# GMP Itemized Development Tasks

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This file current as of 21 Apr 2006. An up-to-date version is available at http://swox.com/gmp/tasks.html. Please send comments about this page to gmp-devel@swox.com.

These are itemized GMP development tasks. Not all the tasks listed here are suitable for volunteers, but many of them are. Please see the projects file for more sizeable projects.

#### Correctness and Completeness

- `_LONG_LONG_LIMB` in gmp.h is not namespace clean. Reported by Patrick Pelissier.
  We sort of mentioned `_LONG_LONG_LIMB` in past releases, so need to be careful about changing it. It used to be a define applications had to set for long long limb systems, but that in particular is no longer relevant now that it's established automatically.
- The various reuse.c tests need to force reallocation by calling `_mpz_realloc` with a small (1 limb) size.
- One reuse case is missing from mpX/tests/reuse.c: `mpz_XXX(a,a,a)`.
- When printing `mpf_t` numbers with exponents >2^53 on machines with 64-bit `mp_exp_t`, the precision of `__mp_bases[base].chars_per_bit_exactly` is insufficient and `mpf_get_str` aborts. Detect and compensate. Alternately, think seriously about using some sort of fixed-point integer value. Avoiding unnecessary floating point is probably a good thing in general, and it might be faster on some CPUs.
- Make the string reading functions allow the `0x' prefix when the base is explicitly 16. They currently only allow that prefix when the base is unspecified (zero).
- `mpf_eq` is not always correct, when one operand is 1000000000... and the other operand is 0111111111..., i.e., extremely close. There is a special case in `mpf_sub` for this situation; put similar code in `mpf_eq`. [In progress.]
- `mpf_eq` doesn't implement what gmp.texi specifies. It should not use just whole limbs, but partial limbs. [In progress.]
- `mpf_set_str` doesn't validate it's exponent, for instance garbage 123.456eX789X is accepted (and an exponent 0 used), and overflow of a `long` is not detected.
- `mpf_add` doesn't check for a carry from truncated portions of the inputs, and in that respect doesn't implement the "infinite precision followed by truncate" specified in the manual.
- Windows DLLs: tests/mpz/reuse.c and tests/mpf/reuse.c initialize global variables with pointers to `mpz_add` etc, which doesn't work when those routines are coming from a DLL (because they're effectively function pointer global variables themselves). Need to rearrange perhaps to a set of calls to a test function rather than iterating over an array.
- `mpz_pow_ui`: Detect when the result would be more memory than a `size_t` can represent and raise some suitable exception, probably an alloc call asking for `SIZE_T_MAX`, and if that somehow succeeds then an `abort`. Various size overflows of this kind are not handled gracefully, probably resulting in segvs.
  In `mpz_n_pow_ui`, detect when the count of low zero bits exceeds an `unsigned long`. There's a (small) chance of this happening but still having enough memory to represent the value. Reported by Winfried Dreckmann in for instance `mpz_ui_pow_ui (x, 4UL, 1431655766UL)`.
- `mpf`: Detect exponent overflow and raise some exception. It'd be nice to allow the full `mp_exp_t` range since that's how it's been in the past, but maybe dropping one bit would make it easier to test if e1+e2 goes out of bounds.

#### Machine Independent Optimization

- `mpf_cmp`: For better cache locality, don't test for low zero limbs until the high limbs fail to give an ordering. Reduce code size by turning the three `mpn_cmp`'s into a single loop stopping when the end of one operand is reached (and then looking for a non-zero in the rest of the other).
- `mpf_mul_2exp`, `mpf_div_2exp`: The use of `mpn_lshift` for any size<=prec means repeated `mul_2exp` and `div_2exp` calls accumulate low zero limbs until size==prec+1 is reached. Those zeros will slow down subsequent operations, especially if the value is otherwise only small. If low bits of the low limb are zero, use `mpn_rshift` so as to not increase the size.
- `mpn_dc_sqrtrem`: Don't use `mpn_addmul_1` with multiplier==2, instead either `mpn_addlsh1_n` when available, or `mpn_lshift`+`mpn_add_n` if not.
- `mpn_dc_sqrtrem`, `mpn_sqrtrem2`: Don't use `mpn_add_1` and `mpn_sub_1` for 1 limb operations, instead `ADDC_LIMB` and `SUBC_LIMB`.
- `mpn_sqrtrem2`: Use plain variables for `sp[0]` and `rp[0]` calculations, so the compiler needn't worry about aliasing between `sp` and `rp`.
- `mpn_sqrtrem`: Some work can be saved in the last step when the remainder is not required, as noted in Paul's paper.
- `mpq_add`, `mpq_add`: The division "op1.den / gcd" is done twice, where of course only once is necessary. Reported by Larry Lambe.
- `mpq_add`, `mpq_sub`: The gcd fits a single limb with high probability and in this case `modlimb_invert` could be used to calculate the inverse just once for the two exact divisions "op1.den / gcd" and "op2.den / gcd", rather than letting `mpn_divexact_1` do it each time. This would require a new `mpn_preinv_divexact_1` interface. Not sure if it'd be worth the trouble.
- `mpq_add`, `mpq_sub`: The use of `mpz_mul(x,y,x)` causes temp allocation or copying in `mpz_mul` which can probably be avoided. A rewrite using `mpn` might be best.
- `mpn_gcdext`: Don't test `count_leading_zeros` for zero, instead check the high bit of the operand and avoid invoking `count_leading_zeros`. This is an optimization on all machines, and significant on machines with slow `count_leading_zeros`, though it's possible an already normalized operand might not be encountered very often.
- Rewrite `umul_ppmm` to use floating-point for generating the most significant limb (if `BITS_PER_MP_LIMB` <= 52 bits). (Peter Montgomery has some ideas on this subject.)
- Improve the default `umul_ppmm` code in longlong.h: Add partial products with fewer operations.
- Consider inlining `mpz_set_ui`. This would be both small and fast, especially for compile-time constants, but would make application binaries depend on having 1 limb allocated to an `mpz_t`, preventing the "lazy" allocation scheme below.
- Consider inlining `mpz_[cft]div_ui` and maybe `mpz_[cft]div_r_ui`. A `__gmp_divide_by_zero` would be needed for the divide by zero test, unless that could be left to `mpn_mod_1` (not sure currently whether all the risc chips provoke the right exception there if using mul-by-inverse).
- Consider inlining: `mpz_fits_s*_p`. The setups for `LONG_MAX` etc would need to go into gmp.h, and on Cray it might, unfortunately, be necessary to forcibly include <limits.h> since there's no apparent way to get `SHRT_MAX` with an expression (since `short` and `unsigned short` can be different sizes).
- `mpz_powm` and `mpz_powm_ui` aren't very fast on one or two limb moduli, due to a lot of function call overheads. These could perhaps be handled as special cases.
- `mpz_powm` and `mpz_powm_ui` want better algorithm selection, and the latter should use REDC. Both could change to use an `mpn_powm` and `mpn_redc`.
- `mpz_powm` REDC should do multiplications by `g[]` using the division method when they're small, since the REDC form of a small multiplier is normally a full size product. Probably would need a new tuned parameter to say what size multiplier is "small", as a function of the size of the modulus.
- `mpz_powm` REDC should handle even moduli if possible. Maybe this would mean for m=n*2^k doing mod n using REDC and an auxiliary calculation mod 2^k, then putting them together at the end.
- `mpn_gcd` might be able to be sped up on small to moderate sizes by improving `find_a`, possibly just by providing an alternate implementation for CPUs with slowish `count_leading_zeros`.
- Toom3 could use a low to high cache localized evaluate and interpolate. The necessary `mpn_divexact_by3c` exists.
- `mpf_set_str` produces low zero limbs when a string has a fraction but is exactly representable, eg. 0.5 in decimal. These could be stripped to save work in later operations.
- `mpz_and`, `mpz_ior` and `mpz_xor` should use `mpn_and_n` etc for the benefit of the small number of targets with native versions of those routines. Need to be careful not to pass size==0. Is some code sharing possible between the `mpz` routines?
- `mpf_add`: Don't do a copy to avoid overlapping operands unless it's really necessary (currently only sizes are tested, not whether r really is u or v).
- `mpf_add`: Under the check for v having no effect on the result, perhaps test for r==u and do nothing in that case, rather than currently it looks like an `MPN_COPY_INCR` will be done to reduce prec+1 limbs to prec.
- `mpf_div_ui`: Instead of padding with low zeros, call `mpn_divrem_1` asking for fractional quotient limbs.
- `mpf_div_ui`: Eliminate `TMP_ALLOC`. When r!=u there's no overlap and the division can be called on those operands. When r==u and is prec+1 limbs, then it's an in-place division. If r==u and not prec+1 limbs, then move the available limbs up to prec+1 and do an in-place there.
- `mpf_div_ui`: Whether the high quotient limb is zero can be determined by testing the dividend for high<divisor. When non-zero, the divison can be done on prec dividend limbs instead of prec+1. The result size is also known before the division, so that can be a tail call (once the `TMP_ALLOC` is eliminated).
- `mpn_divrem_2` could usefully accept unnormalized divisors and shift the dividend on-the-fly, since this should cost nothing on superscalar processors and avoid the need for temporary copying in `mpn_tdiv_qr`.
- `mpf_sqrt`: If r!=u, and if u doesn't need to be padded with zeros, then there's no need for the tp temporary.
- `mpq_cmp_ui` could form the `num1*den2` and `num2*den1` products limb-by-limb from high to low and look at each step for values differing by more than the possible carry bit from the uncalculated portion.
- `mpq_cmp` could do the same high-to-low progressive multiply and compare. The benefits of karatsuba and higher multiplication algorithms are lost, but if it's assumed only a few high limbs will be needed to determine an order then that's fine.
- `mpn_add_1`, `mpn_sub_1`, `mpn_add`, `mpn_sub`: Internally use `__GMPN_ADD_1` etc instead of the functions, so they get inlined on all compilers, not just gcc and others with `inline` recognised in gmp.h. `__GMPN_ADD_1` etc are meant mostly to support application inline `mpn_add_1` etc and if they don't come out good for internal uses then special forms can be introduced, for instance many internal uses are in-place. Sometimes a block of code is executed based on the carry-out, rather than using it arithmetically, and those places might want to do their own loops entirely.
- `__gmp_extract_double` on 64-bit systems could use just one bitfield for the mantissa extraction, not two, when endianness permits. Might depend on the compiler allowing `long long` bit fields when that's the only actual 64-bit type.
- tal-notreent.c could keep a block of memory permanently allocated. Currently the last nested `TMP_FREE` releases all memory, so there's an allocate and free every time a top-level function using `TMP` is called. Would need `mp_set_memory_functions` to tell tal-notreent.c to release any cached memory when changing allocation functions though.
- `__gmp_tmp_alloc` from tal-notreent.c could be partially inlined. If the current chunk has enough room then a couple of pointers can be updated. Only if more space is required then a call to some sort of `__gmp_tmp_increase` would be needed. The requirement that `TMP_ALLOC` is an expression might make the implementation a bit ugly and/or a bit sub-optimal.

      #define TMP_ALLOC(n)
        ((ROUND_UP(n) > current->end - current->point ?
           __gmp_tmp_increase (ROUND_UP (n)) : 0),
           current->point += ROUND_UP (n),
           current->point - ROUND_UP (n))

- `__mp_bases` has a lot of data for bases which are pretty much never used. Perhaps the table should just go up to base 16, and have code to generate data above that, if and when required. Naturally this assumes the code would be smaller than the data saved.
- `__mp_bases` field `big_base_inverted` is only used if `USE_PREINV_DIVREM_1` is true, and could be omitted otherwise, to save space.
- `mpz_get_str`, `mtox`: For power-of-2 bases, which are of course fast, it seems a little silly to make a second pass over the `mpn_get_str` output to convert to ASCII. Perhaps combine that with the bit extractions.
- `mpz_gcdext`: If the caller requests only the S cofactor (of A), and A<B, then the code ends up generating the cofactor T (of B) and deriving S from that. Perhaps it'd be possible to arrange to get S in the first place by calling `mpn_gcdext` with A+B,B. This might only be an advantage if A and B are about the same size.
- `mpz_n_pow_ui` does a good job with small bases and stripping powers of 2, but it's perhaps a bit too complicated for what it gains. The simpler `mpn_pow_1` is a little faster on small exponents. (Note some of the ugliness in `mpz_n_pow_ui` is due to supporting `mpn_mul_2`.)
  Perhaps the stripping of 2s in `mpz_n_pow_ui` should be confined to single limb operands for simplicity and since that's where the greatest gain would be.
  Ideally `mpn_pow_1` and `mpz_n_pow_ui` would be merged. The reason `mpz_n_pow_ui` writes to an `mpz_t` is that its callers leave it to make a good estimate of the result size. Callers of `mpn_pow_1` already know the size by separate means (`mp_bases`).
- `mpz_invert` should call `mpn_gcdext` directly.

#### Machine Dependent Optimization

- `invert_limb` on various processors might benefit from the little Newton iteration done for alpha and ia64.
- Alpha 21264: `mpn_addlsh1_n` could be implemented with `mpn_addmul_1`, since that code at 3.5 is a touch faster than a separate `lshift` and `add_n` at 1.75+2.125=3.875. Or very likely some specific `addlsh1_n` code could beat both.
- Alpha 21264: Improve feed-in code for `mpn_mul_1`, `mpn_addmul_1`, and `mpn_submul_1`.
- Alpha 21164: Rewrite `mpn_mul_1`, `mpn_addmul_1`, and `mpn_submul_1` for the 21164. This should use both integer multiplies and floating-point multiplies. For the floating-point operations, the single-limb multiplier should be split into three 21-bit chunks, or perhaps even better in four 16-bit chunks. Probably possible to reach 9 cycles/limb.
- Alpha: GCC 3.4 will introduce `__builtin_ctzl`, `__builtin_clzl` and `__builtin_popcountl` using the corresponding CIX `ct` instructions, and `__builtin_alpha_cmpbge`. These should give GCC more information about sheduling etc than the `asm` blocks currently used in longlong.h and gmp-impl.h.
- Alpha Unicos: Apparently there's no `alloca` on this system, making `configure` choose the slower `malloc-reentrant` allocation method. Is there a better way? Maybe variable-length arrays per notes below.
- Alpha Unicos 21164, 21264: `.align` is not used since it pads with garbage. Does the code get the intended slotting required for the claimed speeds? `.align` at the start of a function would presumably be safe no matter how it pads.
- ARM V5: `count_leading_zeros` can use the `clz` instruction. For GCC 3.4 and up, do this via `__builtin_clzl` since then gcc knows it's "predicable".
- Itanium: GCC 3.4 introduces `__builtin_popcount` which can be used instead of an `asm` block. The builtin should give gcc more opportunities for scheduling, bundling and predication. `__builtin_ctz` similarly (it just uses popcount as per current longlong.h).
- UltraSPARC/64: Optimize `mpn_mul_1`, `mpn_addmul_1`, for s2 < 2^32 (or perhaps for any zero 16-bit s2 chunk). Not sure how much this can improve the speed, though, since the symmetry that we rely on is lost. Perhaps we can just gain cycles when s2 < 2^16, or more accurately, when two 16-bit s2 chunks which are 16 bits apart are zero.
- UltraSPARC/64: Write native `mpn_submul_1`, analogous to `mpn_addmul_1`.
- UltraSPARC/64: Write `umul_ppmm`. Using four "`mulx`"s either with an asm block or via the generic C code is about 90 cycles. Try using fp operations, and also try using karatsuba for just three "`mulx`"s.
- UltraSPARC/32: Rewrite `mpn_lshift`, `mpn_rshift`. Will give 2 cycles/limb. Trivial modifications of mpn/sparc64 should do.
- UltraSPARC/32: Write special mpn_Xmul_1 loops for s2 < 2^16.
- UltraSPARC/32: Use `mulx` for `umul_ppmm` if possible (see commented out code in longlong.h). This is unlikely to save more than a couple of cycles, so perhaps isn't worth bothering with.
- UltraSPARC/32: On Solaris gcc doesn't give us `__sparc_v9__` or anything to indicate V9 support when -mcpu=v9 is selected. See gcc/config/sol2-sld-64.h. Will need to pass something through from ./configure to select the right code in longlong.h. (Currently nothing is lost because `mulx` for multiplying is commented out.)
- UltraSPARC/32: `mpn_divexact_1` and `mpn_modexact_1c_odd` can use a 64-bit inverse and take 64-bits at a time from the dividend, as per the 32-bit divisor case in mpn/sparc64/mode1o.c. This must be done in assembler, since the full 64-bit registers (`%gN`) are not available from C.
- UltraSPARC/32: `mpn_divexact_by3c` can work 64-bits at a time using `mulx`, in assembler. This would be the same as for sparc64.
- UltraSPARC: `modlimb_invert` might save a few cycles from masking down to just the useful bits at each point in the calculation, since `mulx` speed depends on the highest bit set. Either explicit masks or small types like `short` and `int` ought to work.
- Sparc64 HAL R1 `popc`: This chip reputedly implements `popc` properly (see gcc sparc.md). Would need to recognise it as `sparchalr1` or something in configure / config.sub / config.guess. `popc_limb` in gmp-impl.h could use this (per commented out code). `count_trailing_zeros` could use it too.
- PA64: Improve `mpn_addmul_1`, `mpn_submul_1`, and `mpn_mul_1`. The current code runs at 11 cycles/limb. It should be possible to saturate the cache, which will happen at 8 cycles/limb (7.5 for mpn_mul_1). Write special loops for s2 < 2^32; it should be possible to make them run at about 5 cycles/limb.
- PPC601: See which of the power or powerpc32 code runs better. Currently the powerpc32 is used, but only because it's the default for `powerpc*`.
- PPC630: Rewrite `mpn_addmul_1`, `mpn_submul_1`, and `mpn_mul_1`. Use both integer and floating-point operations, possibly two floating-point and one integer limb per loop. Split operands into four 16-bit chunks for fast fp operations. Should easily reach 9 cycles/limb (using one int + one fp), but perhaps even 7 cycles/limb (using one int + two fp).
- PPC630: `mpn_rshift` could do the same sort of unrolled loop as `mpn_lshift`. Some judicious use of m4 might let the two share source code, or with a register to control the loop direction perhaps even share object code.
- Implement `mpn_mul_basecase` and `mpn_sqr_basecase` for important machines. Helping the generic sqr_basecase.c with an `mpn_sqr_diagonal` might be enough for some of the RISCs.
- POWER2/POWER2SC: Schedule `mpn_lshift`/`mpn_rshift`. Will bring time from 1.75 to 1.25 cycles/limb.
- X86: Optimize non-MMX `mpn_lshift` for shifts by 1. (See Pentium code.)
- X86: Good authority has it that in the past an inline `rep movs` would upset GCC register allocation for the whole function. Is this still true in GCC 3? It uses `rep movs` itself for `__builtin_memcpy`. Examine the code for some simple and complex functions to find out. Inlining `rep movs` would be desirable, it'd be both smaller and faster.
- Pentium P54: `mpn_lshift` and `mpn_rshift` can come down from 6.0 c/l to 5.5 or 5.375 by paying attention to pairing after `shrdl` and `shldl`, see mpn/x86/pentium/README.
- Pentium P55 MMX: `mpn_lshift` and `mpn_rshift` might benefit from some destination prefetching.
- PentiumPro: `mpn_divrem_1` might be able to use a mul-by-inverse, hoping for maybe 30 c/l.
- K7: `mpn_lshift` and `mpn_rshift` might be able to do something branch-free for unaligned startups, and shaving one insn from the loop with alternative indexing might save a cycle.
- PPC32: Try using fewer registers in the current `mpn_lshift`. The pipeline is now extremely deep, perhaps unnecessarily deep.
- Fujitsu VPP: Vectorize main functions, perhaps in assembly language.
- Fujitsu VPP: Write `mpn_mul_basecase` and `mpn_sqr_basecase`. This should use a "vertical multiplication method", to avoid carry propagation. splitting one of the operands in 11-bit chunks.
- Pentium: `mpn_lshift` by 31 should use the special rshift by 1 code, and vice versa `mpn_rshift` by 31 should use the special lshift by 1. This would be best as a jump across to the other routine, could let both live in lshift.asm and omit rshift.asm on finding `mpn_rshift` already provided.
- Cray T3E: Experiment with optimization options. In particular, -hpipeline3 seems promising. We should at least up -O to -O2 or -O3.
- Cray: `mpn_com_n` and `mpn_and_n` etc very probably wants a pragma like `MPN_COPY_INCR`.
- Cray vector systems: `mpn_lshift`, `mpn_rshift`, `mpn_popcount` and `mpn_hamdist` are nice and small and could be inlined to avoid function calls.
- Cray: Variable length arrays seem to be faster than the tal-notreent.c scheme. Not sure why, maybe they merely give the compiler more information about aliasing (or the lack thereof). Would like to modify `TMP_ALLOC` to use them, or introduce a new scheme. Memory blocks wanted unconditionally are easy enough, those wanted only sometimes are a problem. Perhaps a special size calculation to ask for a dummy length 1 when unwanted, or perhaps an inlined subroutine duplicating code under each conditional. Don't really want to turn everything into a dog's dinner just because Cray don't offer an `alloca`.
- Cray: `mpn_get_str` on power-of-2 bases ought to vectorize. Does it? `bits_per_digit` and the inner loop over bits in a limb might prevent it. Perhaps special cases for binary, octal and hex would be worthwhile (very possibly for all processors too).
- S390: `BSWAP_LIMB_FETCH` looks like it could be done with `lrvg`, as per glibc sysdeps/s390/s390-64/bits/byteswap.h. This is only for 64-bit mode or something is it, since 32-bit mode has other code? Also, is it worth using for `BSWAP_LIMB` too, or would that mean a store and re-fetch? Presumably that's what comes out in glibc.
- Improve `count_leading_zeros` for 64-bit machines:

      	   if ((x >> 32) == 0) { x <<= 32; cnt += 32; }
      	   if ((x >> 48) == 0) { x <<= 16; cnt += 16; }
      	   ...

- IRIX 6 MIPSpro compiler has an `__inline` which could perhaps be used in `__GMP_EXTERN_INLINE`. What would be the right way to identify suitable versions of that compiler?
- IRIX `cc` is rumoured to have an `_int_mult_upper` (in `<intrinsics.h>` like Cray), but it didn't seem to exist on some IRIX 6.5 systems tried. If it does actually exist somewhere it would very likely be an improvement over a function call to umul.asm.
- `mpn_get_str` final divisions by the base with `udiv_qrnd_unnorm` could use some sort of multiply-by-inverse on suitable machines. This ends up happening for decimal by presenting the compiler with a run-time constant, but the same for other bases would be good. Perhaps use could be made of the fact base<256.
- `mpn_umul_ppmm`, `mpn_udiv_qrnnd`: Return a structure like `div_t` to avoid going through memory, in particular helping RISCs that don't do store-to-load forwarding. Clearly this is only possible if the ABI returns a structure of two `mp_limb_t`s in registers.
  On PowerPC, structures are returned in memory on AIX and Darwin. In SVR4 they're returned in registers, except that draft SVR4 had said memory, so it'd be prudent to check which is done. We can jam the compiler into the right mode if we know how, since all this is purely internal to libgmp. (gcc has an option, though of course gcc doesn't matter since we use inline asm there.)

#### New Functionality

- Maybe add `mpz_crr` (Chinese Remainder Reconstruction).
- Let `0b' and `0B' mean binary input everywhere.
- `mpz_init` and `mpq_init` could do lazy allocation. Set `ALLOC(var)` to 0 to indicate nothing allocated, and let `_mpz_realloc` do the initial alloc. Set `z->_mp_d` to a dummy that `mpz_get_ui` and similar can unconditionally fetch from. Niels Möller has had a go at this.
  The advantages of the lazy scheme would be:
  - Initial allocate would be the size required for the first value stored, rather than getting 1 limb in `mpz_init` and then more or less immediately reallocating.
  - `mpz_init` would only store magic values in the `mpz_t` fields, and could be inlined.
  - A fixed initializer could even be used by applications, like `mpz_t z = MPZ_INITIALIZER;`, which might be convenient for globals.
  The advantages of the current scheme are:
  - `mpz_set_ui` and other similar routines needn't check the size allocated and can just store unconditionally.
  - `mpz_set_ui` and perhaps others like `mpz_tdiv_r_ui` and a prospective `mpz_set_ull` could be inlined.
- Add `mpf_out_raw` and `mpf_inp_raw`. Make sure format is portable between 32-bit and 64-bit machines, and between little-endian and big-endian machines. A format which MPFR can use too would be good.
- `mpn_and_n` ... `mpn_copyd`: Perhaps make the mpn logops and copys available in gmp.h, either as library functions or inlines, with the availability of library functions instantiated in the generated gmp.h at build time.
- `mpz_set_str` etc variants taking string lengths rather than null-terminators.
- `mpz_andn`, `mpz_iorn`, `mpz_nand`, `mpz_nior`, `mpz_xnor` might be useful additions, if they could share code with the current such functions (which should be possible).
- `mpz_and_ui` etc might be of use sometimes. Suggested by Niels Möller.
- `mpf_set_str` and `mpf_inp_str` could usefully accept 0x, 0b etc when base==0. Perhaps the exponent could default to decimal in this case, with a further 0x, 0b etc allowed there. Eg. 0xFFAA@0x5A. A leading "0" for octal would match the integers, but probably something like "0.123" ought not mean octal.
- `GMP_LONG_LONG_LIMB` or some such could become a documented feature of gmp.h, so applications could know whether to `printf` a limb using `%lu` or `%Lu`.
- `GMP_PRIdMP_LIMB` and similar defines following C99 <inttypes.h> might be of use to applications printing limbs. But if `GMP_LONG_LONG_LIMB` or whatever is added then perhaps this can easily enough be left to applications.
- `gmp_printf` could accept `%b` for binary output. It'd be nice if it worked for plain `int` etc too, not just `mpz_t` etc.
- `gmp_printf` in fact could usefully accept an arbitrary base, for both integer and float conversions. A base either in the format string or as a parameter with `*` should be allowed. Maybe `&13b` (b for base) or something like that.
- `gmp_printf` could perhaps accept `mpq_t` for float conversions, eg. `"%.4Qf"`. This would be merely for convenience, but still might be useful. Rounding would be the same as for an `mpf_t` (ie. currently round-to-nearest, but not actually documented). Alternately, perhaps a separate `mpq_get_str_point` or some such might be more use. Suggested by Pedro Gimeno.
- `mpz_rscan0` or `mpz_revscan0` or some such searching towards the low end of an integer might match `mpz_scan0` nicely. Likewise for `scan1`. Suggested by Roberto Bagnara.
- `mpz_bit_subset` or some such to test whether one integer is a bitwise subset of another might be of use. Some sort of return value indicating whether it's a proper or non-proper subset would be good and wouldn't cost anything in the implementation. Suggested by Roberto Bagnara.
- `mpf_get_ld`, `mpf_set_ld`: Conversions between `mpf_t` and `long double`, suggested by Dan Christensen. Other `long double` routines might be desirable too, but `mpf` would be a start.
  `long double` is an ANSI-ism, so everything involving it would need to be suppressed on a K&R compiler.
  There'd be some work to be done by `configure` to recognise the format in use, MPFR has a start on this. Often `long double` is the same as `double`, which is easy but pretty pointless. A single float format detector macro could look at `double` then `long double`
  Sometimes there's a compiler option for the size of a `long double`, eg. xlc on AIX can use either 64-bit or 128-bit. It's probably simplest to regard this as a compiler compatibility issue, and leave it to users or sysadmins to ensure application and library code is built the same.
- `mpz_sqrt_if_perfect_square`: When `mpz_perfect_square_p` does its tests it calculates a square root and then discards it. For some applications it might be useful to return that root. Suggested by Jason Moxham.
- `mpz_get_ull`, `mpz_set_ull`, `mpz_get_sll`, `mpz_get_sll`: Conversions for `long long`. These would aid interoperability, though a mixture of GMP and `long long` would probably not be too common. Since `long long` is not always available (it's in C99 and GCC though), disadvantages of using `long long` in libgmp.a would be
  - Library contents vary according to the build compiler.
  - gmp.h would need an ugly `#ifdef` block to decide if the application compiler could take the `long long` prototypes.
  - Some sort of `LIBGMP_HAS_LONGLONG` might be wanted to indicate whether the functions are available. (Applications using autoconf could probe the library too.)
  It'd be possible to defer the need for `long long` to application compile time, by having something like `mpz_set_2ui` called with two halves of a `long long`. Disadvantages of this would be,
  - Bigger code in the application, though perhaps not if a `long long` is normally passed as two halves anyway.
  - `mpz_get_ull` would be a rather big inline, or would have to be two function calls.
  - `mpz_get_sll` would be a worse inline, and would put the treatment of `-0x10..00` into applications (see `mpz_get_si` correctness above).
  - Although having libgmp.a independent of the build compiler is nice, it sort of sacrifices the capabilities of a good compiler to uniformity with inferior ones.
  Plain use of `long long` is probably the lesser evil, if only because it makes best use of gcc. In fact perhaps it would suffice to guarantee `long long` conversions only when using GCC for both application and library. That would cover free software, and we can worry about selected vendor compilers later.
  In C++ the situation is probably clearer, we demand fairly recent C++ so `long long` should be available always. We'd probably prefer to have the C and C++ the same in respect of `long long` support, but it would be possible to have it unconditionally in gmpxx.h, by some means or another.
- `mpz_strtoz` parsing the same as `strtol`. Suggested by Alexander Kruppa.

#### Configuration

- Alpha ev7, ev79: Add code to config.guess to detect these. Believe ev7 will be "3-1307" in the current switch, but need to verify that. (On OSF, current configfsf.guess identifies ev7 using psrinfo, we need to do it ourselves for other systems.)
- Alpha OSF: Libtool (version 1.5) doesn't seem to recognise this system is "pic always" and ends up running gcc twice with the same options. This is wasteful, but harmless. Perhaps a newer libtool will be better.
- ARM: `umul_ppmm` in longlong.h always uses `umull`, but is that available only for M series chips or some such? Perhaps it should be configured in some way.
- HPPA: config.guess should recognize 7000, 7100, 7200, and 8x00.
- HPPA: gcc 3.2 introduces a `-mschedule=7200` etc parameter, which could be driven by an exact hppa cpu type.
- Mips: config.guess should say mipsr3000, mipsr4000, mipsr10000, etc. "hinv -c processor" gives lots of information on Irix. Standard config.guess appends "el" to indicate endianness, but `AC_C_BIGENDIAN` seems the best way to handle that for GMP.
- PowerPC: The function descriptor nonsense for AIX is currently driven by `*-*-aix*`. It might be more reliable to do some sort of feature test, examining the compiler output perhaps. It might also be nice to merge the aix.m4 files into powerpc-defs.m4.
- config.m4 is generated only by the configure script, it won't be regenerated by config.status. Creating it as an `AC_OUTPUT` would work, but it might upset "make" to have things like `L$` get into the Makefiles through `AC_SUBST`. `AC_CONFIG_COMMANDS` would be the alternative. With some careful m4 quoting the `changequote` calls might not be needed, which might free up the order in which things had to be output.
- Automake: Latest automake has a `CCAS`, `CCASFLAGS` scheme. Though we probably wouldn't be using its assembler support we could try to use those variables in compatible ways.
- `GMP_LDFLAGS` could probably be done with plain `LDFLAGS` already used by automake for all linking. But with a bit of luck the next libtool will pass pretty much all `CFLAGS` through to the compiler when linking, making `GMP_LDFLAGS` unnecessary.
- mpn/Makeasm.am uses `-c` and `-o` together in the .S and .asm rules, but apparently that isn't completely portable (there's an autoconf `AC_PROG_CC_C_O` test for it). So far we've not had problems, but perhaps the rules could be rewritten to use "foo.s" as the temporary, or to do a suitable "mv" of the result. The only danger from using foo.s would be if a compile failed and the temporary foo.s then looked like the primary source. Hopefully if the `SUFFIXES` are ordered to have .S and .asm ahead of .s that wouldn't happen. Might need to check.

#### Random Numbers

- `_gmp_rand` is not particularly fast on the linear congruential algorithm and could stand various improvements.
  - Make a second seed area within `gmp_randstate_t` (or `_mp_algdata` rather) to save some copying.
  - Make a special case for a single limb `2exp` modulus, to avoid `mpn_mul` calls. Perhaps the same for two limbs.
  - Inline the `lc` code, to avoid a function call and `TMP_ALLOC` for every chunk.
  - Perhaps the `2exp` and general LC cases should be split, for clarity (if the general case is retained).
- `gmp_randstate_t` used for parameters perhaps should become `gmp_randstate_ptr` the same as other types.
- Some of the empirical randomness tests could be included in a "make check". They ought to work everywhere, for a given seed at least.

#### C++

- `mpz_class(string)`, etc: Use the C++ global locale to identify whitespace.
  `mpf_class(string)`: Use the C++ global locale decimal point, rather than the C one.
  Consider making these variant `mpz_set_str` etc forms available for `mpz_t` too, not just `mpz_class` etc.
- `mpq_class operator+=`: Don't emit an unnecssary `mpq_set(q,q)` before `mpz_addmul` etc.
- Put various bits of gmpxx.h into libgmpxx, to avoid excessive inlining. Candidates for this would be,
  - `mpz_class(const char *)`, etc: since they're normally not fast anyway, and we can hide the exception `throw`.
  - `mpz_class(string)`, etc: to hide the `cstr` needed to get to the C conversion function.
  - `mpz_class string, char*` etc constructors: likewise to hide the throws and conversions.
  - `mpz_class::get_str`, etc: to hide the `char*` to `string` conversion and free. Perhaps `mpz_get_str` can write directly into a `string`, to avoid copying.
    Consider making such `string` returning variants available for use with plain `mpz_t` etc too.

#### Miscellaneous

- `mpz_gcdext` and `mpn_gcdext` ought to document what range of values the generated cofactors can take, and preferably ensure the definition uniquely specifies the cofactors for given inputs. A basic extended Euclidean algorithm or multi-step variant leads to |x|<|b| and |y|<|a| or something like that, but there's probably two solutions under just those restrictions.
- demos/factorize.c: use `mpz_divisible_ui_p` rather than `mpz_tdiv_qr_ui`. (Of course dividing multiple primes at a time would be better still.)
- The various test programs use quite a bit of the main `libgmp`. This establishes good cross-checks, but it might be better to use simple reference routines where possible. Where it's not possible some attention could be paid to the order of the tests, so a `libgmp` routine is only used for tests once it seems to be good.
- `MUL_FFT_THRESHOLD` etc: the FFT thresholds should allow a return to a previous k at certain sizes. This arises basically due to the step effect caused by size multiples effectively used for each k. Looking at a graph makes it fairly clear.
- `__gmp_doprnt_mpf` does a rather unattractive round-to-nearest on the string returned by `mpf_get_str`. Perhaps some variant of `mpf_get_str` could be made which would better suit.

#### Aids to Development

- Add `ASSERT`s at the start of each user-visible mpz/mpq/mpf function to check the validity of each `mp?_t` parameter, in particular to check they've been `mp?_init`ed. This might catch elementary mistakes in user programs. Care would need to be taken over `MPZ_TMP_INIT`ed variables used internally. If nothing else then consistency checks like size<=alloc, ptr not `NULL` and ptr+size not wrapping around the address space, would be possible. A more sophisticated scheme could track `_mp_d` pointers and ensure only a valid one is used. Such a scheme probably wouldn't be reentrant, not without some help from the system.
- tune/time.c could try to determine at runtime whether `getrusage` and `gettimeofday` are reliable. Currently we pretend in configure that the dodgy m68k netbsd 1.4.1 `getrusage` doesn't exist. If a test might take a long time to run then perhaps cache the result in a file somewhere.
- tune/time.c could choose the default precision based on the `speed_unittime` determined, independent of the method in use.
- Cray vector systems: CPU frequency could be determined from `sysconf(_SC_CLK_TCK)`, since it seems to be clock cycle based. Is this true for all Cray systems? Would like some documentation or something to confirm.

#### Documentation

- `mpz_inp_str` (etc) doesn't say when it stops reading digits.
- `mpn_get_str` isn't terribly clear about how many digits it produces. It'd probably be possible to say at most one leading zero, which is what both it and `mpz_get_str` currently do. But want to be careful not to bind ourselves to something that might not suit another implementation.
- `va_arg` doesn't do the right thing with `mpz_t` etc directly, but instead needs a pointer type like `MP_INT*`. It'd be good to show how to do this, but we'd either need to document `mpz_ptr` and friends, or perhaps fallback on something slightly nasty with `void*`.

#### Bright Ideas

The following may or may not be feasible, and aren't likely to get done in the near future, but are at least worth thinking about.

- Reorganize longlong.h so that we can inline the operations even for the system compiler. When there is no such compiler feature, make calls to stub functions. Write such stub functions for as many machines as possible.
- longlong.h could declare when it's using, or would like to use, `mpn_umul_ppmm`, and the corresponding umul.asm file could be included in libgmp only in that case, the same as is effectively done for `__clz_tab`. Likewise udiv.asm and perhaps cntlz.asm. This would only be a very small space saving, so perhaps not worth the complexity.
- longlong.h could be built at configure time by concatenating or #including fragments from each directory in the mpn path. This would select CPU specific macros the same way as CPU specific assembler code. Code used would no longer depend on cpp predefines, and the current nested conditionals could be flattened out.
- `mpz_get_si` returns 0x80000000 for -0x100000000, whereas it's sort of supposed to return the low 31 (or 63) bits. But this is undocumented, and perhaps not too important.
- `mpz_init_set*` and `mpz_realloc` could allocate say an extra 16 limbs over what's needed, so as to reduce the chance of having to do a reallocate if the `mpz_t` grows a bit more. This could only be an option, since it'd badly bloat memory usage in applications using many small values.
- `mpq` functions could perhaps check for numerator or denominator equal to 1, on the assumption that integers or denominator-only values might be expected to occur reasonably often.
- `count_trailing_zeros` is used on more or less uniformly distributed numbers in a couple of places. For some CPUs `count_trailing_zeros` is slow and it's probably worth handling the frequently occurring 0 to 2 trailing zeros cases specially.
- `mpf_t` might like to let the exponent be undefined when size==0, instead of requiring it 0 as now. It should be possible to do size==0 tests before paying attention to the exponent. The advantage is not needing to set exp in the various places a zero result can arise, which avoids some tedium but is otherwise perhaps not too important. Currently `mpz_set_f` and `mpf_cmp_ui` depend on exp==0, maybe elsewhere too.
- `__gmp_allocate_func`: Could use GCC `__attribute__ ((malloc))` on this, though don't know if it'd do much. GCC 3.0 allows that attribute on functions, but not function pointers (see info node "Attribute Syntax"), so would need a new autoconf test. This can wait until there's a GCC that supports it.
- `mpz_add_ui` contains two `__GMPN_COPY`s, one from `mpn_add_1` and one from `mpn_sub_1`. If those two routines were opened up a bit maybe that code could be shared. When a copy needs to be done there's no carry to append for the add, and if the copy is non-empty no high zero for the sub.

#### Old and Obsolete Stuff

The following tasks apply to chips or systems that are old and/or obsolete. It's unlikely anything will be done about them unless anyone is actively using them.

- Sparc32: The integer based udiv_nfp.asm used to be selected by `configure --nfp` but that option is gone now that autoconf is used. The file could go somewhere suitable in the mpn search if any chips might benefit from it, though it's possible we don't currently differentiate enough exact cpu types to do this properly.
- VAX D and G format `double` floats are straightforward and could perhaps be handled directly in `__gmp_extract_double` and maybe in `mpn_get_d`, rather than falling back on the generic code. (Both formats are detected by `configure`.)