21 KiB
Nums
Tomo has two floating point number types: Num (64-bit, AKA double) and
Num32 (32-bit, AKA float). Num literals can have a decimal point (e.g.
5.), a scientific notation suffix (e.g. 1e8) or a percent sign. Numbers
that end in a percent sign are divided by 100 at compile time (i.e. 5% == 0.05). Numbers can also use the deg suffix to represent degrees, which
are converted to radians at compile time (i.e. 180deg == Nums.PI).
Nums support the standard math operations (x+y, x-y, x*y, x/y) as well as
powers/exponentiation (x^y) and modulus (x mod y and x mod1 y).
32-bit numbers can be constructed using the type name: Num32(123.456).
NaN
IEEE-754 floating point numbers define a concept call NaN (Not a Number),
which is the result value used to signal various operations (e.g. 0/0) that
have no mathematically defined result value. NaNs are implemented at the
hardware level and propagate through floating point operations. This allows you
to perform many chained operations on the assumption that it's unlikely to have
NaN values, and only perform checks at the end of the chain of operations,
instead of performing checks after each operation. Unfortunately, it's also
easy to forget to perform any checks at all because most type systems don't
differentiate between possibly-NaN values and definitely-not-NaN values.
Tomo has a separate concept for expressing the lack of a defined value:
optional types. Consequently, Tomo has merged these two concepts, so NaN is
called none and has the type Num? or Num32?. In this way, it's no
different from optional integers or optional arrays. This means that if a
variable has type Num, it is guaranteed to not hold a NaN value. This also
means that operations which may produce NaN values have a result type of
Num?. For example, division can take two non-NaN values and return a result
that is NaN (zero divided by zero). Similarly, multiplication can produce NaN
values (zero times infinity), and many math functions like sqrt() can return
NaN for some inputs.
Unfortunately, one of the big downsides of optional types is that explicit
none handling can be very verbose. To make Nums actually usable, Tomo applies
very liberal use of type coercion and implicit none checks when values are
required to be non-none. Here are a few examples:
>> x := 0.0
= 0 : Num
y := 1.0
# Division might produce none:
>> x / y
= 0 : Num?
>> x / x
= none : Num?
# Optional types and none values propagate:
>> x/y + 1 + 2
= 3 : Num?
>> x/x + 1 + 2
= none : Num?
# Optional Nums can be handled explicitly using `or` and `!`:
>> x/x or -123
= -123 : Num
# This would raise a runtime error if `x` and `y` were zero:
>> (x/y)!
= 0 : Num
# Assigning to a non-optional variable will do an implicit check for none and
# raise a runtime error if the value is none, essentially the same as an
# implicit `!`:
x = x/y
func doop(x:Num -> Num):
# If a function's return type is non-optional and an optional value is
# used in a return statement, an implicit none check will be inserted and
# will error if the value is none:
return x / 2
# Function arguments are also implicitly checked for none if the given value
# is optional and the function needs a non-optional value:
>> doop(x/y)
= 0 : Num
Hopefully the end result of this system is one where users can take advantage of the performance benefits of hardware NaN propagation, while still having the compiler enforce checking for undefined values. Users who don't want automatic NaN-checking can use optional types and explicit checks where necessary. By default, automatic NaN-checking happens at interface boundaries (function arguments, return values, and variable assignments), so NaN values should be caught early when an error message would have helpful context, while eliminating conditional branching inside of compound math expressions. Users should also be able to write code that can safely assume that all values provided are not NaN.
Constants
1_PI:\frac{1}{\pi}2_PI:2 \times \pi2_SQRTPI:2 \times \sqrt{\pi}E: Base of natural logarithms (e)INF: Positive infinityLN10: Natural logarithm of 10LN2: Natural logarithm of 2LOG2E: Logarithm base 2 ofePI: Pi (\pi)PI_2:\frac{\pi}{2}PI_4:\frac{\pi}{4}SQRT1_2:\sqrt{\frac{1}{2}}SQRT2:\sqrt{2}TAU: Tau (2 \times \pi)
Functions
Each Num type has its own version of the following functions. Functions can be
called either on the type itself: Num.sqrt(x) or as a method call:
x:sqrt(). Method call syntax is preferred.
func abs(n: Num -> Num)func acos(x: Num -> Num)func acosh(x: Num -> Num)func asin(x: Num -> Num)func asinh(x: Num -> Num)func atan(x: Num -> Num)func atan2(x: Num, y: Num -> Num)func atanh(x: Num -> Num)func cbrt(x: Num -> Num)func ceil(x: Num -> Num)func clamped(x, low, high: Num -> Num)func copysign(x: Num, y: Num -> Num)func cos(x: Num -> Num)func cosh(x: Num -> Num)func erf(x: Num -> Num)func erfc(x: Num -> Num)func exp(x: Num -> Num)func exp2(x: Num -> Num)func expm1(x: Num -> Num)func fdim(x: Num, y: Num -> Num)func floor(x: Num -> Num)func format(n: Num, precision: Int = 0 -> Text)func hypot(x: Num, y: Num -> Num)func isfinite(n: Num -> Bool)func isinf(n: Num -> Bool)func j0(x: Num -> Num)func j1(x: Num -> Num)func log(x: Num -> Num)func log10(x: Num -> Num)func log1p(x: Num -> Num)func log2(x: Num -> Num)func logb(x: Num -> Num)func mix(amount: Num, x: Num, y: Num -> Num)func near(x: Num, y: Num, ratio: Num = 1e-9, min_epsilon: Num = 1e-9 -> Bool)func nextafter(x: Num, y: Num -> Num)func parse(text: Text -> Num?)func percent(n: Num -> Text)func rint(x: Num -> Num)func round(x: Num -> Num)func scientific(n: Num, precision: Int = 0 -> Text)func significand(x: Num -> Num)func sin(x: Num -> Num)func sinh(x: Num -> Num)func sqrt(x: Num -> Num)func tan(x: Num -> Num)func tanh(x: Num -> Num)func tgamma(x: Num -> Num)func trunc(x: Num -> Num)func y0(x: Num -> Num)func y1(x: Num -> Num)
abs
Calculates the absolute value of a number.
func abs(n: Num -> Num)
n: The number whose absolute value is to be computed.
Returns:
The absolute value of n.
Example:
>> -3.5:abs()
= 3.5
acos
Computes the arc cosine of a number.
func acos(x: Num -> Num)
x: The number for which the arc cosine is to be calculated.
Returns:
The arc cosine of x in radians.
Example:
>> 0.0:acos() // -> (π/2)
= 1.5708
acosh
Computes the inverse hyperbolic cosine of a number.
func acosh(x: Num -> Num)
x: The number for which the inverse hyperbolic cosine is to be calculated.
Returns:
The inverse hyperbolic cosine of x.
Example:
>> 1.0:acosh()
= 0
asin
Computes the arc sine of a number.
func asin(x: Num -> Num)
x: The number for which the arc sine is to be calculated.
Returns:
The arc sine of x in radians.
Example:
>> 0.5:asin() // -> (π/6)
= 0.5236
asinh
Computes the inverse hyperbolic sine of a number.
func asinh(x: Num -> Num)
x: The number for which the inverse hyperbolic sine is to be calculated.
Returns:
The inverse hyperbolic sine of x.
Example:
>> 0.0:asinh()
= 0
atan
Computes the arc tangent of a number.
func atan(x: Num -> Num)
x: The number for which the arc tangent is to be calculated.
Returns:
The arc tangent of x in radians.
Example:
>> 1.0:atan() // -> (π/4)
= 0.7854
atan2
Computes the arc tangent of the quotient of two numbers.
func atan2(x: Num, y: Num -> Num)
x: The numerator.y: The denominator.
Returns:
The arc tangent of x/y in radians.
Example:
>> Num.atan2(1, 1) // -> (π/4)
= 0.7854
atanh
Computes the inverse hyperbolic tangent of a number.
func atanh(x: Num -> Num)
x: The number for which the inverse hyperbolic tangent is to be calculated.
Returns:
The inverse hyperbolic tangent of x.
Example:
>> 0.5:atanh()
= 0.5493
cbrt
Computes the cube root of a number.
func cbrt(x: Num -> Num)
x: The number for which the cube root is to be calculated.
Returns:
The cube root of x.
Example:
>> 27.0:cbrt()
= 3
ceil
Rounds a number up to the nearest integer.
func ceil(x: Num -> Num)
x: The number to be rounded up.
Returns:
The smallest integer greater than or equal to x.
Example:
>> 3.2:ceil()
= 4
clamped
Returns the given number clamped between two values so that it is within that range.
clamped(x, low, high: Num -> Num)
x: The number to clamp.low: The lowest value the result can take.high: The highest value the result can take.
Returns:
The first argument clamped between the other two arguments.
Example:
>> 2.5:clamped(5.5, 10.5)
= 5.5
copysign
Copies the sign of one number to another.
func copysign(x: Num, y: Num -> Num)
x: The number whose magnitude will be copied.y: The number whose sign will be copied.
Returns:
A number with the magnitude of x and the sign of y.
Example:
>> 3.0:copysign(-1)
= -3
cos
Computes the cosine of a number (angle in radians).
func cos(x: Num -> Num)
x: The angle in radians.
Returns:
The cosine of x.
Example:
>> 0.0:cos()
= 1
cosh
Computes the hyperbolic cosine of a number.
func cosh(x: Num -> Num)
x: The number for which the hyperbolic cosine is to be calculated.
Returns:
The hyperbolic cosine of x.
Example:
>> 0.0:cosh()
= 1
erf
Computes the error function of a number.
func erf(x: Num -> Num)
x: The number for which the error function is to be calculated.
Returns:
The error function of x.
Example:
>> 0.0:erf()
= 0
erfc
Computes the complementary error function of a number.
func erfc(x: Num -> Num)
x: The number for which the complementary error function is to be calculated.
Returns:
The complementary error function of x.
Example:
>> 0.0:erfc()
= 1
exp
Computes the exponential function e^x for a number.
func exp(x: Num -> Num)
x: The exponent.
Returns:
The value of e^x.
Example:
>> 1.0:exp()
= 2.7183
exp2
Computes 2^x for a number.
func exp2(x: Num -> Num)
x: The exponent.
Returns:
The value of 2^x.
Example:
>> 3.0:exp2()
= 8
expm1
Computes e^x - 1 for a number.
func expm1(x: Num -> Num)
x: The exponent.
Returns:
The value of e^x - 1.
Example:
>> 1.0:expm1()
= 1.7183
fdim
Computes the positive difference between two numbers.
func fdim(x: Num, y: Num -> Num)
x: The first number.y: The second number.
Returns:
The positive difference \max(0, x - y).
Example:
fd
>> 5.0:fdim(3)
= 2
floor
Rounds a number down to the nearest integer.
func floor(x: Num -> Num)
x: The number to be rounded down.
Returns:
The largest integer less than or equal to x.
Example:
>> 3.7:floor()
= 3
format
Formats a number as a text with a specified precision.
func format(n: Num, precision: Int = 0 -> Text)
n: The number to be formatted.precision: The number of decimal places. Default is0.
Returns: A text representation of the number with the specified precision.
Example:
>> 3.14159:format(precision=2)
= "3.14"
hypot
Computes the Euclidean norm, \sqrt{x^2 + y^2}, of two numbers.
func hypot(x: Num, y: Num -> Num)
x: The first number.y: The second number.
Returns:
The Euclidean norm of x and y.
Example:
>> Num.hypot(3, 4)
= 5
isfinite
Checks if a number is finite.
func isfinite(n: Num -> Bool)
n: The number to be checked.
Returns:
yes if n is finite, no otherwise.
Example:
>> 1.0:isfinite()
= yes
>> Num.INF:isfinite()
= no
isinf
Checks if a number is infinite.
func isinf(n: Num -> Bool)
n: The number to be checked.
Returns:
yes if n is infinite, no otherwise.
Example:
>> Num.INF:isinf()
= yes
>> 1.0:isinf()
= no
j0
Computes the Bessel function of the first kind of order 0.
func j0(x: Num -> Num)
x: The number for which the Bessel function is to be calculated.
Returns:
The Bessel function of the first kind of order 0 of x.
Example:
>> 0.0:j0()
= 1
j1
Computes the Bessel function of the first kind of order 1.
func j1(x: Num -> Num)
x: The number for which the Bessel function is to be calculated.
Returns:
The Bessel function of the first kind of order 1 of x.
Example:
>> 0.0:j1()
= 0
log
Computes the natural logarithm (base e) of a number.
func log(x: Num -> Num)
x: The number for which the natural logarithm is to be calculated.
Returns:
The natural logarithm of x.
Example:
>> Num.E:log()
= 1
log10
Computes the base-10 logarithm of a number.
func log10(x: Num -> Num)
x: The number for which the base-10 logarithm is to be calculated.
Returns:
The base-10 logarithm of x.
Example:
>> 100.0:log10()
= 2
log1p
Computes \log(1 + x) for a number.
func log1p(x: Num -> Num)
x: The number for which\log(1 + x)is to be calculated.
Returns:
The value of \log(1 + x).
Example:
>> 1.0:log1p()
= 0.6931
log2
Computes the base-2 logarithm of a number.
func log2(x: Num -> Num)
x: The number for which the base-2 logarithm is to be calculated.
Returns:
The base-2 logarithm of x.
Example:
>> 8.0:log2()
= 3
logb
Computes the binary exponent (base-2 logarithm) of a number.
func logb(x: Num -> Num)
x: The number for which the binary exponent is to be calculated.
Returns:
The binary exponent of x.
Example:
>> 8.0:logb()
= 3
mix
Interpolates between two numbers based on a given amount.
func mix(amount: Num, x: Num, y: Num -> Num)
amount: The interpolation factor (between0and1).x: The starting number.y: The ending number.
Returns:
The interpolated number between x and y based on amount.
Example:
>> 0.5:mix(10, 20)
= 15
>> 0.25:mix(10, 20)
= 12.5
near
Checks if two numbers are approximately equal within specified tolerances. If two numbers are within an absolute difference or the ratio between the two is small enough, they are considered near each other.
func near(x: Num, y: Num, ratio: Num = 1e-9, min_epsilon: Num = 1e-9 -> Bool)
x: The first number.y: The second number.ratio: The relative tolerance. Default is1e-9.min_epsilon: The absolute tolerance. Default is1e-9.
Returns:
yes if x and y are approximately equal within the specified tolerances, no otherwise.
Example:
>> 1.0:near(1.000000001)
= yes
>> 100.0:near(110, ratio=0.1)
= yes
>> 5.0:near(5.1, min_epsilon=0.1)
= yes
nextafter
Computes the next representable value after a given number towards a specified direction.
func nextafter(x: Num, y: Num -> Num)
x: The starting number.y: The direction towards which to find the next representable value.
Returns:
The next representable value after x in the direction of y.
Example:
>> 1.0:nextafter(1.1)
= 1.0000000000000002
parse
Converts a text representation of a number into a floating-point number.
func parse(text: Text -> Num?)
text: The text containing the number.
Returns:
The number represented by the text or none if the entire text can't be parsed
as a number.
Example:
>> Num.parse("3.14")
= 3.14
>> Num.parse("1e3")
= 1000
percent
Convert a number into a percentage text with a percent sign.
func percent(n: Num, precision: Int = 0 -> Text)
n: The number to be converted to a percent.precision: The number of decimal places. Default is0.
Returns: A text representation of the number as a percentage with a percent sign.
Example:
>> 0.5:percent()
= "50%"
>> (1./3.):percent(2)
= "33.33%"
rint
Rounds a number to the nearest integer, with ties rounded to the nearest even integer.
func rint(x: Num -> Num)
x: The number to be rounded.
Returns:
The nearest integer value of x.
Example:
>> 3.5:rint()
= 4
>> 2.5:rint()
= 2
round
Rounds a number to the nearest whole number integer.
func round(x: Num -> Num)
x: The number to be rounded.
Returns:
The nearest integer value of x.
Example:
>> 2.3:round()
= 2
>> 2.7:round()
= 3
scientific
Formats a number in scientific notation with a specified precision.
func scientific(n: Num, precision: Int = 0 -> Text)
n: The number to be formatted.precision: The number of decimal places. Default is0.
Returns: A text representation of the number in scientific notation with the specified precision.
Example:
>> 12345.6789:scientific(precision=2)
= "1.23e+04"
significand
Extracts the significand (or mantissa) of a number.
func significand(x: Num -> Num)
x: The number from which to extract the significand.
Returns:
The significand of x.
Example:
>> 1234.567:significand()
= 0.1234567
sin
Computes the sine of a number (angle in radians).
func sin(x: Num -> Num)
x: The angle in radians.
Returns:
The sine of x.
Example:
>> 0.0:sin()
= 0
sinh
Computes the hyperbolic sine of a number.
func sinh(x: Num -> Num)
x: The number for which the hyperbolic sine is to be calculated.
Returns:
The hyperbolic sine of x.
Example:
>> 0.0:sinh()
= 0
sqrt
Computes the square root of a number.
func sqrt(x: Num -> Num)
x: The number for which the square root is to be calculated.
Returns:
The square root of x.
Example:
>> 16.0:sqrt()
= 4
tan
Computes the tangent of a number (angle in radians).
func tan(x: Num -> Num)
x: The angle in radians.
Returns:
The tangent of x.
Example:
>> 0.0:tan()
= 0
tanh
Computes the hyperbolic tangent of a number.
func tanh(x: Num -> Num)
x: The number for which the hyperbolic tangent is to be calculated.
Returns:
The hyperbolic tangent of x.
Example:
>> 0.0:tanh()
= 0
tgamma
Computes the gamma function of a number.
func tgamma(x: Num -> Num)
x: The number for which the gamma function is to be calculated.
Returns:
The gamma function of x.
Example:
>> 1.0:tgamma()
= 1
trunc
Truncates a number to the nearest integer towards zero.
func trunc(x: Num -> Num)
x: The number to be truncated.
Returns:
The integer part of x towards zero.
Example:
>> 3.7:trunc()
= 3
>> (-3.7):trunc()
= -3
y0
Computes the Bessel function of the second kind of order 0.
func y0(x: Num -> Num)
x: The number for which the Bessel function is to be calculated.
Returns:
The Bessel function of the second kind of order 0 of x.
Example:
>> 1.0:y0()
= -0.7652
y1
Computes the Bessel function of the second kind of order 1.
func y1(x: Num -> Num)
x: The number for which the Bessel function is to be calculated.
Returns:
The Bessel function of the second kind of order 1 of x.
Example:
>> 1.0:y1()
= 0.4401