Racket

quad-fp: quadruple-precision floating point🔗ℹ

Shaobo He

package: quad-fp

A toy Racket FFI binding to GCC’s libquadmath, exposing IEEE 754 quadruple-precision (128-bit, __float128) floating point — roughly 34 significant decimal digits, versus the ~16 of Racket’s native double flonums.

Values are opaque: a quad is shuttled across the FFI boundary as 128 bits and is only ever produced or consumed by the operations below.

The binding loads a native shared library (libquadf) that is compiled from source at install time, so a C toolchain with libquadmath must be present. See the package README for details.

1 Predicate🔗ℹ

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

Returns #t if v is a quad-precision value produced by this library, #f otherwise.

2 Conversion🔗ℹ

procedure
(double-flonum->quad-flonum x) → Quad?
x : flonum?
procedure
(quad-flonum->double-flonum q) → flonum?
q : Quad?

Convert between a Racket double flonum? and a quad. Widening a double is exact; narrowing back to a double rounds to nearest.

procedure
(string->quad-flonum s) → Quad?
  s : string?
procedure
(quad-flonum->string q [precision]) → string?
  q : Quad?
  precision : exact-integer? = 36

Parse a decimal string into a quad (via strtoflt128), and render a quad to a decimal string with precision significant digits (via quadmath_snprintf). The default of 36 digits round-trips binary128 exactly.

procedure
(quad-flonum->bytes q) → bytes?
q : Quad?

Returns the 16-byte little-endian representation of q’s __float128 bit pattern.

3 Arithmetic🔗ℹ

procedure
(qf+ a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qf- a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qf* a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qf/ a b) → Quad?
  a : Quad?
  b : Quad?

Quad-precision addition, subtraction, multiplication, and division.

4 Elementary functions🔗ℹ

procedure
(qfabs a) → Quad?
  a : Quad?
procedure
(qfsqrt a) → Quad?
  a : Quad?
procedure
(qfcbrt a) → Quad?
  a : Quad?
procedure
(qfpow a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfhypot a b) → Quad?
  a : Quad?
  b : Quad?

Absolute value, square and cube roots, a raised to b, and (qfsqrt (qf+ (qf* a a) (qf* b b))) without overflow.

procedure
(qfexp a) → Quad?
  a : Quad?
procedure
(qfexp2 a) → Quad?
  a : Quad?
procedure
(qfexpm1 a) → Quad?
  a : Quad?
procedure
(qflog a) → Quad?
  a : Quad?
procedure
(qflog2 a) → Quad?
  a : Quad?
procedure
(qflog10 a) → Quad?
  a : Quad?
procedure
(qflog1p a) → Quad?
  a : Quad?

Exponentials (qfexpm1 computes ex-1 accurately near 0) and logarithms (qflog1p computes log(1+x)).

procedure
(qfsin a) → Quad?
  a : Quad?
procedure
(qfcos a) → Quad?
  a : Quad?
procedure
(qftan a) → Quad?
  a : Quad?
procedure
(qfasin a) → Quad?
  a : Quad?
procedure
(qfacos a) → Quad?
  a : Quad?
procedure
(qfatan a) → Quad?
  a : Quad?
procedure
(qfatan2 a b) → Quad?
  a : Quad?
  b : Quad?

Trigonometric functions and their inverses; qfatan2 is the two-argument arctangent.

procedure
(qfsinh a) → Quad?
  a : Quad?
procedure
(qfcosh a) → Quad?
  a : Quad?
procedure
(qftanh a) → Quad?
  a : Quad?
procedure
(qfasinh a) → Quad?
  a : Quad?
procedure
(qfacosh a) → Quad?
  a : Quad?
procedure
(qfatanh a) → Quad?
  a : Quad?

Hyperbolic functions and their inverses.

5 Rounding🔗ℹ

procedure
(qfceil a) → Quad?
  a : Quad?
procedure
(qffloor a) → Quad?
  a : Quad?
procedure
(qftrunc a) → Quad?
  a : Quad?
procedure
(qfround a) → Quad?
  a : Quad?
procedure
(qfrint a) → Quad?
  a : Quad?
procedure
(qfnearbyint a) → Quad?
  a : Quad?

Round toward +∞, -∞, zero, and nearest (ties away from zero for qfround; using the current rounding mode for qfrint and qfnearbyint).

6 Special functions🔗ℹ

procedure
(qferf a) → Quad?
  a : Quad?
procedure
(qferfc a) → Quad?
  a : Quad?
procedure
(qflgamma a) → Quad?
  a : Quad?
procedure
(qftgamma a) → Quad?
  a : Quad?
procedure
(qfj0 a) → Quad?
  a : Quad?
procedure
(qfj1 a) → Quad?
  a : Quad?
procedure
(qfy0 a) → Quad?
  a : Quad?
procedure
(qfy1 a) → Quad?
  a : Quad?

Error functions, log-gamma and gamma, and Bessel functions of the first (qfj0, qfj1) and second (qfy0, qfy1) kind, orders 0 and 1.

7 Other binary functions🔗ℹ

procedure
(qffmod a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfremainder a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfcopysign a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qffdim a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfmax a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfmin a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qfnextafter a b) → Quad?
  a : Quad?
  b : Quad?
procedure
(qflogb a) → Quad?
  a : Quad?
procedure
(qffma a b c) → Quad?
  a : Quad?
  b : Quad?
  c : Quad?

Floating-point remainder, IEEE remainder, sign copying, positive difference, maximum, minimum, next representable value toward b, unbiased exponent, and the fused multiply-add (qf+ (qf* a b) c) computed with a single rounding.

8 Comparison🔗ℹ

procedure
(qf= a b) → boolean?
  a : Quad?
  b : Quad?
procedure
(qf< a b) → boolean?
  a : Quad?
  b : Quad?
procedure
(qf<= a b) → boolean?
  a : Quad?
  b : Quad?
procedure
(qf> a b) → boolean?
  a : Quad?
  b : Quad?
procedure
(qf>= a b) → boolean?
  a : Quad?
  b : Quad?

Quad-precision comparisons, returning Racket booleans.

9 Classification🔗ℹ

procedure
(qfnan? a) → boolean?
  a : Quad?
procedure
(qfinfinite? a) → boolean?
  a : Quad?
procedure
(qffinite? a) → boolean?
  a : Quad?
procedure
(qfsignbit? a) → boolean?
  a : Quad?
procedure
(qfsignaling? a) → boolean?
  a : Quad?

Test for NaN, infinity, finiteness, a set sign bit, and signaling NaN.

10 Constants🔗ℹ

value
quad-pi : Quad?
value
quad-pi/2 : Quad?
value
quad-pi/4 : Quad?
value
quad-1/pi : Quad?
value
quad-2/pi : Quad?
value
quad-2/sqrt-pi : Quad?
value
quad-e : Quad?
value
quad-log2e : Quad?
value
quad-log10e : Quad?
value
quad-ln2 : Quad?
value
quad-ln10 : Quad?
value
quad-sqrt2 : Quad?
value
quad-1/sqrt2 : Quad?

The mathematical constants from libquadmath’s M_*q macros (π, π/2, π/4, 1/π, 2/π, 2/√π, e, log₂e, log₁₀e, ln 2, ln 10, √2, 1/√2).

value
quad-max : Quad?
value
quad-min : Quad?
value
quad-epsilon : Quad?
value
quad-denorm-min : Quad?

Largest finite, smallest positive normal, machine epsilon, and smallest positive subnormal (FLT128_MAX, FLT128_MIN, FLT128_EPSILON, FLT128_DENORM_MIN).

value
quad-mant-dig : exact-integer?
value
quad-decimal-dig : exact-integer?
value
quad-min-exp : exact-integer?
value
quad-max-exp : exact-integer?
value
quad-min-10-exp : exact-integer?
value
quad-max-10-exp : exact-integer?

Format characteristics: mantissa bits (113), round-trippable decimal digits (33), and the binary and decimal exponent ranges.

11 Example🔗ℹ

(require quad-fp)
(define a (double-flonum->quad-flonum 1.0))
(define b (double-flonum->quad-flonum 1e-20))
; In double, 1.0 + 1e-20 == 1.0; in quad the addend survives:
(qf= (qf+ a b) a)
; => #f
(quad-flonum->string (qf* quad-pi quad-pi))
; => "9.86960440108935861883449099987615081"

1	Predicate
2	Conversion
3	Arithmetic
4	Elementary functions
5	Rounding
6	Special functions
7	Other binary functions
8	Comparison
9	Classification
10	Constants
11	Example