Library Reference

version 2.112.0

Internal API

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std.math.algebraic

This is a submodule ofstd.math.

It contains classical algebraic functions likeabs,sqrt, andpoly.

License:

Boost License 1.0.

Authors:

Walter Bright, Don Clugston, Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger

Sourcestd/math/algebraic.d

pure nothrow @nogc autoabs(Num)(Numx) if (isIntegral!Num || is(typeof(Num.init >= 0)) && is(typeof(-Num.init)));

Calculates the absolute value of a number.

Parameters:

Num	(template parameter) type of number
Num`x`	real number value

Returns:

The absolute value of the number. If floating-point or integral, the return type will be the same as the input.

LimitationsWhen x is a signed integral equal toNum.min the value of x will be returned instead. Note for 2's complement;-Num.min (=Num.max + 1) is not representable due to overflow.

Examples:

import std.math.traits : isIdentical, isNaN;assert(isIdentical(abs(-0.0L), 0.0L));assert(isNaN(abs(real.nan)));writeln(abs(-real.infinity));// real.infinitywriteln(abs(-56));// 56writeln(abs(2321312L));// 2321312Lwriteln(abs(23u));// 23u

pure nothrow @nogc @safe realfabs(realx); pure nothrow @nogc @safe doublefabs(doublex); pure nothrow @nogc @safe floatfabs(floatx);

Returns |x|

Special Values
x	fabs(x)
±0.0	+0.0
±∞	+∞

Examples:

import std.math.traits : isIdentical;assert(isIdentical(fabs(0.0f), 0.0f));assert(isIdentical(fabs(-0.0f), 0.0f));writeln(fabs(-10.0f));// 10.0fassert(isIdentical(fabs(0.0), 0.0));assert(isIdentical(fabs(-0.0), 0.0));writeln(fabs(-10.0));// 10.0assert(isIdentical(fabs(0.0L), 0.0L));assert(isIdentical(fabs(-0.0L), 0.0L));writeln(fabs(-10.0L));// 10.0L

pure nothrow @nogc @safe floatsqrt(floatx); pure nothrow @nogc @safe doublesqrt(doublex); pure nothrow @nogc @safe realsqrt(realx);

Compute square root of x.

Special Values
x	sqrt(x)	invalid?
-0.0	-0.0	no
<0.0	NAN	yes
+∞	+∞	no

Examples:

import std.math.operations : feqrel;import std.math.traits : isNaN;assert(sqrt(2.0).feqrel(1.4142) > 16);assert(sqrt(9.0).feqrel(3.0) > 16);assert(isNaN(sqrt(-1.0f)));assert(isNaN(sqrt(-1.0)));assert(isNaN(sqrt(-1.0L)));

pure nothrow @nogc @trusted realcbrt(realx);

Calculates the cube root of x.

Special Values
x	cbrt(x)	invalid?
±0.0	±0.0	no
NAN	NAN	yes
±∞	±∞	no

Examples:

import std.math.operations : feqrel;assert(cbrt(1.0).feqrel(1.0) > 16);assert(cbrt(27.0).feqrel(3.0) > 16);assert(cbrt(15.625).feqrel(2.5) > 16);

pure nothrow @nogc @safe Thypot(T)(const Tx, const Ty) if (isFloatingPoint!T);

Calculates the length of the hypotenuse of a right-angled triangle with sides of length x and y. The hypotenuse is the value of the square root of the sums of the squares of x and y:

sqrt(x² + y²)

Note that hypot(x, y), hypot(y, x) and hypot(x, -y) are equivalent.

Special Values
x	y	hypot(x, y)	invalid?
x	±0.0	\|x\|	no
±∞	y	+∞	no
±∞	NAN	+∞	no
NAN	y != ±∞	NAN	no

Examples:

import std.math.operations : feqrel;import std.math.traits : isNaN;assert(hypot(1.0, 1.0).feqrel(1.4142) > 16);assert(hypot(3.0, 4.0).feqrel(5.0) > 16);writeln(hypot(real.infinity, 1.0L));// real.infinitywriteln(hypot(1.0L,real.infinity));// real.infinitywriteln(hypot(real.infinity,real.nan));// real.infinitywriteln(hypot(real.nan,real.infinity));// real.infinityassert(hypot(real.nan, 1.0L).isNaN);assert(hypot(1.0L,real.nan).isNaN);

pure nothrow @nogc @safe Thypot(T)(const Tx, const Ty, const Tz) if (isFloatingPoint!T);

Calculates the distance of the point (x, y, z) from the origin (0, 0, 0) in three-dimensional space. The distance is the value of the square root of the sums of the squares of x, y, and z:

sqrt(x² + y² + z²)

Note that the distance between two points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space can be calculated as hypot(x2-x1, y2-y1, z2-z1).

Parameters:

T`x`	floating point value
T`y`	floating point value
T`z`	floating point value

Returns:

The square root of the sum of the squares of the given arguments.

Examples:

import std.math.operations : isClose;assert(isClose(hypot(1.0, 2.0, 2.0), 3.0));assert(isClose(hypot(2.0, 3.0, 6.0), 7.0));assert(isClose(hypot(1.0, 4.0, 8.0), 9.0));

pure nothrow @nogc @trusted Unqual!(CommonType!(T1, T2))poly(T1, T2)(T1x, in T2[]A) if (isFloatingPoint!T1 && isFloatingPoint!T2); pure nothrow @nogc @safe Unqual!(CommonType!(T1, T2))poly(T1, T2, int N)(T1x, ref const T2[N]A) if (isFloatingPoint!T1 && isFloatingPoint!T2 && (N > 0) && (N <= 10));

Evaluate polynomial A(x) = a₀ + a₁x + a₂x² + a₃x³; ...

Uses Horner's rule A(x) = a₀ + x(a₁ + x(a₂ + x(a₃ + ...)))

Parameters:

T1`x`	the value to evaluate.
T2[]`A`	array of coefficients a₀, a₁, etc.

Examples:

realx = 3.1L;staticreal[] pp = [56.1L, 32.7L, 6];writeln(poly(x, pp));// (56.1L + (32.7L + 6.0L * x) * x)

TnextPow2(T)(const Tval) if (isIntegral!T); TnextPow2(T)(const Tval) if (isFloatingPoint!T);

Gives the next power of two afterval.T can be any built-in numerical type.

If the operation would lead to an over/underflow, this function will return0.

Parameters:

Tval any number

Returns:

the next power of two afterval

Examples:

writeln(nextPow2(2));// 4writeln(nextPow2(10));// 16writeln(nextPow2(4000));// 4096writeln(nextPow2(-2));// -4writeln(nextPow2(-10));// -16writeln(nextPow2(uint.max));// 0writeln(nextPow2(uint.min));// 0writeln(nextPow2(size_t.max));// 0writeln(nextPow2(size_t.min));// 0writeln(nextPow2(int.max));// 0writeln(nextPow2(int.min));// 0writeln(nextPow2(long.max));// 0writeln(nextPow2(long.min));// 0

Examples:

writeln(nextPow2(2.1));// 4.0writeln(nextPow2(-2.0));// -4.0writeln(nextPow2(0.25));// 0.5writeln(nextPow2(-4.0));// -8.0writeln(nextPow2(double.max));// 0.0writeln(nextPow2(double.infinity));// double.infinity

TtruncPow2(T)(const Tval) if (isIntegral!T); TtruncPow2(T)(const Tval) if (isFloatingPoint!T);

Gives the last power of two beforeval.<> can be any built-in numerical type.

Parameters:

Tval any number

Returns:

the last power of two beforeval

Examples:

writeln(truncPow2(3));// 2writeln(truncPow2(4));// 4writeln(truncPow2(10));// 8writeln(truncPow2(4000));// 2048writeln(truncPow2(-5));// -4writeln(truncPow2(-20));// -16writeln(truncPow2(uint.max));// int.max + 1writeln(truncPow2(uint.min));// 0writeln(truncPow2(ulong.max));// long.max + 1writeln(truncPow2(ulong.min));// 0writeln(truncPow2(int.max));// (int.max / 2) + 1writeln(truncPow2(int.min));// int.minwriteln(truncPow2(long.max));// (long.max / 2) + 1writeln(truncPow2(long.min));// long.min

Examples:

writeln(truncPow2(2.1));// 2.0writeln(truncPow2(7.0));// 4.0writeln(truncPow2(-1.9));// -1.0writeln(truncPow2(0.24));// 0.125writeln(truncPow2(-7.0));// -4.0writeln(truncPow2(double.infinity));// double.infinity

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Library Reference

std.math.algebraic