52-bit | 15.95 | 2.23E-308 | 1.8E308 |

It is better to be approximately (vaguely) right than exactly wrong. Carveth Read

- Do not use FPU in interrupt service routines.
- Tasks/Threads with FPU operations need much more return stack depth.
- Rounding is not always working properly. Not useful for precision more than 3.

0.1005e fs. 1.01E-1 ok. 0.1005e fm. 101m ok. 4 set-precision 0.100005e fs. 1.0000E-1 ok. 0.100005e fm. 100.00m ok. 1.00005e f>x x. 1,00004994869232177734375000000000 ok. 1,00005 x. 1,00004999991506338119506835937500 ok.

- https://forth-standard.org/standard/float
- https://en.wikipedia.org/wiki/IEEE_754
- https://en.wikipedia.org/wiki/Single-precision_floating-point_format
- What Every Computer Scientist Should Know About Floating-Point Arithmetic
- Fixed Point, https://en.wikipedia.org/wiki/Q_(number_format)

- https://www.complang.tuwien.ac.at/forth/gforth/Docs-html/Number-Conversion.html#Number-Conversion
- https://www.complang.tuwien.ac.at/forth/gforth/Docs-html/Floating-Point.html#Floating-Point
- https://forth-standard.org/proposals/recognizer#contribution-142
- https://interrupt.memfault.com/blog/cortex-m-rtos-context-switching ARM Cortex-M RTOS Context Switching
- https://mcuoneclipse.com/2019/03/29/be-aware-floating-point-operations-on-arm-cortex-m4f/
- http://support.raisonance.com/content/how-should-i-use-floating-point-unit-fpu-cortex-m4
- Newlib
- The Forth Scientific Library e.g. for ANS Forth Complex Arithmetic Lexicon

No separate floating-point stack. A single precision floating-point number is one cell. The 32-bit base-2 format is officially referred to as binary32 IEEE 754-2008.

f+ ( r1 r2 -- r3 ) Add r1 to r2 giving the sum r3 f- ( r1 r2 -- r3 ) Subtract r2 from r1, giving r3 f* ( r1 r2 -- r3 ) Multiply r1 by r2 giving r3 f/ ( r1 r2 -- r3 ) Divide r1 by r2, giving the quotient r3 fsqrt ( r1 -- r2 ) r2 is the square root of r1 fabs ( r1 -- r2 ) r2 is the absolute value of r1 fnegate ( r1 -- r2 ) r2 is the negation of r1 fround ( r1 -- r2 ) round r1 to an integral value using the "round to nearest" rule, giving r2 floor ( r1 -- r2 ) round r1 to an integral value using the "round toward negative infinity" rule, giving r2 ftrunc ( r1 -- r2 ) round r1 to an integral value using the "round towards zero" rule, giving r2. 10**>f ( n -- r ) raise 10 to the power n, giving product r flog>n ( r -- n ) n is the base-ten logarithm of r fflags@ ( -- u ) get the current value of the Floating Point Status/Control register FPSCR fflags! ( u -- ) assign the given value to the Floating Point Status/Control register FPSCR f0= ( r -- ? ) flag is true if r is equal to zero f0< ( r -- ? ) flag is true if r is less than zero f< ( r1 r2 -- ? ) flag is true if r1 is less than r2 f~ ( r1 r2 r3 -- ? ) If r3 is positive, flag is true if the absolute value of (r1 minus r2) is less than r3 If r3 is zero, flag is true if the implementation-dependent encoding of r1 and r2 are exactly identical (positive and negative zero are unequal if they have distinct encodings). If r3 is negative, flag is true if the absolute value of (r1 minus r2) is less than the absolute value of r3 times the sum of the absolute values of r1 and r2. f>s ( r -- n ) n is the single-cell signed-integer equivalent of the integer portion of r s>f ( n -- r ) r is the floating-point equivalent of the single-cell value n f>x ( r -- x ) x is the fixed-point equivalent of the floating-point r x>f ( x -- r ) r is the floating-point equivalent of the fixed-point x pi ( -- r ) r is pi, approx. 3.14159274101257324 e ( -- r ) r is e, approx. 2.7182818 fnumber (a # -- r u ) convert the specified string by a and # to float r, on success u is 1, otherwise 0 >float (a # -- r ? ) convert the specified string by a and # to float r, on success flag is true f. ( r -- ) display, with a trailing space, the floating-point number r in fixed-point notation fs. ( r -- ) display, with a trailing space, the floating-point number r in scientific notation fe. ( r -- ) display, with a trailing space, the floating-point number r in engineering notation fm. ( r -- ) display, with a trailing space, the floating-point number r in metric unit prefix notation precision ( -- u ) return the number of significant digits currently used by f., fs., fe., or fm. as u set-precision ( u -- ) set the number of significant digits currently used by f., fs., fe., or fm. to u

FPU support without trigonometric, hyperbolic and exponential functions is not even half the battle. Fortunately there is the GNU C math library. C mathematical functions @ Wikipedia

fsin ( r1 -- r2 ) r2 is the sine of the radian angle r1 fcos ( r1 -- r2 ) r2 is the cosine of the radian angle r1 ftan ( r1 -- r2 ) r2 is the principal radian angle whose tangent is r1 fasin ( r1 -- r2 ) r2 is the principal radian angle whose sine is r1 facos ( r1 -- r2 ) r2 is the principal radian angle whose cosine is r1 fatan ( r1 -- r2 ) r2 is the principal radian angle whose tangent is r1 fsincos ( r1 -- r2 r3 ) r2 is the sine of the radian angle r1. r3 is the cosine of the radian angle r1 fatan2 ( r1 r2 -- r3 ) r3 is the principal radian angle (between -π and π) whose tangent is r1/r2 fsinh ( r1 -- r2 ) r2 is the hyperbolic sine of r1 fcosh ( r1 -- r2 ) r2 is the hyperbolic cosine of r1 ftanh ( r1 -- r2 ) r2 is the hyperbolic tangent of r1 fasinh ( r1 -- r2 ) r2 is the floating-point value whose hyperbolic sine is r1 facosh ( r1 -- r2 ) r2 is the floating-point value whose hyperbolic cosine is r1 fatanh ( r1 -- r2 ) r2 is the floating-point value whose hyperbolic tangent is r1 fceil ( r1 -- r2 ) return the smallest integral value that is not less than r1 ffloor ( r1 -- r2 ) Round r1 to an integral value using the "round toward negative infinity" rule, giving r2 fexp ( r1 -- r2 ) raise e to the power r1, giving r2. f** ( r1 r2 -- r3 ) raise r1 to the power r2, giving the product r3 fln ( r1 -- r2 ) r2 is the natural logarithm of r1 flog ( r1 -- r2 ) r2 is the base-ten logarithm of r1

Fixed-point numbers (s31.32) are stored ( n-comma n-whole ) and can be handled like signed double numbers. Because of the name conflict with the floating-point words I changed the names of the fixed-point word and use for fixed-point words x instead of f.

All angles are in degrees.

d+ ( r1 r2 -- r3 ) add r1 to r2 giving the sum r3 d- ( r1 r2 -- r3 ) subtract r2 from r1, giving r3 x* ( r1 r2 -- r3 ) multiply r1 by r2 giving r3 x/ ( r1 r2 -- r3 ) divide r1 by r2, giving the quotient r3 x. ( r -- ) display, with a trailing space, the fixed-point number r x.n ( r n -- ) print a fixed-point number r with n fractional digits (truncated) x#S ( n1 -- n2 ) Adds 32 comma-digits to number output x# ( n1 -- n2 ) Adds one comma-digit to number output

Words from fixpt-mat-lib.fs

sqrt ( r1 -- r2 ) r2 is the square root of r1 sin cos tan asin acos atan log2 log10 ln pow2 pow10 exp floor deg2rad ( deg -- rad ) rad2deg ( rad -- deg ) pi pi/2 pi/4 +inf -inf

Calculation of two parallel resistors:

: f|| ( r1 r2 -- r3) 2dup f* -rot f+ f/ ; 27k 100k f|| fm. 21.3k ok.

2.2n 47k f* dup fm. 103u ok.cutoff frequency

2e pi f* f* 1e swap f/ fm. 1.54k ok.

Mecrisp-Cube has the word `f.`

defined as an assembler routine in fpu.s, but the example here is written in Forth. I use a dot for the decimal separator. Terry Porter "because those crazy Europeans use a comma instead of a decimal point". Not all europeans are crazy, at least the Swiss are an exception ;-), they use decimal points (but not always, for details see https://en.wikipedia.org/wiki/Decimal_separator).

: f. ( r -- ) \ display, with a trailing space, the floating-point number r in fixed-point notation dup f0< if 45 emit fabs then dup $3F000000 \ .5 precision 0 do $41200000 f/ \ 10.0 / loop f+ \ round f>x <# 0 #s 2drop \ integer part 46 hold< \ decimal point precision 0 do x# \ fract digit loop dup #> type space ;

All measurements and calculation are based on the **Cortex M4F MCU STM32WB55 @ 32 MHz**.

Simple test program to estimate execution time of `fsin`

and `fsqrt`

:

: test-fpu ( -- n ) \ test 1000 times sin return n in ms osKernelGetTickCount cr pi 2e f* 1000e f/ \ 2*pi/1000 cr 1000 0 do \ dup i s>f f* drop dup i s>f f* fsin drop \ i . dup i s>f f* fsin fs. cr \ i . dup i s>f f* fsin hex. cr loop drop osKernelGetTickCount swap - ;

With `fsin`

it takes about 7 ms, without about 1 ms for 1000 iterations. Therefore a `fsin`

word takes about 6 us. For the **!STM32F405 @ 168 MHz**, the `fsin`

takes about 2 us.

`fsqrt`

takes also about 2 ms for 1000 iterations. Therefore a `fsqrt`

word takes about 1 us or less (the same time as `f/`

, see below).

Basic operations like `f/`

are defined as inline. First check `fsin`

and `f/`

with the builtin disassembler:

see fsin 08007BE8: B500 push { lr } 08007BEA: 4630 mov r0 r6 08007BEC: F025 bl 0802D694 08007BEE: FD52 08007BF0: 4606 mov r6 r0 08007BF2: BD00 pop { pc } ok.The FPU instructions are unknown to the disassembler

see f/ 0800745A: EE00 0800745C: 6A90 0800745E: CF40 ldmia r7 { r6 } 08007460: EE00 08007462: 6A10 08007464: EE80 08007466: 0A20 08007468: EE10 0800746A: 6A10 0800746C: 4770 bx lrFrom fpu.s on GitHub

@ ----------------------------------------------------------------------------- Wortbirne Flag_foldable_2|Flag_inline, "f/" f_slash: @ ( r1 r2 -- r3 ) Divide r1 by r2, giving the quotient r3. @ ----------------------------------------------------------------------------- vmov s1, tos 1 drop ldmia r7 { r6 } 1 vmov s0, tos 1 vdiv.f32 s0, s0, s1 14 vmov tos, s0 1 bx lr cycles 18About 20 cycles (625 ns @ 32 MHz) for a division, 10 (300 ns) for multiplication, and 5 (150 ns) for +/-.

`vsqrt.f32`

has 14 cycles.

include /fsr/fixpt-math-lib.fs ok. : test-fix ( -- n ) \ test 1000 times fixed-point sin return n in ms osKernelGetTickCount cr \ pi 2e f* 1000e f/ \ 2*pi/1000 360,0 1000,0 x/ cr 1000 0 do \ 2dup i 0 swap x* 2drop \ 2dup i 0 swap x* sin 2drop 2dup i 0 swap x* sqrt 2drop \ i . 2dup i 0 swap x* sin x. cr \ i . 2dup i 0 swap x* sqrt x. cr \ i . 2dup i 0 swap x* sin hex. hex. cr loop 2drop osKernelGetTickCount swap - ; test-fix . 323With

`sqrt`

it takes about 323 ms (`sin`

is not working for me), without about 6 ms. Therefore a `sqrt`

word takes about 317 us, with FPU it takes less than 1 us. A simple multiplication about 6 us (FPU 300 ns).
Only addition and subtraction are comparable:

see d+ 080008B6: CF07 ldmia r7 { r0 r1 r2 } 1 080008B8: 1812 adds r2 r2 r0 1 080008BA: 414E adcs r6 r1 1 080008BC: 3F04 subs r7 #4 1 080008BE: 603A str r2 [ r7 #0 ] 1 080008C0: 4770 bx lr Cycles 5 @ ----------------------------------------------------------------------------- Wortbirne Flag_foldable_2|Flag_inline, "f+" f_add: @ ( r1 r2 -- r3 ) Add r1 to r2 giving the sum r3. @ ----------------------------------------------------------------------------- vmov s1, tos 1 drop 1 vmov s0, tos 1 vadd.f32 s0, s1 1 vmov tos, s0 1 bx lr Cycles 5

**Swift-Forth** on a 64 bit Windows PC @ 3.4 GHz, HW FPU

: test ( -- ) \ test 1'000 times sin, displays time in us ucounter cr pi 2e f* 1000e f/ \ 2*pi/1000 cr 1000 0 do \ fdup i s>f f* fdrop fdup i s>f f* fsin fdrop \ i . fdup i s>f f* fsin fs. cr \ i . fdup i s>f f* fsin hex. hex. cr loop fdrop utimer ;91 us, 28 us -> 63 ns for

`fsin`

. 2 magnitudes faster than Mecrisp-Cube M4F @ 32 MHz
**Gforth** on a 64 bit Linux PC @ Intel I7 8 cores 2.2 GHz, HW FPU

: test ( -- ) \ test 1'000 times sin, displays time in us utime cr pi 2e f* 1000e f/ \ 2*pi/1000 cr 1000 0 do \ fdup i s>f f* fdrop fdup i s>f f* fsin fdrop \ i . fdup i s>f f* fsin fs. cr \ i . fdup i s>f f* fsin hex. hex. cr loop fdrop utime 2swap d- ;64 us, 13 us -> 51 ns for

`fsin`

. 2 magnitudes faster than Mecrisp-Cube M4F @ 32 MHz

As long as you do only elementary arithmetic, fixed- and floating-point have comparable execution time (but division and multiplication is a magnitude slower). But for more elaborate calculation (trigonomteric, exponential functions) the execution time is for fixed-point at least two magnitudes slower.

If time is not an issue in either development or execution, you can easily do without the FPU.

This work by Peter Schmid is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

I | Attachment | History | Action | Size | Date | Who | Comment |
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png | Float_example-header.png | r1 | manage | 28.2 K | 2022-11-05 - 10:45 | PeterSchmid | |

png | buergi-sin.png | r1 | manage | 680.6 K | 2022-12-30 - 16:50 | PeterSchmid | |

png | ieee-754.png | r1 | manage | 12.7 K | 2022-11-02 - 11:52 | PeterSchmid |

Topic revision: r44 - 2022-12-30 - PeterSchmid

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