spigot(1)

spigot(1) 💖 by Simon Tatham

okay bestie, welcome to the exact real calculator that literally never stops giving digits unless you interrupt it. no cap.

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✨ NAME ✨

spigot — command-line exact real calculator 🎀

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🌸 SYNOPSIS 🌸

spigot [ options ] expression

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🦋 DESCRIPTION 🦋

okay so spigot is an exact real calculator: you give it a mathematical expression to evaluate, and it computes it to any desired precision. by default it just prints digits to standard output until you interrupt it. yes, forever. it's giving infinite main character energy 💅

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. slay.

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

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

🫶 options that 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 printing leading zeroes if necessary.
• --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.

🌈 rounding options 🌈

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 ✨

okay bestie, spigot's expression language is lowkey a whole vibe, and it supports these operator tiers, listed from lowest priority to highest priority — so the higher ones grab tighter, iykyk:

• + and - — addition and subtraction. Left-associative.
• *, /, %, mod, rem — multiplication, division and remainder. Left-associative. % and mod are synonyms, and 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.
• ^ and ** — power. Right-associative.

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🫶 Let expressions 🫶

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.

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💅 Built-in functions 💅

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.

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🌸 Built-in constants 🌸

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.

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🦋 Number formats 🦋

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.

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💖 RETURN VALUE 💖

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.

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😭 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.)

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🎀 LICENCE 🎀

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

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💖 Shoutout to Simon Tatham for this iconic page, dated 2014‐12‐17, for spigot(1). Periodt. 💖