5 Arbitrary-Precision Floating-Point Numbers (Bigfloats)

▼ |  | Math Library

1 | Constants and Elementary Functions
2 | Flonums
3 | Special Functions
4 | Number Theory
5 | Arbitrary-Precision Floating-Point Numbers (Bigfloats)
6 | Arrays
7 | Matrices and Linear Algebra
8 | Statistics Functions
9 | Probability Distributions
10 | Stuff That Doesn’t Belong Anywhere Else

► | 5 | Arbitrary-Precision Floating-Point Numbers (Bigfloats)

5.1 | Quick Start
5.2 | Fictionally Asked Questions
5.3 | Type and Constructors
5.4 | Accessors and Conversion Functions
5.5 | Parameters
5.6 | Constants
5.7 | Predicates
5.8 | Rounding
5.9 | Mathematical Operations
5.10 | Low-level Functions

On this page:

5.1`` Quick Start
5.2`` Fictionally Asked Questions
5.3`` Type and Constructors
Bigfloat
bigfloat?
bf
bfrandom
bfcopy
5.4`` Accessors and Conversion Functions
bigfloat-precision
bigfloat-signbit
bigfloat-significand
bigfloat-exponent
bigfloat->sig+ exp
bigfloat->integer
bigfloat->rational
bigfloat->real
bigfloat->flonum
bigfloat->string
string->bigfloat
5.5`` Parameters
bf-precision
bf-rounding-mode
bf-min-precision
bf-max-precision
5.6`` Constants
pi.bf
phi.bf
gamma.bf
catalan.bf
log2.bf
-inf.bf
-max.bf
-min.bf
-0.bf
0.bf
+ min.bf
+ max.bf
+ inf.bf
+ nan.bf
epsilon.bf
-10.bf
-9.bf
-8.bf
-7.bf
-6.bf
-5.bf
-4.bf
-3.bf
-2.bf
-1.bf
1.bf
2.bf
3.bf
4.bf
5.bf
6.bf
7.bf
8.bf
9.bf
10.bf
5.7`` Predicates
bfzero?
bfpositive?
bfnegative?
bfinteger?
bfeven?
bfodd?
bfrational?
bfinfinite?
bfnan?
bf=
bf>
bf<
bf>=
bf<=
5.8`` Rounding
bftruncate
bffloor
bfceiling
bfround
bffrac
bfrint
5.9`` Mathematical Operations
bfmax
bfmin
bf+
bf*
bf-
bf/
bfsqr
bfabs
bfsgn
bfsqrt
bf1/ sqrt
bfcbrt
bfroot
bfhypot
bflog
bflog2
bflog10
bfexp
bfexp2
bfexp10
bflog1p
bfexpm1
bfexpt
bfsin
bfcos
bftan
bfasin
bfacos
bfatan
bfatan2
bfsinh
bfcosh
bftanh
bfasinh
bfacosh
bfatanh
bfsec
bfcsc
bfcot
bfsech
bfcsch
bfcoth
bfsin+ cos
bfsinh+ cosh
bffactorial
bfgamma
bflog-gamma
bflog-gamma/ sign
bfpsi0
bfeint
bfli2
bfzeta
bferf
bferfc
bfbesj0
bfbesj1
bfbesj
bfbesy0
bfbesy1
bfbesy
bfagm
5.10`` Low-level Functions
bigfloat->ordinal
ordinal->bigfloat
bigfloats-between
bfstep
bfnext
bfprev
bfshift
bfcanonicalize

5.3.6
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### 5`` Arbitrary-Precision Floating-Point Numbers (Bigfloats)

Neil Toronto <ntoronto@racket-lang.org>

(require math/bigfloat)

This library provides a Typed Racket interface to MPFR, a C library that provides

- A C type of arbitrary-precision floating-point numbers.

- Elementary and special functions that are efficient and proved correct.

- Well-defined semantics that correspond with the latest IEEE 754 standard.

The arbitrary-precision floating-point numbers MPFR provides and operates on are represented by the Typed Racket type Bigfloat and identified by the predicate bigfloat?.

With a few noted exceptions, bigfloat functions regard their arguments as if they were exact, regardless of their precision. Conceptually, they compute exact results using infinitely many bits, and return results with (bf-precision) bits by rounding them using (bf-rounding-mode). In practice, they use finite algorithms that have been painstakingly proved to be equivalent to that conceptual, infinite process.

MPFR is free and license-compatible with commercial software. It is distributed with Racket for Windows and Mac OS X, is installed on most Linux systems, and is easy to install on major Unix-like platforms.

5.1 Quick Start
5.2 Fictionally Asked Questions
5.3 Type and Constructors
5.4 Accessors and Conversion Functions
5.5 Parameters
5.6 Constants
5.7 Predicates
5.8 Rounding
5.9 Mathematical Operations
5.10 Low-level Functions

#### 5.1`` Quick Start

1. Set the bigfloat function result precision using (bf-precision <some-number-of-bits>).

2. Use bf to convert real values and well-formed strings to bigfloats.

3. Operate on bigfloats using bf-prefixed functions like bf+ and bfsin.

4. Convert bigfloats to real values using bigfloat->real, bigfloat->flonum, and bigfloat->integer. Format them for display using bigfloat->string.

For examples, continue through the FAQ.

#### 5.2`` Fictionally Asked Questions

Why use math/bigfloat?

There are a few reasons.

Reason: Flonums have either too much or too little precision for your application.

Examples:
> > (flsqrt 3.0)
> 1.7320508075688772
> > pi
> 3.141592653589793
> > (bf-precision 16)
> > (bfsqrt (bf 3))
> (bf #e1.73206)
> > (bf-precision 179)
> > pi.bf
> (bf #e3.141592653589793238462643383279502884197169399375105819)

A flonum has a 53-bit significand (we’ll say it has 53 bits of precision) and an 11-bit exponent. A bigfloat has an arbitrary precision of at least 2 bits and a 31-bit exponent.

Reason: To compute ridiculously large or small numbers with confidence.

Examples:
> > (bf-precision 128)
> > (bfexp (bfexp (bfexp (bf 3))))
> (bf "2.050986436051648895105860942072054674579e229520860")
> > (bflog (bflog (bflog (bfexp (bfexp (bfexp (bf 3)))))))
> (bf 3)

Reason: To verify your floating-point hardware.

IEEE 754-2008 stipulates that conforming implementations must correctly round the results of all operations. Roughly speaking, results can’t be more than half a bit off, where the bit in question is the least significant in the significand.

Of course, implementations don’t always adhere to standards. For example, on my old laptop, evaluating (exp 400) results in 5.221469689764346e+173. Note the last four decimal digits in the significand: 4346. But they should be 4144:

> > (bf-precision 53)
> > (bigfloat->flonum (bfexp (bf 400)))
> 5.221469689764144e+173

My new laptop computes 5.221469689764144e+173 as it should.

Reason: To control rounding of the least significant bit.

IEEE 754 provides for different rounding modes for the smallest bit of a flonum result, such as round to even and round toward zero. We might use this to implement interval arithmetic correctly, by rounding lower bounds downward and upper bounds upward. But there isn’t a portable way to set the rounding mode!

MPFR allows the rounding mode to be different for any operation, and math/bigfloat exposes this capability using the parameter bf-rounding-mode.

When shouldn’t I use math/bigfloat?

When you need raw speed. Bigfloat functions can be hundreds to thousands of times slower than flonum functions.

That’s not to say that they’re inefficient. For example, bflog implements the algorithm with the best known asymptotic complexity. It just doesn’t run directly on hardware, and it can’t take fixed-precision-only shortcuts.

Why are there junk digits on the end of (bf 1.1)?

That’s approximately the value of the flonum 1.1. Use (bf #e1.1) or (bf "1.1") to make the junk go away. In general, you should prefer to convert exact rationals and strings to bigfloats.

Why is the last digit of pi.bf not rounded correctly?

All the bits but the last is exact, and the last bit is correctly rounded. This doesn’t guarantee that the last digit will be.

A decimal digit represents at most log(10)/log(2) ≈ 3.3 bits. This is an irrational number, so the decimal/bit boundary never lines up except at the decimal point. Thus, the last decimal digit of any bigfloat must represent fewer than 3.3 bits, so it’s wrong more often than not. But it’s the last bit that counts.

#### 5.3`` Type and Constructors

> syntax Bigfloat
> procedure (bigfloat? v) → Boolean
> v : Any

An opaque type that represents an arbitrary-precision floating-point number, or a bigfloat, and the opaque type’s predicate.

> procedure (bf x) → Bigfloat
> x : (U String Real)
> (bf sig exp) → Bigfloat
> sig : Integer
> exp : Integer

The one-argument variant converts a string or real x to a bigfloat.

> > (bf-precision 128)
> > (bf 4)
> (bf 4)
> > (bf 1/7)
> (bf #e0.1428571428571428571428571428571428571426)
> > (bf 41/10)
> (bf #e4.099999999999999999999999999999999999995)
> > (bf "not a number")
> bf: expected a well-formed decimal number; given "not a number"
> > (bf "15e200000000")
> (bf "1.499999999999999999999999999999999999998e200000001")

In the last example, the result of (bf "15e200000000") is displayed as a string conversion because the exact rational number would be very large.

> > > * It can be a good idea if you’re testing a flonum implementation of a function against a bigfloat implementation.

Converting from flonum literals is usually a bad idea* because flonums have only 53 bits precision. Prefer to pass exact rationals and strings to bf.

The two-argument variant converts a signed significand sig and a power of 2 exp to a bigfloat. Generally, (bf sig exp) = (bf (* sig (expt 2 exp))), but the two-argument variant is much faster, especially for large exp.

Examples:
> > (bf 200 56)
> (bf 14411518807585587200)
> > (bf (* 200 (expt 2 56)))
> (bf 14411518807585587200)

The bfrandom function generates random bigfloats between 0 and 1 using the two-argument variant in this way:

> > (require (only-in math/base random-bits))
> > (bf-precision 64)
> > (bf (random-bits 64) -64)
> (bf #e0.416872969910248753552)

> procedure (bfrandom) → Bigfloat

Returns a uniformly distributed random bigfloat in the interval [0,1].

> procedure (bfcopy x) → Bigfloat
> x : Bigfloat

Returns the (bf-precision)-bit bigfloat closest to x, under the current bf-rounding-mode.

A common pattern to compute bigfloats in higher precision is

> > (bf-precision 64)
> > (bfcopy
> (parameterize ([bf-precision (+ (bf-precision) 10)])
> (bf/ (bf+ 1.bf (bfsqrt 5.bf)) 2.bf)))
> (bf #e1.61803398874989484821)

This example computes the golden ratio (phi.bf) with 10 bits more than requested, to make up for triple rounding error.

#### 5.4`` Accessors and Conversion Functions

> procedure (bigfloat-precision x) → Exact-Positive-Integer
> x : Bigfloat

Returns the number of bits in the significand of x. This is almost always the value of (bf-precision) when x was created.

> procedure (bigfloat-signbit x) → (U 0 1)
> x : Bigfloat

Returns the sign bit of the significand of x.

Examples:
> > (bigfloat-signbit -1.bf)
> 1
> > (bigfloat-signbit 0.bf)
> 0
> > (bigfloat-signbit -0.bf)
> 1
> > (bigfloat-signbit -inf.bf)
> 1

> procedure (bigfloat-significand x) → Integer
> x : Bigfloat
> procedure (bigfloat-exponent x) → Integer
> x : Bigfloat

Return the signed significand or exponent of x.

To access the significand and exponent at the same time, use bigfloat->sig+exp.

> procedure (bigfloat->sig+exp x) → (Values Integer Integer)
> x : Bigfloat

Returns the signed significand and exponent of x.

If (values sig exp) = (bigfloat->sig+exp x), its value as an exact rational is (* sig (expt 2 exp)). In fact, bigfloat->rational converts bigfloats to rationals in exactly this way, after ensuring that (bfrational? x) is #t.

This function and the two-argument variant of bf are mutual inverses.

> procedure (bigfloat->integer x) → Integer
> x : Bigfloat
> procedure (bigfloat->rational x) → Exact-Rational
> x : Bigfloat
> procedure (bigfloat->real x) → (U Exact-Rational Flonum)
> x : Bigfloat
> procedure (bigfloat->flonum x) → Flonum
> x : Bigfloat

Convert bigfloats to integer, exact rational, real and flonum values respectively.

bigfloat->integer, bigfloat->rational and bigfloat->real return values that can be converted exactly back to x using bf. For the first two, this is done by raising an error if x is not respectively integer or rational. On the other hand, bigfloat->real returns +inf.0, -inf.0 or +nan.0 when x is not a rational bigfloat.

bigfloat->flonum rounds x to 53 bits precision to fit the value into a flonum, using the current value of bf-rounding-mode.

> > (bf-precision 64)
> > (bigfloat->integer (bf 21/10))
> bigfloat->integer: contract violation
> expected: bfinteger?
> given: (bf #e2.09999999999999999991)
> > (bigfloat->integer (bfround (bf 21/10)))
> 2
> > (define x (bf 1/7))
> > (bigfloat->flonum x)
> 0.14285714285714285
> > (bigfloat->rational x)
> 10540996613548315209/73786976294838206464
> > (rationalize (bigfloat->rational x) (expt 2 (- (bf-precision))))
> 1/7
> > (bf= x (bf (bigfloat->rational x)))
> #t

Be careful with exact conversions. Bigfloats with large exponents may not fit in memory as integers or exact rationals. Worse, they might fit, but have all your RAM and swap space for lunch.

> procedure (bigfloat->string x) → String
> x : Bigfloat
> procedure (string->bigfloat s) → (U Bigfloat False)
> s : String

Convert a bigfloat x to a string s and back.

The string returned by bigfloat->string includes enough digits that string->bigfloat can reconstruct the bigfloat precisely. In other words, string->bigfloat is a left inverse of bigfloat->string.

If s isn’t a well-formed decimal number with an optional exponent part, string->bigfloat returns #f. (In contrast, (bf s) raises an error.)

Examples:
> > (bf-precision 64)
> > (bigfloat->string (bf 4))
> "4"
> > (bigfloat->string (bf 1/10000))
> "1.00000000000000000001e-4"
> > (string->bigfloat "0.14285714285714285714")
> (bf #e0.142857142857142857141)
> > (string->bigfloat "square root of two")
> #f
> > (string->bigfloat (bigfloat->string pi.bf))
> (bf #e3.14159265358979323851)
> > pi.bf
> (bf #e3.14159265358979323851)

#### 5.5`` Parameters

> parameter (bf-precision) → Integer
> (bf-precision bits) → void?
> bits : Integer

A parameter that determines the precision of bigfloats returned from most bigfloat functions. Exceptions are noted in the documentation for functions that do not use bf-precision.

For nonzero, rational bigfloats, the number of bits bits includes the leading one bit. For example, to simulate 64-bit floating point, use (bf-precision 53) even though flonums have a 52-bit significand, because the one bit is implicit in a flonum.

This parameter has a guard that ensures (bf-precision) is between bf-min-precision and bf-max-precision.

> parameter (bf-rounding-mode) → (U 'nearest 'zero 'up 'down)
> (bf-rounding-mode mode) → void?
> mode : (U 'nearest 'zero 'up 'down)

A parameter that determines the mode used to round the results of most bigfloat functions. Conceptually, rounding is applied to infinite-precision results to fit them into (bf-precision) bits.

> value bf-min-precision : Exact-Positive-Integer

Equal to 2, because single-bit bigfloats can’t be correctly rounded.

> value bf-max-precision : Exact-Positive-Integer

The largest value of (bf-precision). This is platform-dependent, and probably much larger than you’ll ever need.

#### 5.6`` Constants

Most bigfloat “constants” are actually identifier macros that expand to the application of a zero-argument function. This allows, for example, pi.bf to depend on the current value of bf-precision, and allows all of them to be constructed lazily. Most constants are memoized, possibly at multiple precisions.

> value pi.bf : Bigfloat
> value phi.bf : Bigfloat
> value gamma.bf : Bigfloat
> value catalan.bf : Bigfloat
> value log2.bf : Bigfloat

Approximations of π, φ, γ, G and log(2).

Examples:
> > (bf-precision 10)
> > pi.bf
> (bf #e3.1406)
> > (bf-precision 179)
> > pi.bf
> (bf #e3.141592653589793238462643383279502884197169399375105819)
> > phi.bf
> bf #e1.618033988749894848204586834365638117720309179805762863)
> > gamma.bf
> (bf #e0.5772156649015328606065120900824024310421593359399235988)
> > catalan.bf
> (bf #e0.9159655941772190150546035149323841107741493742816721343)
> > log2.bf
> (bf #e0.6931471805599453094172321214581765680755001343602552545)

> value -inf.bf : Bigfloat
> value -max.bf : Bigfloat
> value -min.bf : Bigfloat
> value -0.bf : Bigfloat
> value 0.bf : Bigfloat
> value +min.bf : Bigfloat
> value +max.bf : Bigfloat
> value +inf.bf : Bigfloat
> value +nan.bf : Bigfloat
> value epsilon.bf : Bigfloat

Bigfloat constants corresponding to -inf.0, -max.0 -min.0, -0.0, 0.0, +min.0, +max.0, +inf.0, +nan.0 and epsilon.0.

The constants -inf.bf, -0.bf, 0.bf, +inf.bf, and +nan.bf have fixed precision.

> value -10.bf : Bigfloat
> value -9.bf : Bigfloat
> value -8.bf : Bigfloat
> value -7.bf : Bigfloat
> value -6.bf : Bigfloat
> value -5.bf : Bigfloat
> value -4.bf : Bigfloat
> value -3.bf : Bigfloat
> value -2.bf : Bigfloat
> value -1.bf : Bigfloat
> value 1.bf : Bigfloat
> value 2.bf : Bigfloat
> value 3.bf : Bigfloat
> value 4.bf : Bigfloat
> value 5.bf : Bigfloat
> value 6.bf : Bigfloat
> value 7.bf : Bigfloat
> value 8.bf : Bigfloat
> value 9.bf : Bigfloat
> value 10.bf : Bigfloat

More fixed-precision bigfloat constants.

#### 5.7`` Predicates

> procedure (bfzero? x) → Boolean
> x : Bigfloat
> procedure (bfpositive? x) → Boolean
> x : Bigfloat
> procedure (bfnegative? x) → Boolean
> x : Bigfloat
> procedure (bfinteger? x) → Boolean
> x : Bigfloat
> procedure (bfeven? x) → Boolean
> x : Bigfloat
> procedure (bfodd? x) → Boolean
> x : Bigfloat
> procedure (bfrational? x) → Boolean
> x : Bigfloat
> procedure (bfinfinite? x) → Boolean
> x : Bigfloat
> procedure (bfnan? x) → Boolean
> x : Bigfloat

Unary predicates corresponding to zero?, positive?, negative?, integer?, even?, odd?, rational?, infinite? and nan?.

> procedure (bf= x y) → Boolean
> x : Bigfloat
> y : Bigfloat
> procedure (bf> x y) → Boolean
> x : Bigfloat
> y : Bigfloat
> procedure (bf< x y) → Boolean
> x : Bigfloat
> y : Bigfloat
> procedure (bf>= x y) → Boolean
> x : Bigfloat
> y : Bigfloat
> procedure (bf<= x y) → Boolean
> x : Bigfloat
> y : Bigfloat

Standard comparison functions. As is usual, infinities are either greater or less than any other bigfloat, and every comparison returns #f when either argument is +nan.bf.

#### 5.8`` Rounding

> procedure (bftruncate x) → Bigfloat
> x : Bigfloat
> procedure (bffloor x) → Bigfloat
> x : Bigfloat
> procedure (bfceiling x) → Bigfloat
> x : Bigfloat
> procedure (bfround x) → Bigfloat
> x : Bigfloat

Like truncate, floor, ceiling and round, but for bigfloats.

Rounding is to the nearest integer, with ties broken by rounding to even.

Examples:
> > (bfround (bf 1.5))
> (bf 2)
> > (bfround (bf 2.5))
> (bf 2)
> > (bfround (bf -1.5))
> (bf -2)
> > (bfround (bf -2.5))
> (bf -2)

> procedure (bffrac x) → Bigfloat
> x : Bigfloat

Returns the fractional part of x, with the same sign as x.

> procedure (bfrint x) → Bigfloat
> x : Bigfloat

Rounds x to the nearest integer bigfloat, in the direction specified by (bf-rounding-mode).

#### 5.9`` Mathematical Operations

> procedure (bfmax x ...) → Bigfloat
> x : Bigfloat
> procedure (bfmin x ...) → Bigfloat
> x : Bigfloat

Return the maximum and minimum of their arguments, respectively.

When given no arguments, bfmin returns +inf.bf, and bfmax returns -inf.bf.

> procedure (bf+ x ...) → Bigfloat
> x : Bigfloat
> procedure (bf* x ...) → Bigfloat
> x : Bigfloat
> procedure (bf- x y ...) → Bigfloat
> x : Bigfloat
> y : Bigfloat
> procedure (bf/ x y ...) → Bigfloat
> x : Bigfloat
> y : Bigfloat
> procedure (bfsqr x) → Bigfloat
> x : Bigfloat
> procedure (bfabs x) → Bigfloat
> x : Bigfloat
> procedure (bfsgn x) → Bigfloat
> x : Bigfloat

Standard arithmetic functions, corresponding to +, *, -, /, sqr, abs and sgn.

When bf+ and bf- are given more than two arguments, they compute the answers in a way that incurs rounding error only once.

> procedure (bfsqrt x) → Bigfloat
> x : Bigfloat
> procedure (bf1/sqrt x) → Bigfloat
> x : Bigfloat
> procedure (bfcbrt x) → Bigfloat
> x : Bigfloat

Return the square root, reciprocal square root, and cube root of x.

> procedure (bfroot x n) → Bigfloat
> x : Bigfloat
> n : Integer

Returns the nth root of x. n must be a nonnegative fixnum.

> procedure (bfhypot x y) → Bigfloat
> x : Bigfloat
> y : Bigfloat

Computes (bfsqrt (bf+ (bfsqr x) (bfsqr y))) without uncessary overflow, incurring rounding error only once. See flhypot for an example using flonums.

> procedure (bflog x) → Bigfloat
> x : Bigfloat
> procedure (bflog2 x) → Bigfloat
> x : Bigfloat
> procedure (bflog10 x) → Bigfloat
> x : Bigfloat

Return the log of x in base e, 2 and 10.

> procedure (bfexp x) → Bigfloat
> x : Bigfloat
> procedure (bfexp2 x) → Bigfloat
> x : Bigfloat
> procedure (bfexp10 x) → Bigfloat
> x : Bigfloat

Return the exponential of x in base e, 2 and 10.

> procedure (bflog1p x) → Bigfloat
> x : Bigfloat
> procedure (bfexpm1 x) → Bigfloat
> x : Bigfloat

Like (bflog (bf+ 1.bf x)) and (bf- (bfexp x) 1.bf), but correct when x is near zero. See fllog1p for motivation and examples.

> procedure (bfexpt x y) → Bigfloat
> x : Bigfloat
> y : Bigfloat

Computes xy. See flexpt and expt.

> procedure (bfsin x) → Bigfloat
> x : Bigfloat
> procedure (bfcos x) → Bigfloat
> x : Bigfloat
> procedure (bftan x) → Bigfloat
> x : Bigfloat
> procedure (bfasin x) → Bigfloat
> x : Bigfloat
> procedure (bfacos x) → Bigfloat
> x : Bigfloat
> procedure (bfatan x) → Bigfloat
> x : Bigfloat
> procedure (bfatan2 x y) → Bigfloat
> x : Bigfloat
> y : Bigfloat

Standard trigonometric functions and their inverses.

> procedure (bfsinh x) → Bigfloat
> x : Bigfloat
> procedure (bfcosh x) → Bigfloat
> x : Bigfloat
> procedure (bftanh x) → Bigfloat
> x : Bigfloat
> procedure (bfasinh x) → Bigfloat
> x : Bigfloat
> procedure (bfacosh x) → Bigfloat
> x : Bigfloat
> procedure (bfatanh x) → Bigfloat
> x : Bigfloat

Standard hyperbolic functions and their inverses.

> procedure (bfsec x) → Bigfloat
> x : Bigfloat
> procedure (bfcsc x) → Bigfloat
> x : Bigfloat
> procedure (bfcot x) → Bigfloat
> x : Bigfloat

Standard reciprocal trigonometric functions. MPFR does not implement their inverses.

> procedure (bfsech x) → Bigfloat
> x : Bigfloat
> procedure (bfcsch x) → Bigfloat
> x : Bigfloat
> procedure (bfcoth x) → Bigfloat
> x : Bigfloat

Standard reciprocal hyperbolic functions. MPFR does not implement their inverses.

> procedure (bfsin+cos x) → (Values Bigfloat Bigfloat)
> x : Bigfloat

Simultaneously computes the sine and cosine of x.

> procedure (bfsinh+cosh x) → (Values Bigfloat Bigfloat)
> x : Bigfloat

Simultaneously computes the hyperbolic sine and cosine of x.

> procedure (bffactorial x) → Bigfloat
> x : Integer

Returns the factorial of x.

> procedure (bfgamma x) → Bigfloat
> x : Bigfloat

Computes the gamma function, a generalization of the factorial function.

> procedure (bflog-gamma x) → Bigfloat
> x : Bigfloat
> procedure (bflog-gamma/sign x) → (Values Bigfloat (U -1 1))
> x : Bigfloat

Computes the log-gamma function, or the log of the absolute value of the gamma function. bflog-gamma/sign additionally returns the sign of (bfgamma x).

> procedure (bfpsi0 x) → Bigfloat
> x : Bigfloat

Computes the digamma function, the logarithmic derivative of the gamma function.

> procedure (bfeint x) → Bigfloat
> x : Bigfloat

Returns the exponential integral of x.

> procedure (bfli2 x) → Bigfloat
> x : Bigfloat

Returns the dilogarithm of x, or the polylogarithm of order 2.

> procedure (bfzeta x) → Bigfloat
> x : Bigfloat

Computes the Riemann zeta function.

> procedure (bferf x) → Bigfloat
> x : Bigfloat
> procedure (bferfc x) → Bigfloat
> x : Bigfloat

Compute the error function and complementary error function, respectively.

> procedure (bfbesj0 x) → Bigfloat
> x : Bigfloat
> procedure (bfbesj1 x) → Bigfloat
> x : Bigfloat
> procedure (bfbesj n x) → Bigfloat
> n : Integer
> x : Bigfloat
> procedure (bfbesy0 x) → Bigfloat
> x : Bigfloat
> procedure (bfbesy1 x) → Bigfloat
> x : Bigfloat
> procedure (bfbesy n x) → Bigfloat
> n : Integer
> x : Bigfloat

These compute Bessel functions.

A “j” in the name indicates that a function computes a Bessel function of the first kind. A “y” indicates the second kind.

The “j” or “y” is followed by the order: zero, one, or n (user-specified).

> procedure (bfagm x y) → bigfloat
> x : Bigfloat
> y : Bigfloat

Returns the arithmetic-geometric mean of x and y. Typically, this isn’t directly useful, but it’s used in some asymptotically fast algorithms such as the one that computes bflog.

#### 5.10`` Low-level Functions

> procedure (bigfloat->ordinal x) → Integer
> x : Bigfloat
> procedure (ordinal->bigfloat n) → Bigfloat
> n : Integer
> procedure (bigfloats-between x y) → Integer
> x : Bigfloat
> y : Bigfloat
> procedure (bfstep x n) → Bigfloat
> x : Bigfloat
> n : Integer
> procedure (bfnext x) → Bigfloat
> x : Bigfloat
> procedure (bfprev x) → Bigfloat
> x : Bigfloat

Like flonum->ordinal, ordinal->flonum, flonums-between, flstep, flnext and flprev, but for bigfloats.

The major difference is that these operate using (bf-precision) bits. Additionally, unlike other bigfloat functions, all of these convert their bigfloat arguments to (bf-precision) bits.

> procedure (bfshift x n) → Bigfloat
> x : Bigfloat
> n : Integer

Like arithmetic-shift, but for bigfloats. More precisely, this returns (bf* x (bfexpt (bf 2) (bf n))), but is much faster.

> procedure (bfcanonicalize x) → Bigfloat
> x : Bigfloat

Bigfloats are canonicalized before hashing, to ensure that equality implies an equal hash. If x is nonzero and rational, returns a new bigfloat with no more bits of precision than are necessary to encode x exactly, by removing all low-order zeros from the significand and adjusting the exponent.

For zero or non-rational x, returns -inf.bf, -0.bf, 0.bf, +inf.bf, or +nan.bf, depending on the value of x.

Two nonzero, rational bigfloats are equal? if and only if their canonicalized significands and exponents are equal. Two zero or non-rational bigfloats are equal? if and only if their canonicalizations are eq?.

Canonicalizing bigfloats won’t change answers computed from them.

Examples:
> > (bf-precision 64)
> > (define x (bf 1 -2))
> > x
> (bf #e0.25)
> > (bfcanonicalize x)
> (bf #e0.25)
> > (bigfloat-precision x)
> 64
> > (bigfloat-precision (bfcanonicalize x))
> 2

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