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X!XX!Xi16::MAX as f64S S STT?TS' S'SS S'Si32::MIN as f64 - 1.0T T TTT?TT' T'TT T'Ti32::MAX as f64 + 1.0W WW WWWWi32::MIN as f64X XX! X!XX!Xi32::MAX as f64S S STT?TS' S'SS S'Si64::MIN as f64 - 1.0T T TTT?TT' T'TT T'Ti64::MAX as f64 + 1.0W WW WWWWi64::MIN as f64X XX! X!XX!Xi64::MAX as f64S S STT?TS' S'SS S'Si128::MIN as f64 - 1.0T T TTT?TT' T'TT T'Ti128::MAX as f64 + 1.0W WW WWWWi128::MIN as f64X XX! X!XX!Xi128::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^usize::MAX as f64 + 1.0a aa! a!aa!ausize::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^u8::MAX as f64 + 1.0a aa! a!aa!au8::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^u16::MAX as f64 + 1.0a aa! a!aa!au16::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^u32::MAX as f64 + 1.0a aa! a!aa!au32::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^u64::MAX as f64 + 1.0a aa! a!aa!au64::MAX as f64^ ^ ^^^?^^' ^'^^ ^'^u128::MAX as f64 + 1.0a aa! a!aa!au128::MAX as f64  ˘!˘!ۘ˘! 2139095040  !!!8388607  ӝ"ӝ"ӝ" 2147483648))ݔBBB(57.2957795130823208767981548141051703f32رر±)±)ұ±)9218868437227405312)))4503599627370495ڶ*ڶ*ڶ*9223372036854775808Ԧ' Numeric traits for generic mathematics* ## Compatibility< The `num-traits` crate is tested for rustc 1.8 and greater.?  https://docs.rs/num-traits/0.2 9    $B            >           "           u                 $       $     J         d              .      II$$I))ID114= has_i1285/IK114= 5/KIQ114= 5/QI\ IccIhhImmIrrIvvI IIIIII* Numbers which have upper and lower bounds-]]; Returns the smallest finite number this type can represent> ]] ^: Returns the largest finite number this type can represent= ]] _ Numbers which have lower bounds# ``> `` a#Icc] bb d Numbers which have upper bounds# ee= ee f#Ihh gg i (IllWrapping Џ kk m kk n  r'r5r<<rAAr\IILU MGrcIILU MGcriIILU MGirt r{{rrrrr rrrrrr4 A generic trait for converting a value to a number.7B A value can be represented by the target type when it lies withinE3 the range of scalars supported by the target type.6E For example, a negative integer cannot be represented by an unsignedHC integer type, and an `i64` with a very high magnitude might not beF convertible to an `i32`.J On the other hand, conversions with possible precision loss or truncationMG are admitted, like an `f32` with a decimal part to an integer type, orJ1 even a large `f64` saturating to `f32` infinity.4 #C Converts the value of `self` to an `isize`. If the value cannot beF4 represented by an `isize`, then `None` is returned.7  @ Converts the value of `self` to an `i8`. If the value cannot beC1 represented by an `i8`, then `None` is returned.4    A Converts the value of `self` to an `i16`. If the value cannot be D2 represented by an `i16`, then `None` is returned. 5    A Converts the value of `self` to an `i32`. If the value cannot be D2 represented by an `i32`, then `None` is returned. 5    A Converts the value of `self` to an `i64`. If the value cannot be D2 represented by an `i64`, then `None` is returned.5 !B Converts the value of `self` to an `i128`. If the value cannot beEH represented by an `i128` (`i64` under the default implementation), thenK `None` is returned.K This method is only available with feature `i128` enabled on Rust >= 1.26.NK The default implementation converts through `to_i64()`. Types implementingNN this trait should override this method if they can represent a greater range.Q  #B Converts the value of `self` to a `usize`. If the value cannot beE3 represented by a `usize`, then `None` is returned.6 ? Converts the value of `self` to a `u8`. If the value cannot beB0 represented by a `u8`, then `None` is returned.3 @ Converts the value of `self` to a `u16`. If the value cannot beC1 represented by a `u16`, then `None` is returned.4 @ Converts the value of `self` to a `u32`. If the value cannot beC1 represented by a `u32`, then `None` is returned.4  @ Converts the value of `self` to a `u64`. If the value cannot beC1 represented by a `u64`, then `None` is returned.4  !A Converts the value of `self` to a `u128`. If the value cannot beDG represented by a `u128` (`u64` under the default implementation), thenJNK The default implementation converts through `to_u64()`. Types implementingNQ          "H Converts the value of `self` to an `f32`. Overflows may map to positive!KH or negative inifinity, otherwise `None` is returned if the value cannot!K be represented by an `f32`.""# 'H Converts the value of `self` to an `f64`. Overflows may map to positive#K$K be represented by an `f64`.%%D The default implementation tries to convert through `to_i64()`, and%GF failing that through `to_u64()`. Types implementing this trait should%I< override this method if they can represent a greater range.&?'' ()(,*,0"09*9<+<@#@I-I L#LLLL has_to_int_uncheckedLLLO2P%Z4Z'c$co4 A generic trait for converting a number to a value.j7kkEk6lHlFmmMnJn4o oop'E Converts an `isize` to return an optional value of this type. If theoHC value cannot be represented by this type, then `None` is returned.pFp  r!B Converts an `i8` to return an optional value of this type. If theqErFs t#C Converts an `i16` to return an optional value of this type. If thesFtFu w#C Converts an `i32` to return an optional value of this type. If theuFvFw x$C Converts an `i64` to return an optional value of this type. If thewFxFx |%D Converts an `i128` to return an optional value of this type. If theyGyFzzN{M The default implementation converts through `from_i64()`. Types implementing{P{Q|||| |||  ~'D Converts a `usize` to return an optional value of this type. If the}G~F~  !B Converts an `u8` to return an optional value of this type. If theEF #C Converts an `u16` to return an optional value of this type. If theFF #C Converts an `u32` to return an optional value of this type. If theFF $C Converts an `u64` to return an optional value of this type. If theFF %D Converts an `u128` to return an optional value of this type. If theGFˆʈNM The default implementation converts through `from_u64()`. Types implementingPQ ފ  #B Converts a `f32` to return an optional value of this type. If theۋEF #B Converts a `f64` to return an optional value of this type. If theލEFF The default implementation tries to convert through `from_i64()`, andIH failing that through `from_u64()`. Types implementing this trait shouldɏK?  'ä0r )4r 6) Cast from one machine scalar to another., # Examples ``` # use num_traits as num;, let twenty: f32 = num::cast(0x14).unwrap();/ assert_eq!(twenty, 20f32);б &2 An interface for casting between machine scalars.ֲ5  .? Creates a number from another value that can be converted intoBG a primitive via the `ToPrimitive` trait. If the source value cannot beJ9 represented by the target type, then `None` is returned.ϴ<E6HFַMJ4  (r- (A A generic interface for casting between machine scalars with theD: `as` operator, which admits narrowing and precision loss.=H Implementers of this trait `AsPrimitive` should behave like a primitiveK@ numeric type (e.g. a newtype around another primitive), and theC% intended conversion must never fail.( # use num_traits::AsPrimitive;"( let three: i32 = (3.14159265f32).as_();+ assert_eq!(three, 3); # Safety [ **In Rust versions before 1.45.0**, some uses of the `as` operator were not entirely safe.^, In particular, it was undefined behavior if/E a truncated floating point value could not fit in the target integerHA type ([#10184](https://github.com/rust-lang/rust/issues/10184)).D ```ignore "$ let x: u8 = (1.04E+17).as_(); // UB' ϴ5 Convert a value to another, using the `as` operator.8     .::??DDO U^ dnoott} } KC Generic trait for floating point numbers that works with `no_std`.F5 This trait implements a subset of the `Float` trait.8  ϴ ͹ ϴ  Returns positive infinity." use num_traits::float::FloatCore;% use std::{f32, f64}; fn check(x: T) {"! assert!(T::infinity() == x);$ } check(f32::INFINITY); check(f64::INFINITY);  Returns negative infinity.ٺ%"% assert!(T::neg_infinity() == x);( check(f32::NEG_INFINITY); check(f64::NEG_INFINITY);    Returns NaN.    ٺ %  fn check() {  let n = T::nan();  assert!(n != n);    check::();  check::();     Returns `-0.0`.     ٺ %   fn check(n: T) { " let z = T::neg_zero();  assert!(z.is_zero());  assert!(T::one() / z == n); #    @ Returns the smallest finite value that this type can represent.Cٺ%"" assert!(T::min_value() == x);% check(f32::MIN); check(f64::MIN);   N Returns the smallest positive, normalized value that this type can represent.Qٺ%"+ assert!(T::min_positive_value() == x);. check(f32::MIN_POSITIVE); check(f64::MIN_POSITIVE); ) Returns epsilon, a small positive value.,ٺ%" assert!(T::epsilon() == x);# check(f32::EPSILON); check(f64::EPSILON); ? Returns the largest finite value that this type can represent.Bٺ%"" assert!(T::max_value() == x);% check(f32::MAX); check(f64::MAX);  % Returns `true` if the number is NaN.(ٺ%( fn check(x: T, p: bool) {+ assert!(x.is_nan() == p);! check(f32::NAN, true); check(f32::INFINITY, false);  check(f64::NAN, true); check(0.0f64, false); #* Returns `true` if the number is infinite.-ٺ%   +# assert!(x.is_infinite() == p); &!! check(f32::INFINITY, true);! check(f32::NEG_INFINITY, true);!# check(f32::NAN, false);! check(f64::INFINITY, true);! check(f64::NEG_INFINITY, true);"#""#  '9 Returns `true` if the number is neither infinite or NaN.#<$$$$ٺ$%%%%+! assert!(x.is_finite() == p);%$&&&  check(f32::MAX, true);&! check(f64::NEG_INFINITY, false);&$ check(f64::MIN_POSITIVE, true);'# check(f64::NAN, false);'''  ,J Returns `true` if the number is neither zero, infinite, subnormal or NaN.(M))))ٺ)%)**+! assert!(x.is_normal() == p);*$*** ++$+#,,,  3 H Returns the floating point category of the number. If only one property-KB is going to be tested, it is generally faster to use the specific-E predicate instead......ٺ/%/ use std::num::FpCategory;//. fn check(x: T, c: FpCategory) {/1 assert!(x.classify() == c);0#00, check(f32::INFINITY, FpCategory::Infinite);0/% check(f32::MAX, FpCategory::Normal);1(" check(f64::NAN, FpCategory::Nan);1%. check(f64::MIN_POSITIVE, FpCategory::Normal);117 check(f64::MIN_POSITIVE / 2.0, FpCategory::Subnormal);2:! check(0.0f64, FpCategory::Zero);2$33 8< Returns the largest integer less than or equal to a number.3?4444ٺ4%44% fn check(x: T, y: T) {4( assert!(x.floor() == y);5 55% check(f32::INFINITY, f32::INFINITY);5( check(0.9f32, 0.0);6 check(1.0f32, 1.0);6 check(1.1f32, 1.0);6 check(-0.0f64, 0.0);6 check(-0.9f64, -1.0);7 check(-1.0f64, -1.0);7 check(-1.1f64, -2.0);7 check(f64::MIN, f64::MIN);778 >@ Returns the smallest integer greater than or equal to a number.:C::::ٺ;%;;;( assert!(x.ceil() == y);;<<<( check(0.9f32, 1.0);<< check(1.1f32, 2.0);== check(-0.9f64, -0.0);== check(-1.1f64, -1.0);>>>> EO Returns the nearest integer to a number. Round half-way cases away from `0.0`.@RAAAAٺA%BBB( assert!(x.round() == y);B CCC( check(0.4f32, 0.0);C check(0.5f32, 1.0);C check(0.6f32, 1.0);C check(-0.4f64, 0.0);D check(-0.5f64, -1.0);D check(-0.6f64, -1.0);DDEE M% Return the integer part of a number.I(IIIJٺJ%JJJ( assert!(x.trunc() == y);K KKK(KLLLLLMMMM S) Returns the fractional part of a number.O,OOOOٺO%PPP( assert!(x.fract() == y);P QQ check(f32::MAX, 0.0);Q check(0.75f32, 0.75);Q check(1.0f32, 0.0);Q check(1.25f32, 0.25);QR check(-0.75f64, -0.75);R check(-1.0f64, 0.0);R check(-1.25f64, -0.25);R check(f64::MIN, 0.0);SSS YI Computes the absolute value of `self`. Returns `FloatCore::nan()` if theTL number is `FloatCore::nan()`.U!UUUUٺV%VVV( assert!(x.abs() == y);WWWW(W check(0.0f64, 0.0);WX check(-1.0f64, 1.0);X check(f64::MIN, f64::MAX);XXY `5 Returns a number that represents the sign of `self`.Z8[E - `1.0` if the number is positive, `+0.0` or `FloatCore::infinity()`[HJ - `-1.0` if the number is negative, `-0.0` or `FloatCore::neg_infinity()`[M9 - `FloatCore::nan()` if the number is `FloatCore::nan()`\<\]]]ٺ]%]]]( assert!(x.signum() == y);^!^^ check(f32::INFINITY, 1.0);^ check(3.0f32, 1.0);_ check(0.0f32, 1.0);_ check(-0.0f64, -1.0);_ check(-3.0f64, -1.0);_ check(f64::MIN, -1.0);_`` g!; Returns `true` if `self` is positive, including `+0.0` andb>2 `FloatCore::infinity()`, and since Rust 1.20 alsob5 `FloatCore::nan()`.bccccٺc%cdd+( assert!(x.is_sign_positive() == p);d+deee check(0.0f32, true);e check(-0.0f64, false);ef$f# check(-f64::NAN, false);ffg l!; Returns `true` if `self` is negative, including `-0.0` andg>6 `FloatCore::neg_infinity()`, and since Rust 1.20 alsoh9 `-FloatCore::nan()`.hiiiiٺi%iij+( assert!(x.is_sign_negative() == p);j+jjj  check(f32::MAX, false);k check(0.0f32, false);k check(-0.0f64, true);kk#! check(f64::MIN_POSITIVE, false);l$llm r!( Returns the minimum of the two numbers.m+nE If one of the arguments is NaN, then the other argument is returned.nHnnooٺo%oo- fn check(x: T, y: T, min: T) {o0 assert!(x.min(y) == min);p!pp check(1.0f32, 2.0, 1.0);p check(f32::NAN, 2.0, 2.0);q check(1.0f64, -2.0, -2.0);q check(1.0f64, f64::NAN, 1.0);q!qr x!( Returns the maximum of the two numbers.t+ttHuuuuٺu%uv- fn check(x: T, y: T, max: T) {v0 assert!(x.max(y) == max);v!vv check(1.0f32, 2.0, 2.0);v check(1.0f32, f32::NAN, 1.0);w! check(-1.0f64, 2.0, 2.0);w check(-1.0f64, f64::NAN, -1.0);w#xx ~? Returns the reciprocal (multiplicative inverse) of the number.zBzz{{ٺ{%{{{( assert!(x.recip() == y);|  assert!(y.recip() == x);| || check(f32::INFINITY, 0.0);| check(2.0f32, 0.5);} check(-0.25f64, -4.0);}# check(-0.0f64, f64::NEG_INFINITY);}&~~ '$ Raise a number to an integer power.~': Using this function is generally faster than using `powf`=ٺ%2 fn check(x: T, exp: i32, powi: T) {5" assert!(x.powi(exp) == powi);% check(9.0f32, 2, 81.0); check(1.0f32, -2, 1.0);ʁ check(10.0f64, 20, 1e20); check(4.0f64, -2, 0.0625);$ check(-1.0f64, std::i32::MIN, 1.0);'ۂ 8 Converts to degrees, assuming the number is in radians.;ٺ%χ) fn check(rad: T, deg: T) {,& assert!(rad.to_degrees() == deg);)ӈ݈ check(0.0f32, 0.0); check(f32::consts::PI, 180.0);"% check(f64::consts::FRAC_PI_4, 45.0);(% check(f64::INFINITY, f64::INFINITY);Չ(  8 Converts to radians, assuming the number is in degrees.;ٺ%ɋ) fn check(deg: T, rad: T) {,& assert!(deg.to_radians() == rad);)͌׌ߌ check(180.0, f32::consts::PI);"% check(45.0, f64::consts::FRAC_PI_4);(ύ(  *K Returns the mantissa, base 2 exponent, and sign as integers, respectively.NJ The original number can be recovered by `sign * mantissa * 2 ^ exponent`.MϏ׏ٺ%Ő5 fn check(x: T, m: u64, e: i16, s:i8) {͐89 let (mantissa, exponent, sign) = x.integer_decode();< assert_eq!(mantissa, m);ˑ  assert_eq!(exponent, e);  assert_eq!(sign, s); check(2.0f32, 1 << 23, -22, 1);Ȓ#" check(-2.0f32, 1 << 23, -22, -1);%' check(f32::INFINITY, 1 << 23, 105, 1);*, check(f64::NEG_INFINITY, 1 << 52, 972, -1);ɓ/ )     ˘ј!    ӝٝ       ڡڡݡ ޡ  ء  )     ±ȱ!    ڶ      ̻̻ϻ лӻ Իܻ ޻ʻ  / ۭ/ޭ  ))..01 Defines an additive identity element for `Self`.54j # Lawsn y ```{.text}} a + 0 = a ∀ a ∈ Self" 0 + a = a ∀ a ∈ Self"  6 Returns the additive identity element of `Self`, `0`.9 # Purity G This function should return the same result at all times regardless ofJ? external mutable state, for example values stored in TLS or inB `static mut`s. = Sets `self` to the additive identity element of `Self`, `0`.@ < Returns `true` if `self` is equal to the additive identity.?   "                 /6 Defines a multiplicative identity element for `Self`. 9 ּ a * 1 = a ∀ a ∈ Self" 1 * a = a ∀ a ∈ Self"  < Returns the multiplicative identity element of `Self`, `1`.? JB C Sets `self` to the multiplicative identity element of `Self`, `1`.F ;B Returns `true` if `self` is equal to the multiplicative identity.E? For performance reasons, it's best to implement this manually.B; After a semver bump, this method will be required, and the>/ `where Self: PartialEq` bound will be removed.2     $ Returns the additive identity, `0`.' * Returns the multiplicative identity, `1`.-  1'',,11<DQ fw & Generic trait for primitive integers.)\ The `PrimInt` trait is an abstraction over the builtin primitive integer types (e.g., `u8`,_Y `u32`, `isize`, `i128`, ...). It inherits the basic numeric traits and extends them with\/ bitwise operators and non-wrapping arithmetic.2Z The trait explicitly inherits `Copy`, `Eq`, `Ord`, and `Sized`. The intention is that all]^ types implementing this trait behave like primitive types that are passed by value by defaulta\ and behave like builtin integers. Furthermore, the types are expected to expose the integer_^ value in binary representation and support bitwise operators. The standard bitwise operationsa] (e.g., bitwise-and, bitwise-or, right-shift, left-shift) are inherited and the trait extends`^ these with introspective queries (e.g., `PrimInt::count_ones()`, `PrimInt::leading_zeros()`),aW bitwise combinators (e.g., `PrimInt::rotate_left()`), and endianness converters (e.g.,Z `PrimInt::to_be()`).] All `PrimInt` types are expected to be fixed-width binary integers. The width can be queried`U via `T::zero().count_zeros()`. The trait currently lacks a way to query the width at X compile-time.  _ While a default implementation for all builtin primitive integers is provided, the trait is in bY no way restricted to these. Other integer types that fulfil the requirements are free to \ implement the trait was well. ! ^ This trait and many of the method names originate in the unstable `core::num::Int` trait from a] the rust standard library. The original trait was never stabilized and thus removed from the ` standard library. ϴ͹]            ϴ͹            C Returns the number of ones in the binary representation of `self`.F use num_traits::PrimInt; let n = 0b01001100u8; assert_eq!(n.count_ones(), 3);"  D Returns the number of zeros in the binary representation of `self`.G assert_eq!(n.count_zeros(), 5);#  @ Returns the number of leading ones in the binary representationC of `self`. let n = 0xF00Du16;! assert_eq!(n.leading_ones(), 4);$  A Returns the number of leading zeros in the binary representationD let n = 0b0101000u16;# assert_eq!(n.leading_zeros(), 10);&  A Returns the number of trailing ones in the binary representationD let n = 0xBEEFu16;" assert_eq!(n.trailing_ones(), 4);%  B Returns the number of trailing zeros in the binary representationE# assert_eq!(n.trailing_zeros(), 3);& "%A Shifts the bits to the left by a specified amount, `n`, wrapping D8 the truncated bits to the end of the resulting integer. ;!!!!!! let n = 0x0123456789ABCDEFu64;!" let m = 0x3456789ABCDEF012u64;"""" assert_eq!(n.rotate_left(12), m);"%"#  &&B Shifts the bits to the right by a specified amount, `n`, wrapping#E> the truncated bits to the beginning of the resulting integer.#A$$$$$%%" let m = 0xDEF0123456789ABCu64;%"%# assert_eq!(n.rotate_right(12), m);%&&&  )$@ Shifts the bits to the left by a specified amount, `n`, filling&C% zeros in the least significant bits.'(', This is bitwise equivalent to signed `Shl`.'/'((((((" let m = 0x3456789ABCDEF000u64;(")! assert_eq!(n.signed_shl(12), m);)$))  -$A Shifts the bits to the right by a specified amount, `n`, copying*DE the "sign bit" in the most significant bits even for unsigned types.*H+, This is bitwise equivalent to signed `Shr`.+/++++,, let n = 0xFEDCBA9876543210u64;," let m = 0xFFFFEDCBA9876543u64;,"-! assert_eq!(n.signed_shr(12), m);-$--  1&-C.(.. This is bitwise equivalent to unsigned `Shl`..1////// let n = 0x0123456789ABCDEFi64;/" let m = 0x3456789ABCDEF000i64;0"0# assert_eq!(n.unsigned_shl(12), m);0&01  4&A Shifts the bits to the right by a specified amount, `n`, filling1D$ zeros in the most significant bits.1'2. This is bitwise equivalent to unsigned `Shr`.21222333 let n = -8i8; // 0b111110003 let m = 62i8; // 0b0011111033" assert_eq!(n.unsigned_shr(2), m);4%44  6( Reverses the byte order of the integer.4+5555555" let m = 0xEFCDAB8967452301u64;6"6 assert_eq!(n.swap_bytes(), m);6"66  :+ Reverses the order of bits in the integer.7.7Y The least significant bit becomes the most significant bit, second least-significant bit7\* becomes second most-significant bit, etc.8-889999 let n = 0x12345678u32;9 let m = 0x1e6a2c48u32;9:! assert_eq!(n.reverse_bits(), m);:$$ assert_eq!(0u32.reverse_bits(), 0);:'::  ?? Convert an integer from big endian to the target's endianness.;B;G On big endian this is a no-op. On little endian the bytes are swapped.& } else {> 0 assert_eq!(u64::from_be(n), n.swap_bytes())>3>?? CB Convert an integer from little endian to the target's endianness.?E?G On little endian this is a no-op. On big endian the bytes are swapped.@J@@@@AAA"A$ if cfg!(target_endian = "little") {A'# assert_eq!(u64::from_le(n), n)B&ܕB 0 assert_eq!(u64::from_le(n), n.swap_bytes())B3BCC F; Convert `self` to big endian from the target's endianness.C>C͓CJDDDDDEE"EE$ assert_eq!(n.to_be(), n)E ܕF * assert_eq!(n.to_be(), n.swap_bytes())F-FFF J> Convert `self` to little endian from the target's endianness.GAGԗGJHHHHHII"II' assert_eq!(n.to_le(), n)I ܕJ * assert_eq!(n.to_le(), n.swap_bytes())J-JJJ LE Raises self to the power of `exp`, using exponentiation by squaring.KHKKKKKL assert_eq!(2i32.pow(4), 16);L LL L"L MĤM ĤMMP/PPҤP ĤPPTU  .$$))..6D Performs addition that returns `None` instead of wrapping around on5G overflow.}    0H Adds two numbers, checking for overflow. If overflow happens, `None` isK returned.     6R Performs subtraction that returns `None` instead of wrapping around on underflow. U        0E Subtracts two numbers, checking for underflow. If underflow happens, H    6W Performs multiplication that returns `None` instead of wrapping around on underflow orZ    0I Multiplies two numbers, checking for underflow or overflow. If underflowL) or overflow happens, `None` is returned.,  6^ Performs division that returns `None` instead of panicking on division by zero and instead ofa+ wrapping around on underflow and overflow..  ر 0F Divides two numbers, checking for underflow, overflow and division byI2 zero. If any of that happens, `None` is returned.5   6` Performs an integral remainder that returns `None` instead of panicking on division by zero andc6 instead of wrapping around on underflow and overflow.9      ׵ &0[ Finds the remainder of dividing two numbers, checking for underflow, overflow and division ^5 by zero. If any of that happens, `None` is returned.!8"""" use num_traits::CheckedRem;" use std::i32::MIN;""7 assert_eq!(CheckedRem::checked_rem(&10, &7), Some(3));":8 assert_eq!(CheckedRem::checked_rem(&10, &-7), Some(3));#;9 assert_eq!(CheckedRem::checked_rem(&-10, &7), Some(-3));#<: assert_eq!(CheckedRem::checked_rem(&-10, &-7), Some(-3));$=$4 assert_eq!(CheckedRem::checked_rem(&10, &0), None);%7%8 assert_eq!(CheckedRem::checked_rem(&MIN, &1), Some(0));%;6 assert_eq!(CheckedRem::checked_rem(&MIN, &-1), None);&9&& & ++.J Performs negation that returns `None` if the result can't be represented.-M. ..3&\ Negates a number, returning `None` for results that can't be represented, like signed `MIN`._S values that can't be positive, or non-zero unsigned values that can't be negative./V/000 use num_traits::CheckedNeg;0007 assert_eq!(CheckedNeg::checked_neg(&1_i32), Some(-1));0:7 assert_eq!(CheckedNeg::checked_neg(&-1_i32), Some(1));1:1 assert_eq!(CheckedNeg::checked_neg(&MIN), None);1426 assert_eq!(CheckedNeg::checked_neg(&0_u32), Some(0));293 assert_eq!(CheckedNeg::checked_neg(&1_u32), None);2633 3 95@ Performs a left shift that returns `None` on shifts larger than8C the type width.99 999 999 >0= Checked shift left. Computes `self << rhs`, returning `None`:@B if `rhs` is larger than or equal to the number of bits in `self`.:E;; use num_traits::CheckedShl;;; let x: u16 = 0x0001;;;; assert_eq!(CheckedShl::checked_shl(&x, 0), Some(0x0001));;>; assert_eq!(CheckedShl::checked_shl(&x, 1), Some(0x0002));<>; assert_eq!(CheckedShl::checked_shl(&x, 15), Some(0x8000));<>3 assert_eq!(CheckedShl::checked_shl(&x, 16), None);=6=> > >>F5A Performs a right shift that returns `None` on shifts larger thanEDFF FFF FFF J0> Checked shift right. Computes `self >> rhs`, returning `None`FAGEGH use num_traits::CheckedShr;HH let x: u16 = 0x8000;HH; assert_eq!(CheckedShr::checked_shr(&x, 0), Some(0x8000));H>; assert_eq!(CheckedShr::checked_shr(&x, 1), Some(0x4000));I>; assert_eq!(CheckedShr::checked_shr(&x, 15), Some(0x0001));I>3 assert_eq!(CheckedShr::checked_shr(&x, 16), None);J6JJ K      M & . ر6 Q @ ׵[ . ر6 Q @ ׵[ ' E Calculates Euclidean division, the matching method for `rem_euclid`.pH  ( This computes the integer `n` such that+ % `self = n * v + self.rem_euclid(v)`.( D In other words, the result is `self / v` rounded to the integer `n`G  such that `self >= n * v`.      use num_traits::Euclid;   let a: i32 = 7;  let b: i32 = 4; 8 assert_eq!(Euclid::div_euclid(&a, &b), 1); // 7 > 4 * 1; = assert_eq!(Euclid::div_euclid(&-a, &b), -2); // -7 >= 4 * -2@ = assert_eq!(Euclid::div_euclid(&a, &-b), -1); // 7 >= -4 * -1@ = assert_eq!(Euclid::div_euclid(&-a, &-b), 2); // -7 >= -4 * 2@     ' > Calculates the least nonnegative remainder of `self (mod v)`.A  F In particular, the return value `r` satisfies `0.0 <= r < v.abs()` inI D most cases. However, due to a floating point round-off error it canG D result in `r == v.abs()`, violating the mathematical definition, ifG E `self` is much smaller than `v.abs()` in magnitude and `self < 0.0`. H H This result is not an element of the function's codomain, but it is the K H closest floating point number in the real numbers and thus fulfills the K ? property `self == self.div_euclid(v) * v + self.rem_euclid(v)` B  approximatively.                  + assert_eq!(Euclid::rem_euclid(&a, &b), 3); . , assert_eq!(Euclid::rem_euclid(&-a, &b), 1); / , assert_eq!(Euclid::rem_euclid(&a, &-b), 3); / - assert_eq!(Euclid::rem_euclid(&-a, &-b), 1);0                         has_div_euclid      1 $    $    ! ! ! ! !  ! ! !  ! ! !  ! ! ! !  ! !1 !$ ! !  #$ # #  % % % % &7 V Performs euclid division that returns `None` instead of panicking on division by zero%Y : and instead of wrapping around on underflow and overflow.&= & &  (7 Y Finds the euclid remainder of dividing two numbers, checking for underflow, overflow and'\ > division by zero. If any of that happens, `None` is returned.'A ( (  (( ) ,$ , 2% 2 ! )Y U Unary operator for retrieving the multiplicative inverse, or reciprocal, of a value.X c  ( The result after applying the operator.m+   . Returns the multiplicative inverse of `self`.1      use std::f64::INFINITY;  use num_traits::Inv;  " assert_eq!(7.0.inv() * 7.0, 1.0);% % assert_eq!((-0.0).inv(), -INFINITY);(                 " "       " " "       ".6$ E Fused multiply-add. Computes `(self * a) + b` with only one roundingH E error, yielding a more accurate result than an unfused multiply-add.IH  G Using `mul_add` can be more performant than an unfused multiply-add ifJ ? the target architecture has a dedicated `fma` CPU instruction.B  H Note that `A` and `B` are `Self` by default, but this is not mandatory.K   # Example    use std::f32;   let m = 10.0_f32;  let x = 4.0_f32;  let b = 60.0_f32;   // 100.0 : let abs_difference = (m.mul_add(x, b) - (m*x + b)).abs();=  1 assert!(abs_difference <= 100.0 * f32::EPSILON);4   ### #      : The resulting type after applying the fused multiply-add.=  - + Performs the fused multiply-add operation..   * - The fused multiply-add assignment operation.0  ##      ) .        ?G        5 " " % .  &  < " " % .  &  < B " " % .  &  B M T T Y Y ^ ^ c c g g s z z           : , Performs addition with a flag for overflow./        4 g Returns a tuple of the sum along with a boolean indicating whether an arithmetic overflow would occur.j G If an overflow would have occurred then the wrapped value is returned.J     : 0 Performs substraction with a flag for overflow. 3            4 n Returns a tuple of the difference along with a boolean indicating whether an arithmetic overflow would occur. q  J    : 2 Performs multiplication with a flag for overflow.5        4 k Returns a tuple of the product along with a boolean indicating whether an arithmetic overflow would occur.n J    T\         Q Saturating math operations. Deprecated, use `SaturatingAdd`, `SaturatingSub` and!T  `SaturatingMul` instead.v  )  Saturating addition operator.! F Returns a+b, saturating at the numeric bounds instead of overflowing.I   ) ! Saturating subtraction operator.$ F Returns a-b, saturating at the numeric bounds instead of overflowing.I   '     9 O Performs addition that saturates at the numeric bounds instead of overflowing. R           + a Saturating addition. Computes `self + other`, saturating at the relevant high or low boundary of d  the type.     9 R Performs subtraction that saturates at the numeric bounds instead of overflowing.U        + d Saturating subtraction. Computes `self - other`, saturating at the relevant high or low boundary ofg     9 U Performs multiplication that saturates at the numeric bounds instead of overflowing.X        + g Saturating multiplication. Computes `self * other`, saturating at the relevant high or low boundary ofj     hp)))..3388==BBIV 71 Performs addition that wraps around on overflow.4   )Y Wrapping (modular) addition. Computes `self + other`, wrapping around at the boundary of\    74 Performs subtraction that wraps around on overflow. 7        )Y Wrapping (modular) subtraction. Computes `self - other`, wrapping around at the boundary \ of the type.    77 Performs multiplication that wraps around on overflow.:   )\ Wrapping (modular) multiplication. Computes `self * other`, wrapping around at the boundary_   ) Performs a negation that does not panic., "/ Wrapping (modular) negation. Computes `-self`,2- wrapping around at the boundary of the type.06 Since unsigned types do not have negative equivalents9? all applications of this function will wrap (except for `-0`).B@ For values smaller than the corresponding signed type's maximumCB the result is the same as casting the corresponding signed value.EF Any larger values are equivalent to `MAX + 1 - (val - MAX - 1)` whereI2 `MAX` is the corresponding signed type's maximum.5   use num_traits::WrappingNeg;  ( assert_eq!(100i8.wrapping_neg(), -100); +* assert_eq!((-100i8).wrapping_neg(), 100);!-7 assert_eq!((-128i8).wrapping_neg(), -128); // wrapped!!:"" " ' '*8+ Performs a left shift that does not panic.).* *** *** /); Panic-free bitwise shift-left; yields `self << mask(rhs)`,*>= where `mask` removes any high order bits of `rhs` that would+@4 cause the shift to exceed the bitwidth of the type.+7,, use num_traits::WrappingShl;, ,,,7 assert_eq!(WrappingShl::wrapping_shl(&x, 0), 0x0001);-:7 assert_eq!(WrappingShl::wrapping_shl(&x, 1), 0x0002);-:7 assert_eq!(WrappingShl::wrapping_shl(&x, 15), 0x8000);.:7 assert_eq!(WrappingShl::wrapping_shl(&x, 16), 0x0001);.:// / 58, Performs a right shift that does not panic.4/5 555 555 :)< Panic-free bitwise shift-right; yields `self >> mask(rhs)`,5?6@6777 use num_traits::WrappingShr;7 7777 assert_eq!(WrappingShr::wrapping_shr(&x, 0), 0x8000);8:7 assert_eq!(WrappingShr::wrapping_shr(&x, 1), 0x4000);8:7 assert_eq!(WrappingShr::wrapping_shr(&x, 15), 0x0001);9:7 assert_eq!(WrappingShr::wrapping_shr(&x, 16), 0x8000);9::: : @0@@ @@@@A(A A A0BA BAAAB(B B C0DC CCCCD(D D D0EE EEEEEE E F0GF F FFFG(G G H0HH H HHHH(H I  (12 2 >>y0 Binary operator for raising a value to a power.E3,, +'# Returns `self` to the power `rhs`.& use num_traits::Pow;( assert_eq!(Pow::pow(10u32, 2u32), 100);+ (QF Raises a value to the power of exp, using exponentiation by squaring.%I&N Note that `0⁰` (`pow(0, 0)`) returns `1`. Mathematically this is undefined.&Q'' ' ```rust'  use num_traits::pow;'' assert_eq!(pow(2i8, 4), 16);'  assert_eq!(pow(6u8, 3), 216);'!A assert_eq!(pow(0u8, 0), 1); // Be aware if this case affects you(D(((((()  ((0YN Raises a value to the power of exp, returning `None` if an overflow occurred.,Q-\ Note that `0⁰` (`checked_pow(0, 0)`) returns `Some(1)`. Mathematically this is undefined.-_-& Otherwise same as the `pow` function.-).. ..  use num_traits::checked_pow;. .+ assert_eq!(checked_pow(2i8, 4), Some(16));..' assert_eq!(checked_pow(7i8, 8), None);/*3 assert_eq!(checked_pow(7u32, 8), Some(5_764_801));/6O assert_eq!(checked_pow(0u32, 0), Some(1)); // Be aware if this case affect you/R00 0000  0011 (29 HH2I Useful functions for signed numbers (i.e. numbers that can be negative).NL͹ ͹  Computes the absolute value. D For `f32` and `f64`, `NaN` will be returned if the number is `NaN`.GH For signed integers, `::MIN` will be returned if the number is `::MIN`.K (( The positive difference of two numbers.+X Returns `zero` if the number is less than or equal to `other`, otherwise the difference[( between `self` and `other` is returned.+  Returns the sign of the number.# For `f32` and `f64`:8 * `1.0` if the number is positive, `+0.0` or `INFINITY`;= * `-1.0` if the number is negative, `-0.0` or `NEG_INFINITY`@ * `NaN` if the number is `NaN`" For signed integers: * `0` if the number is zero * `1` if the number is positive#! * `-1` if the number is negative$  T Returns true if the number is positive and false if the number is zero or negative.W    T Returns true if the number is negative and false if the number is zero or positive. W     &˹ '      #$" "C For `f32` and `f64`, `NaN` will be returned if the number is `NaN`"F""K### ##%*$+$K Returns zero if `x` is less than or equal to `y`, otherwise the difference$N! between `x` and `y` is returned.%$%%% %%)'&#&&&&;'@'"''(((#($))) ))), A trait for values which cannot be negative)/*͹*͹***+*++,++.? The base trait for numeric types, covering `0` and `1` values,B> comparisons, basic numeric operations, and string conversion.A  P6 Convert from a string and radix (typically `2..=36`).9  use num_traits::Num;5 let result = ::from_str_radix("27", 10);8 assert_eq!(result, Ok(27));6 let result = ::from_str_radix("foo", 10);9 assert!(result.is_err()); # Supported radices[ The exact range of supported radices is at the discretion of each type implementation. For^X primitive integers, this is implemented by the inherent `from_str_radix` methods in the[U standard library, which **panic** if the radix is not in the range from 2 to 36. TheX> implementation in this crate for primitive floats is similar.A\ For third-party types, it is suggested that implementations should follow suit and at least_[ accept `2..=36` without panicking, but an `Err` may be returned for any unsupported radix.^Y It's possible that a type might not even support the common radix 10, nor any, if string\* parsing doesn't make sense for that type.- > Generic trait for types implementing basic numeric operationsAK This is automatically implemented for types which implement the operators.N0 0     .0 !0I The trait for `Num` types which also implement numeric operations takingL! the second operand by reference. $  N!͹!0 0!͹!0!!!!!"0 0"!!""$KN The trait for `Num` references which implement numeric operations, taking the"Q0 second operand either by value or by reference."3#Y This is automatically implemented for all types which implement the operators. It covers#\U every type implementing the operations though, regardless of it being a reference or$X related to `Num`.$%11%0 0%%%0%%%%%% 1%%%0 0&%%%%&&'zO Generic trait for types implementing numeric assignment operators (like `+=`).&R&'N' 0'((((' 'ȅ(߅(((' '($0(()))))((((*'E The trait for `Num` types which also implement assignment operators.)H**N+ ͹++ ͹++ ++++ ++-BK The trait for `NumAssign` types which also implement assignment operations+N( taking the second operand by reference.,+,,N- - 0 0-ی- 0-----. 0 0.--....2 22322333O33 6%  ځ é7477 9699 : ==~:+ A value bounded by a minimum and a maximum|.|2 If input is less than min then this returns min.|55 If input is greater than max then this returns max.}8 Otherwise this returns input.}"}- **Panics** in debug mode if `!(min <= max)`.~0~~~  ~~6# A value bounded by a minimum value&Հـ5"` `clamp_min(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::min(std::f32::NAN, 1.0)`.cN **Panics** in debug mode if `!(min == min)`. (This occurs if `min` is `NAN`.)Q ߕ  6# A value bounded by a maximum value&фՄ8"` `clamp_max(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::max(std::f32::NAN, 1.0)`.cN **Panics** in debug mode if `!(max == max)`. 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R Return `1.0 / π`. R  Return `1.0 / sqrt(2.0)`. R  Return `2.0 / π`. R  Return `2.0 / sqrt(π)`. R Return `π / 2.0`. R  Return `π / 3.0`. R  Return `π / 4.0`. R  Return `π / 6.0`. R  Return `π / 8.0`. R  Return `ln(10.0)`. R Return `ln(2.0)`. R Return `log10(e)`. R Return `log2(e)`. R #Return Archimedes’ constant `π`. %R Return `sqrt(2.0)`. R =%Return the full circle constant `τ`. '0R  AReturn `log10(2.0)`. Rر  AReturn `log2(10.0)`. 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