#### NAME

spigot - command-line exact real calculator

#### SYNOPSIS

spigot [ options ] expression

#### DESCRIPTION

spigot  is  an  exact  real  calculator: that is, you give it a mathematical expression to
evaluate, and it computes it to any desired precision, by default simply  printing  digits
to standard output until it is interrupted.

spigot provides command-line options to control the format of the output, restrict it to a
specified number of digits, and apply rounding at the end of those digits. It can  produce
output  in any base between 2 and 36 (after that it runs out of digit characters), or as a
continued fraction, and it can read input numbers from files in any of  those  formats  as
well.

This  man  page gives only a brief summary of spigot's functionality. For full detail, you
should read the main manual spigot.html; if that is not installed on your system, you  can
find it on the web at

http://www.chiark.greenend.org.uk/~sgtatham/spigot/spigot.html

#### OPTIONS

The following options control spigot's basic output format:

-b base, -B base
Output  the  number  in  base  base,  which  must  be  an  integer between 2 and 36
inclusive. Digits above 9 are represented by lower case or upper case letters,  for
the options -b and -B respectively. The default is -b 10.

-c     Output  the  number  as  a  list  of  continued  fraction  coefficients, as decimal
integers, by default one per line.

-C     Output the number's continued fraction convergents, one per line, in  the  form  of
two decimal integers with a / between them.

-R     Output the number's value as a rational, in the form of two decimal integers with a
/ between them, or just one decimal integer if the number is a rational. If  spigot
does not know the number to be rational immediately, it will start evaluating it to
see if it turns out rational later, so if it  is  not  rational  then  spigot  will
compute for ever.

-S, -D, -Q, -H
Output  the  number  as  a hex representation of an IEEE 754 bit pattern, in 32-bit
single  precision,  64-bit  double,  128-bit  quad   or   16-bit   half   precision
respectively.  If that representation is not exact, a decimal point will be printed
followed by further mantissa digits.

--printf format, --printf=format
Format the number in the same  way  that  printf(3)  would,  given  the  formatting
directive format. format must begin with a % and end with the associated conversion
specifier, which must be a floating-point one (one of efgaEFGA).

The following options modify the details of those output formats:

-d limit
Limit the amount of data output. In -b mode, no more than limit  digits  after  the
decimal  point are printed. In -c or -C mode, no more than limit continued fraction
coefficients or convergents are printed, not counting the initial one  representing
the  number's  integer  part.  In  the  IEEE  754  output modes, no more than limit
additional bits of precision are generated after the end of the official  mantissa.
limit may be negative.

-l     In -c mode, output continued fraction terms all on one line, separated by a ; after
the first term and , after each subsequent term.

-w min-int-digits
In -b mode, output at  least  min-int-digits  of  the  number's  integer  part,  by

--nibble
In  --printf  mode  with  the  `a'  or  `A' conversion specifier, choose the output
exponent to always be a multiple of 4, instead of the default behaviour of choosing
it as large as possible.

-n     In  any  mode  where  spigot  prints  output  on  a single line, suppress the usual
trailing newline if spigot's output terminates.

The following options control rounding, when spigot's output is limited by the -d  option.
(Rounding does not occur in continued fraction modes.)

--rz   Round towards zero. This is the default.

--ri   Round away from zero.

--ru   Round up (towards positive infinity).

--rd   Round down (towards negative infinity).

--rn, --rne
Round to nearest, breaking ties toward an even last digit.

--rno  Round to nearest, breaking ties toward an odd last digit.

--rnz, --rni, --rnu, --rnd
Round  to  nearest,  breaking  ties  as  if  rounding  via --rz, --ri, --ru or --rd
respectively.

Miscellaneous options:

--tentative=state
Control the printing of `tentative output'. Tentative output is printed when spigot
does  not  know for sure what the next digit of the number is because it's starting
to look as if it's exactly on a digit boundary. Tentative output  is  in  red,  and
followed  by an indication of about how many digits spigot has examined beyond that
point (i.e. how close to exact that digit is known to be); spigot will  retract  it
later if it finds out something definite.

state  can  be  `on', `off' or `auto'. `auto' is the default, and means that spigot
should only print tentative output if its output is directed to a terminal device.

-T     If instructed to read from a file descriptor which points to a  terminal,  put  the
terminal into raw mode (turning off ICANON and ECHO modes) while doing so.

#### EXPRESSIONS

spigot's  expression  language  supports  the following options, in order of priority from
lowest to highest:

+ and -

*, /, %, mod, rem
Multiplication, division and remainder. (Left-associative.) % and mod are synonyms,
which  both  return  a  remainder  between  0  and  the  denominator; rem returns a
remainder of either sign, with absolute value at most half that of the denominator,
and ties broken by rounding to even in IEEE 754 style.

Unary - and +
Negation and no-op.

^, **  Power. (Right-associative.)

You  can  define  variables  and  functions  of  your  own in subexpressions using the let
expression, as follows:

let var=value in expression
Defines the name var to refer to the value of the expression value. The  definition
is in scope within expression, but not in any other parts of the spigot input.

let fn(params)=defn in expression
Defines  the  syntax  fn(args)  to  refer to the expression defn with the arguments
substituted in for the  parameters.  params  must  be  a  comma-separated  list  of
identifiers; args is a comma-separated list of expressions.

A  let expression can contain multiple definitions, separated by commas, e.g. `let x=1,y=2
in x+y'. Each definition is in scope for subsequent definitions, so  you  can  write  `let
x=1,y=x+1  in' expr.  But  definitions  are  not  in  scope for themselves; in particular,
functions may not be recursive.

spigot also provides the following built-in functions:

sqrt, cbrt
Square and cube roots.

hypot, atan2 (two arguments)
Rectangular to polar coordinate conversions: the hypotenuse function  (square  root
of the sum of the squared arguments), and two-variable inverse tangent.

sin, cos, tan, asin, acos, atan
Trigonometric functions and their inverses.

sind, cosd, tand, asind, acosd, atand, atan2d
Trigonometric  functions and their inverses, equivalent to the versions without `d'
on the end except that angles are measured in degrees.

sinh, cosh, tanh, asinh, acosh, atanh
Hyperbolic functions and their inverses.

exp, exp2, exp10, log, log2, log10
Exponential and logarithmic functions: raise e, 2 and 10 to a power, or take a  log
with  the  same three bases. You can also provide a base of your choice as a second
argument to log.

expm1, log1p
Shorthands for exp(x)-1 and log(1+x).

pow (two arguments)
Synonym for the ^ operator.

gamma, tgamma, lgamma
Gamma function (gamma and tgamma are  synonyms  for  this),  and  the  log  of  the
absolute value of the gamma function.

erf, erfc, Phi, norm
Error-function  relatives:  the  error function itself, 1 minus the error function,
and Phi and norm are synonyms for the cumulative normal distribution function.

erfinv, erfcinv, Phiinv, norminv
Inverses of the above error-function relatives.

W, Wn  The Lambert W function, i.e. the inverse of x exp(x). W is the branch with value at
least -1, and Wn is the branch with value at most -1.

Ei, En (two arguments), E1, Ein
Exponential  integrals,  i.e.  integrals  of  things  like  exp(x)/x.  Ei(x) is the
indefinite integral of exp(x)/x itself; En(n,x) (for non-negative integer n) is the
result  of  integrating  exp(-x)/x  n  times, flipping the sign each time; E1(x) is
shorthand for En(1,x); and Ein(x) is the integral of (1-exp(-x))/x.

Li, li Logarithmic integrals, i.e. integrals of 1/log(x). Li(x) and  li(x)  are  both  the
indefinite  integral  of  1/log(x);  only their constants differ, in that Li(2) and
li(0) are each defined to be zero.

Si, si, Ci, Cin
Sine and cosine integrals, i.e. integrals of sin(x)/x and cos(x)/x. Si(x) and si(x)
are  both  the  indefinite  integral  of  sin(x)/x, differing only in the constant:
Si(0)=0, but si(x) has limit 0 as x  tends  to  positive  infinity.  Ci(x)  is  the
indefinite  integral of cos(x)/x, also with limit 0 at positive infinity; Cin(x) is
the indefinite integral of (1-cos(x))/x, with Cin(0)=0.

UFresnelS, UFresnelC, FresnelS, FresnelC
Fresnel integrals. UFresnelS and UFresnelC are the indefinite integrals of sin(x^2)
and  cos(x^2);  FresnelS and FresnelC are the `normalised' versions, i.e. integrals
of sin(π x^2/2) and cos(π x^2/2). All are zero at the origin.

zeta   The Riemann zeta function (restricted to the real numbers).

abs    Absolute value.

ceil, floor
Ceiling and floor: smallest integer at least x, and largest integer at most x.

frac   Fractional part, i.e. x - floor(x).

algebraic (variable number of arguments)
Return a root of an arbitrary polynomial with integer coefficients. The  first  two
arguments  are  the rational bounds of an interval to search, and the rest give the
polynomial's coefficients, with constant term first.

spigot supports the following names for built-in constants:

pi, tau
The circle constant π, and the often more useful 2 π.

e      The base of natural logarithms.

phi    The golden ratio, (1+sqrt(5))/2.

eulergamma
The Euler-Mascheroni constant: the limiting difference  between  the  sum  and  the
integral of 1/n.

apery  Apery's constant: the sum of the reciprocals of the cubes.

Numbers can be input in the following formats:

·      Decimal, with an optional C-style e+exponent or e-exponent for scientific notation

·      Hex,  with  the  prefix  0x,  and  an  optional  C99-style p+exponent or p-exponent
representing a power of 2 multiplier

·      In any base between 2 and 36, with a prefix of the form baseN:, e.g. base7:0.123456

·      As an IEEE 754 hex bit pattern, consisting of exactly 4, 8, 16  or  32  hex  digits
with the prefix ieee:, followed by optional decimal point and extra mantissa digits

·      From  a  file  in base notation, by writing baseNfile: followed by a filename, e.g.
base10fd:pi.txt. The filename is taken to be  the  maximal  sequence  of  non-space
characters  following the prefix, unless it starts with ' or ", in which case it is
taken to be everything up to a matching closing quote, with doubled quote marks  in
between representing a literal quote character.

·      From  a  file  in  continued fraction notation, by writing cfracfile: followed by a
filename.

·      Either of the above, but with file: replaced by xfile: to indicate that end of file
should  be taken as the number being exactly represented rather than running out of
precision.

·      From a file descriptor in any of those notations, by writing baseNfd:  or  cfracfd:
followed by an fd number, e.g. base10fd:0 to read from standard input.

#### RETURNVALUE

spigot  returns  0  if  its  output terminates (because the result is exact, or because it
reached the specified -d limit) with no problems.

In case of a parse error, or an invalid operand to a function, or any other kind of  fatal
error, spigot returns 1.

If spigot is unable to generate output to the desired precision because more precision was
needed from a number read from an input file using baseNfile: or cfracfile:,  then  spigot
returns 2, and prints an error message indicating which input file (in case there was more
than one) ran out first.

#### LIMITATIONS

Due to inherent limitations of its exact real arithmetic  strategy,  spigot  is  generally
unable  to recognise when a number it is computing is exactly equal to a specific boundary
value.

One effect of this is that spigot will not behave as you'd like if the output number has a
terminating   representation   in   the   selected   base.   For   example,   asking   for
sin(asin(0.12345)) will not be able to print 0.12345 and exit. Instead, spigot will get as
far  as  printing  `0.1234', and then print tentative output (mentioned above) to indicate
that it thinks the next digit might be exactly 5, but it will never reach  a  point  where
it's sure of that.

Another  effect  is  that  if  you  ask  spigot  to  evaluate  an  expression  in which an
intermediate result is precisely on a point of discontinuity of the function it is  passed
to, then it may never manage to even start producing output. For example, spigot will hang
completely if you ask it for floor(sin(pi)), since sin(pi) = 0 is a point of discontinuity
of  the  floor  function,  and spigot will never be able to work out that the value of the
input to floor is exactly zero, only that it seems to be closer and  closer  to  zero  the
more it computes.

(An  exception  is  numbers  that  spigot  knows from first principles to be rational. For
example, if you ask spigot to evaluate the simpler expressions `0.12345' or `floor(0)', it
will print the complete output and terminate successfully, in both cases.)

#### LICENCE

spigot is free software, distributed under the MIT licence. Type `spigot --licence' to see
the full licence text.