GMP: Multi-precision Arithmetic

8.16.0.1

GMP: Multi-precision Arithmetic🔗ℹ

This library provides safe low-level bindings to GMP, the GNU Multi-Precision Arithmetic library. This manual assumes familiarity with the relevant parts of the GMP manual.

(require gmp)

package: gmp-lib

This package is distributed under the LGPL. Note that the GMP native library is dual-licensed under the LGPLv3 and the GPLv2 (see copying conditions for details).

1 mpz: Multi-precision Integers🔗ℹ

An mpz (mpz_t) is a mutable value containing an arbitrary-precision integer. In this library, such values are created with mpz and recognized with mpz?. Their backing storage is automatically initialized and garbage-collected.

procedure
(mpz? v) → boolean?
v : any/c

Returns #t if v is an mpz value, #f otherwise.

procedure
(mpz [n]) → mpz?
n : (or/c exact-integer? mpz?) = 0

Creates a new mpz initialized with the value n. If n is an mpz, then its value is copied; n is unchanged and does not share storage with the result.

procedure
(mpz-set! z n) → void?
z : mpz?
n : exact-integer?

Sets z to contain the value n.

procedure
(mpz->number z) → exact-integer?
z : mpz?

Returns the integer value stored in z.

procedure
(mpz-zero? z) → boolean?
  z : mpz?
procedure
(mpz-positive? z) → boolean?
  z : mpz?
procedure
(mpz-negative? z) → boolean?
  z : mpz?

Like zero?, positive?, and negative?, respectively.

procedure
(mpz=? z1 z2) → boolean?
z1 : mpz?
z2 : mpz?

Returns #t if z1 and z2 contain equal integers, #f otherwise.

procedure
(mpz->bytes z
size
signed?
[ big-endian?
dest-buf]
dest-start) → bytes?
  z : mpz?
  size : (or/c exact-positive-integer? #f)
  signed? : boolean?
  big-endian? : boolean? = #t
  dest-buf : bytes? = (make-bytes size)
  dest-start : 0
procedure
(bytes->mpz buf signed? [big-endian? start end]) → mpz?
  buf : bytes?
  signed? : boolean?
  big-endian? : boolean? = #t
  start : exact-nonnegative-integer? = 0
  end : exact-nonnegative-integer? = (bytes-length buf)

Like integer->integer-bytes and integer-bytes->integer, respectively. The size argument to mpz->bytes can be any positive integer (as long as it is large enough to represent z), or it may be #f to automatically use the shortest suitable size.

procedure
(mpz-bytes-length z signed?) → exact-positive-integer?
z : mpz?
signed? : boolean?

Computes the number of bytes needed by mpz->bytes to represent z.

1.1 Operations on mpz Values🔗ℹ

The following operations are supported. By GMP convention, destination arguments come first, followed by source operands.

procedure
(mpz_set rop op) → void?
  rop : mpz?
  op : mpz?
procedure
(mpz_set_ui rop op) → void?
  rop : mpz?
  op : unsigned-long
procedure
(mpz_set_si rop op) → void?
  rop : mpz?
  op : signed-long
procedure
(mpz_set_d rop op) → void?
  rop : mpz?
  op : rational?
procedure
(mpz_set_q rop op) → void?
  rop : mpz?
  op : mpq?
procedure
(mpz_set_str rop op) → void?
  rop : mpz
  op : string?
procedure
(mpz_swap rop11 rop2) → void?
  rop11 : mpz?
  rop2 : mpz?

See Assignment Functions in the GMP manual.

procedure
(mpz_get_ui op) → exact-nonnegative-integer?
  op : mpz?
procedure
(mpz_get_si op) → exact-integer?
  op : mpz?
procedure
(mpz_get_d op) → rational?
  op : mpz?
procedure
(mpz_get_d_2exp op) →
rational? exact-integer?
  op : mpz?

See Conversion Functions in the GMP manual.

procedure
(mpz_add rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_add_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_sub rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_sub_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_ui_sub rop op1 op2) → void?
  rop : mpz?
  op1 : unsigned-long
  op2 : mpz?
procedure
(mpz_mul rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_mul_si rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : signed-long
procedure
(mpz_mul_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_addmul rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_addmul_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_submul rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_submul_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_mul_2exp rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_neg rop op) → void?
  rop : mpz?
  op : mpz?
procedure
(mpz_abs rop op) → void?
  rop : mpz?
  op : mpz?

See Arithmetic Functions in the GMP manual.

procedure
(mpz_cdiv_q q n d) → void?
  q : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_cdiv_r r n d) → void?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_cdiv_qr q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_cdiv_q_ui q n d) → void?
  q : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_cdiv_r_ui r n d) → void?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_cdiv_qr_ui q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_cdiv_ui n d) → void?
  n : mpz?
  d : unsigned-long
procedure
(mpz_cdiv_q_2exp q n b) → void?
  q : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_cdiv_r_2exp r n b) → void?
  r : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_fdiv_q q n d) → void?
  q : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_fdiv_r r n d) → void?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_fdiv_qr q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_fdiv_q_ui q n d) → void?
  q : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_fdiv_r_ui r n d) → void?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_fdiv_qr_ui q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_fdiv_ui n d) → void?
  n : mpz?
  d : unsigned-long
procedure
(mpz_fdiv_q_2exp q n b) → void?
  q : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_fdiv_r_2exp r n b) → void?
  r : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_tdiv_q q n d) → void?
  q : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_tdiv_r r n d) → void?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_tdiv_qr q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_tdiv_q_ui q n d) → void?
  q : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_tdiv_r_ui r n d) → void?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_tdiv_qr_ui q r n d) → void?
  q : mpz?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_tdiv_ui n d) → void?
  n : mpz?
  d : unsigned-long
procedure
(mpz_tdiv_q_2exp q n b) → void?
  q : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_tdiv_r_2exp r n b) → void?
  r : mpz?
  n : mpz?
  b : unsigned-long
procedure
(mpz_mod r n d) → void?
  r : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_mod_ui r n d) → void?
  r : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_divexact q n d) → void?
  q : mpz?
  n : mpz?
  d : mpz?
procedure
(mpz_divexact_ui q n d) → void?
  q : mpz?
  n : mpz?
  d : unsigned-long
procedure
(mpz_divisible_p n d) → exact-integer?
  n : mpz?
  d : mpz?
procedure
(mpz_divisible_ui_p n d) → exact-integer?
  n : mpz?
  d : unsigned-long
procedure
(mpz_divisible_2exp_p n b) → exact-integer?
  n : mpz?
  b : unsigned-long
procedure
(mpz_congruent n c d) → exact-integer?
  n : mpz?
  c : mpz?
  d : mpz?
procedure
(mpz_congruent_ui_p n c d) → exact-integer?
  n : mpz?
  c : mpz?
  d : unsigned-long
procedure
(mpz_congruent_2exp_p n c b) → exact-integer?
  n : mpz?
  c : mpz?
  b : unsigned-long

See Division Functions in the GMP manual.

procedure
(mpz_pow_ui rop base exp) → void?
  rop : mpz?
  base : mpz?
  exp : unsigned-long
procedure
(mpz_ui_pow_ui rop base exp) → void?
  rop : mpz?
  base : unsigned-long
  exp : unsigned-long

See Exponentiation Functions in the GMP manual.

procedure
(mpz_root root u n) → void?
  root : mpz?
  u : mpz?
  n : unsigned-long
procedure
(mpz_rootrem root rem u n) → void?
  root : mpz?
  rem : mpz?
  u : mpz?
  n : unsigned-long
procedure
(mpz_sqrt root op) → void?
  root : mpz?
  op : mpz?
procedure
(mpz_sqrtrem root rem op) → void?
  root : mpz?
  rem : mpz?
  op : mpz?
procedure
(mpz_perfect_power_p op) → exact-integer?
  op : mpz?
procedure
(mpz_perfect_square_p op) → exact-integer?
  op : mpz?

See Root Extraction Functions in the GMP manual.

procedure
(mpz_probab_prime_p n reps) → exact-integer?
  n : mpz?
  reps : signed-integer
procedure
(mpz_nextprime rop op) → void?
  rop : mpz?
  op : mpz?
procedure
(mpz_gcd rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_gcd_ui rop op1 op2) → unsigned-long
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_gcdext g s t a b) → void?
  g : mpz?
  s : mpz?
  t : mpz?
  a : mpz?
  b : mpz?
procedure
(mpz_lcm rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_lcm_ui rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_invert rop op1 op2) → exact-integer?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_kronecker a b) → exact-integer?
  a : mpz?
  b : mpz?
procedure
(mpz_kronecker_si a b) → exact-integer?
  a : mpz?
  b : signed-long
procedure
(mpz_kronecker_ui a b) → exact-integer?
  a : mpz?
  b : unsigned-long
procedure
(mpz_si_kronecker a b) → exact-integer?
  a : signed-long
  b : mpz?
procedure
(mpz_ui_kronecker a b) → exact-integer?
  a : unsigned-long
  b : mpz?
procedure
(mpz_remove rop op f) → exact-nonnegative-integer?
  rop : mpz?
  op : mpz?
  f : mpz?
procedure
(mpz_fac_ui rop op) → void?
  rop : mpz?
  op : mpz?
procedure
(mpz_primordial_ui rop n) → void?
  rop : mpz?
  n : unsigned-long
procedure
(mpz_bin_ui rop n k) → void?
  rop : mpz?
  n : mpz?
  k : unsigned-long
procedure
(mpz_bin_uiui rop n k) → void?
  rop : mpz?
  n : unsigned-long
  k : unsigned-long
procedure
(mpz_fib_ui fn n) → void?
  fn : mpz?
  n : unsigned-long
procedure
(mpz_fib2_ui fn fnsub1 n) → void?
  fn : mpz?
  fnsub1 : mpz?
  n : unsigned-long
procedure
(mpz_lucnum_ui ln n) → void?
  ln : mpz?
  n : unsigned-long
procedure
(mpz_lucnum2_ui ln lnsub1 n) → void?
  ln : mpz?
  lnsub1 : mpz?
  n : unsigned-long

See Number Theoretic Functions in the GMP manual.

procedure
(mpz_cmp op1 op2) → exact-integer?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_cmp_d op1 op2) → exact-integer?
  op1 : mpz?
  op2 : real?
procedure
(mpz_cmp_si op1 op2) → exact-integer?
  op1 : mpz?
  op2 : signed-long
procedure
(mpz_cmp_ui op1 op2) → exact-integer?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_cmpabs op1 op2) → exact-integer?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_cmpabs_d op1 op2) → exact-integer?
  op1 : mpz?
  op2 : real?
procedure
(mpz_cmpabs_ui op1 op2) → exact-integer?
  op1 : mpz?
  op2 : unsigned-long
procedure
(mpz_sgn op) → (or/c -1 0 1)
  op : mpz?

See Comparison Functions in the GMP manual.

procedure
(mpz_and rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_ior rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_xor rop op1 op2) → void?
  rop : mpz?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_com rop op) → void?
  rop : mpz?
  op : mpz?
procedure
(mpz_popcount op) → exact-nonnegative-integer?
  op : mpz?
procedure
(mpz_hamdist op1 op2) → exact-nonnegative-integer?
  op1 : mpz?
  op2 : mpz?
procedure
(mpz_scan0 op start-bit) → exact-nonnegative-integer?
  op : mpz?
  start-bit : exact-nonnegative-integer?
procedure
(mpz_scan1 op start-bit) → exact-nonnegative-integer?
  op : mpz?
  start-bit : exact-nonnegative-integer?
procedure
(mpz_clrbit rop bit-index) → void?
  rop : mpz?
  bit-index : exact-nonnegative-integer?
procedure
(mpz_combit rop bit-index) → void?
  rop : mpz?
  bit-index : exact-nonnegative-integer?
procedure
(mpz_tstbit op bit-index) → (or/c 0 1)
  op : mpz?
  bit-index : exact-nonnegative-integer?

See Logical and Bit Manipulation Functions in the GMP manual.

procedure
(mpz_urandomb rop state n) → void?
  rop : mpz?
  state : gmp-randstate?
  n : exact-nonnegative-integer?
procedure
(mpz_urandomm rop state n) → void?
  rop : mpz?
  state : gmp-randstate?
  n : mpz?
procedure
(mpz_rrandomb rop state n) → void?
  rop : mpz?
  state : gmp-randstate?
  n : exact-nonnegative-integer?

See Integer Random Numbers in the GMP manual.

Added in version 1.2 of package gmp-lib.

procedure
(mpz_fits_ulong_p op) → exact-integer?
  op : mpz?
procedure
(mpz_fits_slong_p op) → exact-integer?
  op : mpz?
procedure
(mpz_fits_uint_p op) → exact-integer?
  op : mpz?
procedure
(mpz_fits_sint_p op) → exact-integer?
  op : mpz?
procedure
(mpz_fits_ushort_p op) → exact-integer?
  op : mpz?
procedure
(mpz_fits_sshort_p op) → exact-integer?
  op : mpz?
procedure
(mpz_odd_p op) → exact-integer?
  op : mpz?
procedure
(mpz_even_p op) → exact-integer?
  op : mpz?
procedure
(mpz_sizeinbase op base) → exact-nonnegative-integer?
  op : mpz?
  base : (integer-in 2 62)

See Miscellaneous Functions in the GMP manual.

procedure
(mpz_size op) → exact-nonnegative-integer?
  op : mpz?
procedure
(mpz_getlimbn op n) → exact-nonnegative-integer?
  op : mpz?
  n : unsigned-long

See Special Functions in the GMP manual.

2 mpq: Multi-precision Rational Numbers🔗ℹ

An mpq (mpq_t) is a mutable value containing a rational number with arbitrary-precision numerator and denominator. In this library, such values are created with mpq and recognized with mpq?. Their backing storage is automatically initialized and garbage-collected.

procedure
(mpq? v) → boolean?
v : any/c

Returns #t if v is an mpq value, #f otherwise.

procedure
(mpq x) → mpq?
x : (or/c (and/c rational? exact?) mpz? mpq?)

Creates a new mpq initialized with the value x. If x is an mpz or mpq, then its value is copied; x is unchanged does not share storage with the result.

procedure
(mpq->number q) → (and/c rational? exact?)
q : mpq?

Returns the exact rational value stored in z.

procedure
(mpq-zero? q) → boolean?
  q : mpq?
procedure
(mpq-positive? q) → boolean?
  q : mpq?
procedure
(mpq-negative? q) → boolean?
  q : mpq?

Like zero?, positive?, and negative?, respectively.

2.1 Operations on mpq Values🔗ℹ

The following operations are supported. By GMP convention, destination arguments come first, followed by source operands.

procedure
(mpq_canonicalize op) → void?
op : mpq?

See Rational Number Functions in the GMP manual.

procedure
(mpq_set rop op) → void?
  rop : mpq?
  op : mpq?
procedure
(mpq_set_z rop op) → void?
  rop : mpq?
  op : mpz?
procedure
(mpq_set_ui rop n d) → void?
  rop : mpq?
  n : unsigned-long
  d : unsigned-long
procedure
(mpq_set_si rop n d) → void?
  rop : mpq?
  n : signed-long
  d : unsigned-long
procedure
(mpq_swap rop1 rop2) → void?
  rop1 : mpq?
  rop2 : mpq?

See Initializing Rationals in the GMP manual.

procedure
(mpq_get_d op) → real?
  op : mpq?
procedure
(mpq_set_d rop op) → void?
  rop : mpq?
  op : real?

See Conversion Functions in the GMP manual.

procedure
(mpq_add rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_sub rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_mul rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_mul_2exp rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : unsigned-long
procedure
(mpq_div rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_div_2exp rop op1 op2) → void?
  rop : mpq?
  op1 : mpq?
  op2 : unsigned-long?
procedure
(mpq_neg rop op) → void?
  rop : mpq?
  op : mpq?
procedure
(mpq_abs rop op) → void?
  rop : mpq?
  op : mpq?
procedure
(mpq_inv rop op) → void?
  rop : mpq?
  op : mpq?

See Arithmetic Functions in the GMP manual.

procedure
(mpq_cmp op1 op2) → exact-integer?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_cmp_z op1 op2) → exact-integer?
  op1 : mpq?
  op2 : mpz?
procedure
(mpq_cmp_ui op1 n2 d2) → exact-integer?
  op1 : mpq?
  n2 : unsigned-long
  d2 : unsigned-long
procedure
(mpq_cmp_si op1 n2 d2) → exact-integer?
  op1 : mpq?
  n2 : signed-long
  d2 : unsigned-long
procedure
(mpq_equal op1 op2) → exact-integer?
  op1 : mpq?
  op2 : mpq?
procedure
(mpq_sgn op) → (or/c -1 0 1)
  op : mpq?

See Comparision Functions in the GMP manual.

procedure
(mpq_get_num num rational) → void?
  num : mpz?
  rational : mpq?
procedure
(mpq_get_den den rational) → void?
  den : mpz?
  rational : mpq?
procedure
(mpq_set_num rational num) → void?
  rational : mpq?
  num : mpz?
procedure
(mpq_set_den rational den) → void?
  rational : mpq?
  den : mpq?

See Applying Integer Functions to Rationals in the GMP manual.

3 Random Number Generation🔗ℹ

A randstate (gmp_randstate_t) represents a random number generation algorithm and its state. In this library, such values are recognized with gmp-randstate? and created with gmp-randstate and gmp-randstate-mt. They are automatically initialized and garbage-collected.

procedure
(gmp-randstate? v) → boolean?
v : any/c

Returns #t if v is a gmp_randstate_t value, #f otherwise.

Added in version 1.2 of package gmp-lib.

procedure
(gmp-randstate) → gmp-randstate?
procedure
(gmp-randstate-mt) → gmp-randstate?

Returns a new gmp_randstate_t initialized with gmp_randinit_default or gmp_randinit_mt, respectively.

Added in version 1.2 of package gmp-lib.

procedure
(gmp_randseed state seed) → void?
  state : gmp-randstate?
  seed : mpz?
procedure
(gmp_randseed_ui state seed) → void?
  state : gmp-randstate?
  seed : exact-nonnegative-integer?

See Random State Seeding in the GMP manual.

Added in version 1.2 of package gmp-lib.

procedure
(gmp_urandomb_ui state n) → exact-nonnegative-integer?
  state : gmp-randstate?
  n : exact-nonnegative-integer?
procedure
(gmp_urandomm_ui state n) → exact-nonnegative-integer?
  state : gmp_randstate?
  n : exact-positive-integer?

See Random State Miscellaneous in the GMP manual.

Added in version 1.2 of package gmp-lib.

1	mpz: Multi-precision Integers
2	mpq: Multi-precision Rational Numbers
3	Random Number Generation