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# std.math.algebraic

This is a submodule of std.math.
It contains classical algebraic functions like abs, sqrt, and poly.
License:
Boost License 1.0.
Authors:
Walter Bright, Don Clugston, Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger

Source std/math/algebraic.d

pure nothrow @nogc auto `abs`(Num)(Num `x`)
if (is(immutable(Num) == immutable(short)) || is(immutable(Num) == immutable(byte)) || 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.

  Limitations Does not work correctly for signed intergal types and value Num.min.

  Examples:
  ditto

      import std.math.traits : isIdentical, isNaN;

      assert(isIdentical(abs(-0.0L), 0.0L));
      assert(isNaN(abs(real.nan)));
      writeln(abs(-real.infinity)); // real.infinity
      writeln(abs(-56)); // 56
      writeln(abs(2321312L)); // 2321312L

pure nothrow @nogc @safe real `fabs`(real `x`);
pure nothrow @nogc @trusted double `fabs`(double `d`);
pure nothrow @nogc @trusted float `fabs`(float `f`);
  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)); // 10.0f

      assert(isIdentical(fabs(0.0), 0.0));
      assert(isIdentical(fabs(-0.0), 0.0));
      writeln(fabs(-10.0)); // 10.0

      assert(isIdentical(fabs(0.0L), 0.0L));
      assert(isIdentical(fabs(-0.0L), 0.0L));
      writeln(fabs(-10.0L)); // 10.0L

pure nothrow @nogc @safe float `sqrt`(float `x`);
pure nothrow @nogc @safe double `sqrt`(double `x`);
pure nothrow @nogc @safe real `sqrt`(real `x`);
  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)));

nothrow @nogc @trusted real `cbrt`(real `x`);
  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 T `hypot`(T)(const T `x`, const T `y`)
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

  Examples:

      import std.math.operations : feqrel;

      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.infinity
      writeln(hypot(real.infinity, real.nan)); // real.infinity

pure nothrow @nogc @safe T `hypot`(T)(const T `x`, const T `y`, const T `z`)
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)(T1 `x`, in T2[] `A`)
if (isFloatingPoint!T1 && isFloatingPoint!T2);
pure nothrow @nogc @safe Unqual!(CommonType!(T1, T2)) `poly`(T1, T2, int N)(T1 `x`, 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:

      real x = 3.1L;
      static real[] pp = [56.1L, 32.7L, 6];

      writeln(poly(x, pp)); // (56.1L + (32.7L + 6.0L * x) * x)

T `nextPow2`(T)(const T `val`)
if (isIntegral!T);
T `nextPow2`(T)(const T `val`)
if (isFloatingPoint!T);
  Gives the next power of two after `val`. T can be any built-in numerical type.
  If the operation would lead to an over/underflow, this function will return 0.
  Parameters:

  T `val` | any number

  Returns:
  the next power of two after `val`
  Examples:

      writeln(nextPow2(2)); // 4
      writeln(nextPow2(10)); // 16
      writeln(nextPow2(4000)); // 4096

      writeln(nextPow2(-2)); // -4
      writeln(nextPow2(-10)); // -16

      writeln(nextPow2(uint.max)); // 0
      writeln(nextPow2(uint.min)); // 0
      writeln(nextPow2(size_t.max)); // 0
      writeln(nextPow2(size_t.min)); // 0

      writeln(nextPow2(int.max)); // 0
      writeln(nextPow2(int.min)); // 0
      writeln(nextPow2(long.max)); // 0
      writeln(nextPow2(long.min)); // 0

  Examples:

      writeln(nextPow2(2.1)); // 4.0
      writeln(nextPow2(-2.0)); // -4.0
      writeln(nextPow2(0.25)); // 0.5
      writeln(nextPow2(-4.0)); // -8.0

      writeln(nextPow2(double.max)); // 0.0
      writeln(nextPow2(double.infinity)); // double.infinity

T `truncPow2`(T)(const T `val`)
if (isIntegral!T);
T `truncPow2`(T)(const T `val`)
if (isFloatingPoint!T);
  Gives the last power of two before `val`. <> can be any built-in numerical type.
  Parameters:

  T `val` | any number

  Returns:
  the last power of two before `val`
  Examples:

      writeln(truncPow2(3)); // 2
      writeln(truncPow2(4)); // 4
      writeln(truncPow2(10)); // 8
      writeln(truncPow2(4000)); // 2048

      writeln(truncPow2(-5)); // -4
      writeln(truncPow2(-20)); // -16

      writeln(truncPow2(uint.max)); // int.max + 1
      writeln(truncPow2(uint.min)); // 0
      writeln(truncPow2(ulong.max)); // long.max + 1
      writeln(truncPow2(ulong.min)); // 0

      writeln(truncPow2(int.max)); // (int.max / 2) + 1
      writeln(truncPow2(int.min)); // int.min
      writeln(truncPow2(long.max)); // (long.max / 2) + 1
      writeln(truncPow2(long.min)); // long.min

  Examples:

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

      writeln(truncPow2(double.infinity)); // double.infinity

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