Classic Maths Libraries

The classic library supports applications being compiled with different maths libraries. Selecting a different library will have implications regarding the size, performance and which machine configuration an application can run in.

Size of floating point number

When compiling with zsdcc a float is always allocated 4 bytes in the C code. However with sccz80, the size of a float depends on which maths library is being used. Traditionally, sccz80 has always allocated a 6 byte block for representing floats, but some maths libraries support a 4 byte (32 bit) float.

-lm - (48 bit genmath) - z80/z80n

Genmath is z88dk's traditional maths library. It provides a 48 bit number, with an 8 bit exponent and a 40 bit mantissa. It utilises a single index register and runs on all of classic's targets (subject to memory). It should be noted that the library makes extensive use of the ixh and ixy registers and as such cannot be used on the Z180 or clones that do not support them,

Genmath can only be used with the sccz80 compiler.

-lmath48 - (48 bit maths by Anders Hejlsberg) - z80/z180/z80n

math48 has been imported into classic from the newlib. It provides a 48 bit number, with an 8 bit exponent and a 40 bit mantissa. It utilises the alternate register set and therefore can't run on all classic targets.

math48 can be used with both sccz80 and zsdcc and is marginally faster than genmath.

BBC Maths - (40 bit maths from BBC BASIC for z80) - z80/z180/z80n

The BBC maths library provides a 32 bit mantissa and 8 bit exponent. It's the same library as the native maths library on the z88. It can be linked with the options: --math-bbc. It can be used on ix/iy switched platforms with -fp-mode=z88 -lbbc_math_iy.

MBF32 - (32 bit maths from Microsoft) - 8080/gbz80/z80/z180/z80n

Support has been added for the (4 byte, 8 bit exponent, 24 bit mantissa) Microsoft single precision library. This is available for machines that run Microsoft BASIC and the appropriate entry points have been located. To date, the following machines are supported:

• VTech Laser 500
• Mitsubishi Multi 8 (10k mode only)
• Mattel Aquarius
• TRS80/Color Genie
• CCE MC-1000

The library can be linked with the following options: --math-mbf32

The source code to the maths subroutines aren't distributed with z88dk, but can be downloaded, compiled and used on any 8080 (or z80 etc) target (See https://github.com/z88dk/z88dk/blob/master/libsrc/math/mbf32/z80/math_mbf32.asm for details).

DAI Math - (32 bit maths extracted from the DAI target) - 8080/z80/z180/z80n

This maths library was extracted from the DAI computer, it offers, a 4 byte, 7 bit exponent, 24 bit mantissa floating point number. It's of particular interest since the whole library is shipped with z88dk and provides an 8080 library.

MBF64 - (64 bit maths from Microsoft) - z80

Support has been added for the (8 byte, 8 bit exponent, 56 bit mantissa) Microsoft double precision library. This is available for machines that run Microsoft BASIC and the appropriate entry points have been located. To date, the following machines are supported:

• VTech Laser 500
• TRS80/Color Genie

mbf64 can be used with sccz80.

-lmath32 - (32 bit maths using IEEE-754 format) - z80/z180/z80n

math32 provides a 32 bit floating point format that is mostly compliant with IEEE-754, which is also the native floating point format of sdcc. The library can be used with both sccz80 and zsdcc.

math32 supports the z180 and ZX Spectrum Next z80n hardware multiply instructions, providing accelerated performance for both these platforms. The z80 CPU is supported through emulation of the hardware multiply format 16_8x8, and also provides good performance.

The intrinsic functions are written in assembler. The higher level functions (trigonometric, exp, pow) are implemented by C functions extracted from the Cephes Maths Library and the Hi-Tech C Floating point library.

More details on the library can be found within the repository.

-lmath16 - (16 bit maths using IEEE-754 format) - z80/z180/z80n

math16 provides a 16 bit floating point format that is mostly compliant with IEEE-754. The library can be used with both sccz80 and zsdcc.

math16 supports the z180 and ZX Spectrum Next z80n hardware multiply instructions, providing accelerated performance for both these platforms. The z80 CPU is supported through emulation of the hardware multiply format 16_8x8, and also provides good performance.

The IEEE 16 bit floating point standard supports a maximum of 3.5 significant digits of accuracy, but it is very fast. It is useful for graphics, as per this Cambridge Z88 example comparing the IEEE16 library with the standard maths library.

The intrinsic functions are written in assembler. The intrinsic type is _Float16 for sccz80, or half_t for both sccz80 or sdcc.

The sccz80 compiler supports _Float16 as a native type, and therefore arithmetic or comparison operations on variables defined as _Float16 or half_t will be done without conversion. The sdcc compiler doesn't support intrinsic 16 bit floating point operations, but functions can still be called as needed.

The math16 library is considered an adjunct library. When printing format conversion and other facilities are required, it is linked together with a main maths library (which provides the conversion routines). The invocation command will typically include --math32 --math16, or -lm --math16 to provide both main and adjunct maths libraries.

More details on the library can be found within the repository.

Machine specific libraries

• -lmathz88 -fp-mode=z88 6 bytes. 8 bit exponent, 32 bit mantissa. Cambridge z88
• -lmz 6 bytes. 8 bit exponent, 32 bit mantissa. ZX Spectrum, ZX81
• -lcpcmath 6 bytes. 8 bit exponent, 32 bit mantissa. Amstrad CPC

All of these libraries use the floating point package located in the machine's ROM. This can result in a compact binary

Maths library aliases

Aliases are provided to make usage of maths libraries straight forward. Including the alias in the zcc invocation will provide all necessary configuration options relevant for each maths library.

classic

• --math-mbf32 is alias for -lmbf32 -pragma-define:CLIB_32BIT_FLOATS=1 -Cc-D__MATH_MBF32 -D__MATH_MBF32
• --math-mbf32_8080 is alias for -lmbf32_8080 -pragma-define:CLIB_32BIT_FLOATS=1 -Cc-D__MATH_MBF32 -D__MATH_MBF32
• --math-mbf64 is alias for -Cc-fp-mode=mbf64 -lmbf64
• --math-bbc is alias for -Cc-fp-mode=z88 -lbbc_math
• --math-dai32 is alias for -Cc-fp-mode=am9511 -ldaimath32 -pragma-define:CLIB_32BIT_FLOATS=1 -Cc-D__MATH_DAI32 -D__MATH_DAI32
• --math-dai32_8080 is alias for -ldaimath32_8080 -pragma-define:CLIB_32BIT_FLOATS=1 -Cc-D__MATH_DAI32 -D__MATH_DAI32

classic + newlib

The --math16 and --math32 aliases are supported across z80, z180, and z80n (SpectrumNext) architectures, and automatically provides each target with the correct hardware multiply opcodes (or for z80 emulation code) to enable best performance.

The math16 library uses 16-bit (integer) memory moves extensively, and therefore optimising code by in-lining memory get and put subroutines has a positive impact on performance, without increasing program size substantially.

• --math32 is alias for -Cc-D__MATH_MATH32 -D__MATH_MATH32 -Cc-fp-mode=ieee -pragma-define:CLIB_32BIT_FLOATS=1 -lmath32
• --math16 is alias for -Cc-D__MATH_MATH16 -D__MATH_MATH16 --opt-code-speed=inlineints -lmath16

Benchmarks

The maths libraries have been lightly benchmarked using a couple of test programs within the repository. These were tested using the classic library and the following compilation line:

zcc +test -DPRINTF [file.c] [maths flags]

As a result, the numbers include the time spent printing, however this isn't particularly time consuming in the overall scheme:

n-body

Library	Compiler	Value 1	Value 2	Ticks
correct values	-->	-0.169075164	-0.169087605
genmath	sccz80	-0.169075164	-0.169087605	3_652_736_949
mbf32 (gbz80)	sccz80	-0.169075083	-0.169086337	2_555_855_304
math48	sccz80	-0.169075164	-0.169087605	2_402_023_498
mbf32 (z80)	sccz80	-0.169075083	-0.169086337	1_936_249_932
daimath32	sccz80	-0.1690751	-0.1690863	1_899_271_269
bbcmath	sccz80	-0.16907516	-0.16908760	1_655_789_776
math32	sccz80	-0.1690752	-0.1690867	1_398_993_950 *
math32_fast	sccz80	-0.1690752	-0.1690867	1_198_780_765 *
math32_z80n (integrated)	sccz80	-0.1690752	-0.1690867	0_576_942_516 *
math32_z180 (integrated)	sccz80	-0.1690752	-0.1690867	0_563_700_933 *

spectral-norm

Library	Compiler	Value	Ticks
correct value	-->	1.274219991
genmath	sccz80	1.274219989	14_817_735_124
math32	zsdcc	1.274219	14_504_079_532
math32	sccz80	1.274219	13_508_416_688
math32_fast	sccz80	1.274219	12_069_049_285*
math48	sccz80	1.274219989	09_035_519_932
bbcmath	sccz80	1.27421999	08_017_859_189
mbf32 (gbz80)	sccz80	1.274220	07_941_220_888
daimath32	sccz80	1.274220	07_483_466_471
mbf32 (z80)	sccz80	1.274220	06_754_491_551
math32_z80n (integrated)	sccz80	1.274219	06_396_544_633
math32_z180 (integrated)	sccz80	1.274219	06_120_760_761

mandelbrot

Library	Compiler	Ticks
genmath	sccz80	3_589_992_847
math48	sccz80	3_323_285_174
math48	zsdcc	3_205_062_412
math32	zsdcc	1_519_179_081
math32	sccz80	1_517_530_881 *
math16	sccz80	1_103_113_465
math32_z80n (integrated)	sccz80	0_922_658_537 *
math32_z180 (integrated)	sccz80	0_892_842_610 *