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# std.math

Contains the elementary mathematical functions (powers, roots, and trigonometric functions), and low-level floating-point operations. Mathematical special functions are available in std.mathspecial.

Category | Members
--- | ---
Constants | E PI PI_2 PI_4 M_1_PI M_2_PI M_2_SQRTPI LN10 LN2 LOG2 LOG2E LOG2T LOG10E SQRT2 SQRT1_2
Classics | abs fabs sqrt cbrt hypot poly nextPow2 truncPow2
Trigonometry | sin cos tan asin acos atan atan2 sinh cosh tanh asinh acosh atanh expi
Rounding | ceil floor round lround trunc rint lrint nearbyint rndtol quantize
Exponentiation & Logarithms | pow exp exp2 expm1 ldexp frexp log log2 log10 logb ilogb log1p scalbn
Modulus | fmod modf remainder
Floating-point operations | approxEqual feqrel fdim fmax fmin fma nextDown nextUp nextafter NaN getNaNPayload cmp
Introspection | isFinite isIdentical isInfinity isNaN isNormal isSubnormal signbit sgn copysign isPowerOf2
Complex Numbers | abs conj sin cos expi
Hardware Control | IeeeFlags FloatingPointControl

The functionality closely follows the IEEE754-2008 standard for floating-point arithmetic, including the use of camelCase names rather than C99-style lower case names. All of these functions behave correctly when presented with an infinity or NaN.
The following IEEE 'real' formats are currently supported:

- 64 bit Big-endian 'double' (eg PowerPC)
- 128 bit Big-endian 'quadruple' (eg SPARC)
- 64 bit Little-endian 'double' (eg x86-SSE2)
- 80 bit Little-endian, with implied bit 'real80' (eg x87, Itanium)
- 128 bit Little-endian 'quadruple' (not implemented on any known processor!)
- Non-IEEE 128 bit Big-endian 'doubledouble' (eg PowerPC) has partial support

Unlike C, there is no global 'errno' variable. Consequently, almost all of these functions are pure nothrow.

Status The semantics and names of feqrel and approxEqual will be revised.

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.d

enum real `E`;
  e = 2.718281...
enum real `LOG2T`;
  log210 = 3.321928...
enum real `LOG2E`;
  log2e = 1.442695...
enum real `LOG2`;
  log102 = 0.301029...
enum real `LOG10E`;
  log10e = 0.434294...
enum real `LN2`;
  ln 2 = 0.693147...
enum real `LN10`;
  ln 10 = 2.302585...
enum real `PI`;
  π = 3.141592...
enum real `PI_2`;
  π / 2 = 1.570796...
enum real `PI_4`;
  π / 4 = 0.785398...
enum real `M_1_PI`;
  1 / π = 0.318309...
enum real `M_2_PI`;
  2 / π = 0.636619...
enum real `M_2_SQRTPI`;
  2 / √π = 1.128379...
enum real `SQRT2`;
  √2 = 1.414213...
enum real `SQRT1_2`;
  √½ = 0.707106...
pure nothrow @safe Num `abs`(Num)(Num `x`)
if ((is(typeof(Num.init >= 0)) && is(typeof(-Num.init)) || (is(Num == short) || is(Num == byte))) && !(is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*))));
pure nothrow @nogc @safe auto `abs`(Num)(Num `z`)
if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*)) || is(Num* : const(creal*)));
pure nothrow @nogc @safe auto `abs`(Num)(Num `y`)
if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*)));
  Calculates the absolute value of a number
  Parameters:

  Num | (template parameter) type of number
  Num `x` | real number value
  Num `z` | complex number value
  Num `y` | imaginary number value

  Returns:
  The absolute value of the number. If floating-point or integral, the return type will be the same as the input; if complex or imaginary, the returned value will be the corresponding floating point type.
  For complex numbers, `abs`(`z`) = sqrt( `z`.re2 + `z`.im2 ) = hypot(`z`.re, `z`.im).
  Examples:
  ditto

      assert(isIdentical(abs(-0.0L), 0.0L));
      assert(isNaN(abs(real.nan)));
      writeln(abs(-real.infinity)); // real.infinity
      writeln(abs(-3.2Li)); // 3.2L
      writeln(abs(71.6Li)); // 71.6L
      writeln(abs(-56)); // 56
      writeln(abs(2321312L)); // 2321312L
      writeln(abs(-1L + 1i)); // sqrt(2.0L)

pure nothrow @nogc @safe auto `conj`(Num)(Num `z`)
if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*)) || is(Num* : const(creal*)));
pure nothrow @nogc @safe auto `conj`(Num)(Num `y`)
if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*)) || is(Num* : const(ireal*)));
  Complex conjugate
  `conj`(x + iy) = x - iy
  Note that `z` * `conj`(`z`) = `z`.re2 - `z`.im2 is always a real number
  Examples:

      creal c = 7 + 3Li;
      writeln(conj(c)); // 7 - 3Li
      ireal z = -3.2Li;
      writeln(conj(z)); // -z

pure nothrow @nogc @safe real `cos`(real `x`);
pure nothrow @nogc @safe double `cos`(double `x`);
pure nothrow @nogc @safe float `cos`(float `x`);
  Returns cosine of `x`. `x` is in radians.

  Special Values
  `x` | `cos`(`x`) | invalid?
  --- | --- | ---
  NAN | NAN | yes
  ±∞ | NAN | yes

  Bugs:
  Results are undefined if |`x`| >= 264.
pure nothrow @nogc @safe real `sin`(real `x`);
pure nothrow @nogc @safe double `sin`(double `x`);
pure nothrow @nogc @safe float `sin`(float `x`);
  Returns sine of `x`. `x` is in radians.

  Special Values
  `x` | `sin`(`x`) | invalid?
  --- | --- | ---
  NAN | NAN | yes
  ±0.0 | ±0.0 | no
  ±∞ | NAN | yes

  Parameters:

  real `x` | angle in radians (not degrees)

  Returns:
  sine of `x`
  See Also:
  cos , tan , asin
  Bugs:
  Results are undefined if |`x`| >= 264.
  Examples:

      import std.math : sin, PI;
      import std.stdio : writefln;

      void someFunc()
      {
        real x = 30.0;
        auto result = sin(x * (PI / 180)); // convert degrees to radians
        writefln("The sine of %s degrees is %s", x, result);
      }

pure nothrow @nogc @safe creal `sin`(creal `z`);
pure nothrow @nogc @safe ireal `sin`(ireal `y`);
  Returns sine for complex and imaginary arguments.
  `sin`(`z`) = `sin`(`z`.re)*cosh(`z`.im) + cos(`z`.re)*sinh(`z`.im)i
  If both `sin`(θ) and cos(θ) are required, it is most efficient to use expi(θ).
  Examples:

      writeln(sin(0.0 + 0.0i)); // 0.0
      writeln(sin(2.0 + 0.0i)); // sin(2.0L)

pure nothrow @nogc @safe creal `cos`(creal `z`);
pure nothrow @nogc @safe real `cos`(ireal `y`);
  cosine, complex and imaginary
  `cos`(`z`) = `cos`(`z`.re)*cosh(`z`.im) - sin(`z`.re)*sinh(`z`.im)i
  Examples:

      writeln(cos(0.0 + 0.0i)); // 1.0
      writeln(cos(1.3L + 0.0i)); // cos(1.3L)
      writeln(cos(5.2Li)); // cosh(5.2L)

pure nothrow @nogc @trusted real `tan`(real `x`);
  Returns tangent of `x`. `x` is in radians.

  Special Values
  `x` | `tan`(`x`) | invalid?
  --- | --- | ---
  NAN | NAN | yes
  ±0.0 | ±0.0 | no
  ±∞ | NAN | yes

pure nothrow @nogc @safe real `acos`(real `x`);
pure nothrow @nogc @safe double `acos`(double `x`);
pure nothrow @nogc @safe float `acos`(float `x`);
  Calculates the arc cosine of `x`, returning a value ranging from 0 to π.

  Special Values
  `x` | `acos`(`x`) | invalid?
  --- | --- | ---
  >1.0 | NAN | yes
  <-1.0 | NAN | yes
  NAN | NAN | yes

pure nothrow @nogc @safe real `asin`(real `x`);
pure nothrow @nogc @safe double `asin`(double `x`);
pure nothrow @nogc @safe float `asin`(float `x`);
  Calculates the arc sine of `x`, returning a value ranging from -π/2 to π/2.

  Special Values
  `x` | `asin`(`x`) | invalid?
  --- | --- | ---
  ±0.0 | ±0.0 | no
  >1.0 | NAN | yes
  <-1.0 | NAN | yes

pure nothrow @nogc @safe real `atan`(real `x`);
pure nothrow @nogc @safe double `atan`(double `x`);
pure nothrow @nogc @safe float `atan`(float `x`);
  Calculates the arc tangent of `x`, returning a value ranging from -π/2 to π/2.

  Special Values
  `x` | `atan`(`x`) | invalid?
  --- | --- | ---
  ±0.0 | ±0.0 | no
  ±∞ | NAN | yes

pure nothrow @nogc @trusted real `atan2`(real `y`, real `x`);
pure nothrow @nogc @safe double `atan2`(double `y`, double `x`);
pure nothrow @nogc @safe float `atan2`(float `y`, float `x`);
  Calculates the arc tangent of `y` / `x`, returning a value ranging from -π to π.

  Special Values
  `y` | `x` | atan(`y`, `x`)
  --- | --- | ---
  NAN | anything | NAN
  anything | NAN | NAN
  ±0.0 | >0.0 | ±0.0
  ±0.0 | +0.0 | ±0.0
  ±0.0 | <0.0 | ±π
  ±0.0 | -0.0 | ±π
  >0.0 | ±0.0 | π/2
  <0.0 | ±0.0 | -π/2
  >0.0 | ∞ | ±0.0
  ±∞ | anything | ±π/2
  >0.0 | -∞ | ±π
  ±∞ | ∞ | ±π/4
  ±∞ | -∞ | ±3π/4

pure nothrow @nogc @safe real `cosh`(real `x`);
pure nothrow @nogc @safe double `cosh`(double `x`);
pure nothrow @nogc @safe float `cosh`(float `x`);
  Calculates the hyperbolic cosine of `x`.

  Special Values
  `x` | `cosh`(`x`) | invalid?
  --- | --- | ---
  ±∞ | ±0.0 | no

pure nothrow @nogc @safe real `sinh`(real `x`);
pure nothrow @nogc @safe double `sinh`(double `x`);
pure nothrow @nogc @safe float `sinh`(float `x`);
  Calculates the hyperbolic sine of `x`.

  Special Values
  `x` | `sinh`(`x`) | invalid?
  --- | --- | ---
  ±0.0 | ±0.0 | no
  ±∞ | ±∞ | no

pure nothrow @nogc @safe real `tanh`(real `x`);
pure nothrow @nogc @safe double `tanh`(double `x`);
pure nothrow @nogc @safe float `tanh`(float `x`);
  Calculates the hyperbolic tangent of `x`.

  Special Values
  `x` | `tanh`(`x`) | invalid?
  --- | --- | ---
  ±0.0 | ±0.0 | no
  ±∞ | ±1.0 | no

pure nothrow @nogc @safe real `acosh`(real `x`);
pure nothrow @nogc @safe double `acosh`(double `x`);
pure nothrow @nogc @safe float `acosh`(float `x`);
  Calculates the inverse hyperbolic cosine of `x`.
  Mathematically, `acosh`(`x`) = log(`x` + sqrt( `x`*`x` - 1))

  Domain X | Range Y
  --- | ---
  1..∞ | 0..∞

  Special Values
  `x` | `acosh`(`x`)
  --- | ---
  NAN | NAN
  <1 | NAN
  1 | 0
  +∞ | +∞

pure nothrow @nogc @safe real `asinh`(real `x`);
pure nothrow @nogc @safe double `asinh`(double `x`);
pure nothrow @nogc @safe float `asinh`(float `x`);
  Calculates the inverse hyperbolic sine of `x`.
  Mathematically,

      asinh(x) =  log( x + sqrt( x*x + 1 )) // if x >= +0
      asinh(x) = -log(-x + sqrt( x*x + 1 )) // if x <= -0

  Special Values
  `x` | `asinh`(`x`)
  --- | ---
  NAN | NAN
  ±0 | ±0
  ±∞ | ±∞

pure nothrow @nogc @safe real `atanh`(real `x`);
pure nothrow @nogc @safe double `atanh`(double `x`);
pure nothrow @nogc @safe float `atanh`(float `x`);
  Calculates the inverse hyperbolic tangent of `x`, returning a value from ranging from -1 to 1.
  Mathematically, `atanh`(`x`) = log( (1+`x`)/(1-`x`) ) / 2

  Domain X | Range Y
  --- | ---
  -∞..∞ | -1 .. 1

  Special Values
  `x` | acosh(`x`)
  --- | ---
  NAN | NAN
  ±0 | ±0
  -∞ | -0

pure nothrow @nogc @safe long `rndtol`(real `x`);
pure nothrow @nogc @safe long `rndtol`(double `x`);
pure nothrow @nogc @safe long `rndtol`(float `x`);
  Returns `x` rounded to a long value using the current rounding mode. If the integer value of `x` is greater than long.max, the result is indeterminate.
real `rndtonl`(real `x`);
  Returns `x` rounded to a long value using the FE_TONEAREST rounding mode. If the integer value of `x` is greater than long.max, the result is indeterminate.
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

pure nothrow @nogc @trusted real `exp`(real `x`);
pure nothrow @nogc @safe double `exp`(double `x`);
pure nothrow @nogc @safe float `exp`(float `x`);
  Calculates e`x`.

  Special Values
  `x` | e`x`
  --- | ---
  +∞ | +∞
  -∞ | +0.0
  NAN | NAN

pure nothrow @nogc @trusted real `expm1`(real `x`);
  Calculates the value of the natural logarithm base (e) raised to the power of `x`, minus 1.
  For very small `x`, `expm1`(`x`) is more accurate than exp(`x`)-1.

  Special Values
  `x` | e`x`-1
  --- | ---
  ±0.0 | ±0.0
  +∞ | +∞
  -∞ | -1.0
  NAN | NAN

pure nothrow @nogc @trusted real `exp2`(real `x`);
  Calculates 2`x`.

  Special Values
  `x` | `exp2`(`x`)
  --- | ---
  +∞ | +∞
  -∞ | +0.0
  NAN | NAN

pure nothrow @nogc @trusted creal `expi`(real `y`);
  Calculate cos(`y`) + i sin(`y`).
  On many CPUs (such as x86), this is a very efficient operation; almost twice as fast as calculating sin(`y`) and cos(`y`) separately, and is the preferred method when both are required.
  Examples:

      writeln(expi(1.3e5L)); // cos(1.3e5L) + sin(1.3e5L) * 1i
      writeln(expi(0.0L)); // 1L + 0.0Li

pure nothrow @nogc @trusted T `frexp`(T)(const T `value`, out int `exp`)
if (isFloatingPoint!T);
  Separate floating point `value` into significand and exponent.
  Returns:
  Calculate and return x and `exp` such that `value` =x*2`exp` and .5 <= |x| < 1.0
  x has same sign as `value`.

  Special Values
  `value` | returns | `exp`
  --- | --- | ---
  ±0.0 | ±0.0 | 0
  +∞ | +∞ | int.max
  -∞ | -∞ | int.min
  ±NAN | ±NAN | int.min

  Examples:

      int exp;
      real mantissa = frexp(123.456L, exp);

      assert(approxEqual(mantissa * pow(2.0L, cast(real) exp), 123.456L));

      assert(frexp(-real.nan, exp) && exp == int.min);
      assert(frexp(real.nan, exp) && exp == int.min);
      assert(frexp(-real.infinity, exp) == -real.infinity && exp == int.min);
      assert(frexp(real.infinity, exp) == real.infinity && exp == int.max);
      assert(frexp(-0.0, exp) == -0.0 && exp == 0);
      assert(frexp(0.0, exp) == 0.0 && exp == 0);

pure nothrow @nogc @trusted int `ilogb`(T)(const T `x`)
if (isFloatingPoint!T);
pure nothrow @nogc @safe int `ilogb`(T)(const T `x`)
if (isIntegral!T && isUnsigned!T);
pure nothrow @nogc @safe int `ilogb`(T)(const T `x`)
if (isIntegral!T && isSigned!T);
  Extracts the exponent of `x` as a signed integral value.
  If `x` is not a special value, the result is the same as cast(int) logb(`x`).

  Special Values
  `x` | `ilogb`(`x`) | Range error?
  --- | --- | ---
  0 | FP_ILOGB0 | yes
  ±∞ | int.max | no
  NAN | FP_ILOGBNAN | no

pure nothrow @nogc @safe real `ldexp`(real `n`, int `exp`);
pure nothrow @nogc @safe double `ldexp`(double `n`, int `exp`);
pure nothrow @nogc @safe float `ldexp`(float `n`, int `exp`);
  Compute `n` * 2`exp`

  References frexp

  Examples:

      import std.meta : AliasSeq;
      foreach (T; AliasSeq!(float, double, real))
      {
          T r;

          r = ldexp(3.0L, 3);
          writeln(r); // 24

          r = ldexp(cast(T) 3.0, cast(int) 3);
          writeln(r); // 24

          T n = 3.0;
          int exp = 3;
          r = ldexp(n, exp);
          writeln(r); // 24
      }

pure nothrow @nogc @safe real `log`(real `x`);
  Calculate the natural logarithm of `x`.

  Special Values
  `x` | `log`(`x`) | divide by 0? | invalid?
  --- | --- | --- | ---
  ±0.0 | -∞ | yes | no
  <0.0 | NAN | no | yes
  +∞ | +∞ | no | no

  Examples:

      writeln(log(E)); // 1

pure nothrow @nogc @safe real `log10`(real `x`);
  Calculate the base-10 logarithm of `x`.

  Special Values
  `x` | `log10`(`x`) | divide by 0? | invalid?
  --- | --- | --- | ---
  ±0.0 | -∞ | yes | no
  <0.0 | NAN | no | yes
  +∞ | +∞ | no | no

  Examples:

      assert(fabs(log10(1000) - 3) < .000001);

pure nothrow @nogc @safe real `log1p`(real `x`);
  Calculates the natural logarithm of 1 + `x`.
  For very small `x`, `log1p`(`x`) will be more accurate than log(1 + `x`).

  Special Values
  `x` | `log1p`(`x`) | divide by 0? | invalid?
  --- | --- | --- | ---
  ±0.0 | ±0.0 | no | no
  -1.0 | -∞ | yes | no
  <-1.0 | NAN | no | yes
  +∞ | -∞ | no | no

pure nothrow @nogc @safe real `log2`(real `x`);
  Calculates the base-2 logarithm of `x`: log2`x`

  Special Values
  `x` | `log2`(`x`) | divide by 0? | invalid?
  --- | --- | --- | ---
  ±0.0 | -∞ | yes | no
  <0.0 | NAN | no | yes
  +∞ | +∞ | no | no

  Examples:

      assert(approxEqual(log2(1024.0L), 10));

nothrow @nogc @trusted real `logb`(real `x`);
  Extracts the exponent of `x` as a signed integral value.
  If `x` is subnormal, it is treated as if it were normalized. For a positive, finite `x`:
  1 <= `x` * FLT_RADIX-`logb`(`x`) < FLT_RADIX

  Special Values
  `x` | `logb`(`x`) | divide by 0?
  --- | --- | ---
  ±∞ | +∞ | no
  ±0.0 | -∞ | yes

nothrow @nogc @trusted real `fmod`(real `x`, real `y`);
  Calculates the remainder from the calculation `x`/`y`.
  Returns:
  The value of `x` - i * `y`, where i is the number of times that `y` can be completely subtracted from `x`. The result has the same sign as `x`.

  Special Values
  `x` | `y` | `fmod`(`x`, `y`) | invalid?
  --- | --- | --- | ---
  ±0.0 | not 0.0 | ±0.0 | no
  ±∞ | anything | NAN | yes
  anything | ±0.0 | NAN | yes
  !=±∞ | ±∞ | `x` | no

nothrow @nogc @trusted real `modf`(real `x`, ref real `i`);
  Breaks `x` into an integral part and a fractional part, each of which has the same sign as `x`. The integral part is stored in `i`.
  Returns:
  The fractional part of `x`.

  Special Values
  `x` | `i` (on input) | `modf`(`x`, `i`) | `i` (on return)
  --- | --- | --- | ---
  ±∞ | anything | ±0.0 | ±∞

nothrow @nogc @trusted real `scalbn`(real `x`, int `n`);
  Efficiently calculates `x` * 2`n`.
  `scalbn` handles underflow and overflow in the same fashion as the basic arithmetic operators.

  Special Values
  `x` | scalb(`x`)
  --- | ---
  ±∞ | ±∞
  ±0.0 | ±0.0

  Examples:

      writeln(scalbn(-real.infinity, 5)); // -real.infinity

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

pure nothrow @nogc @safe real `fabs`(real `x`);
pure nothrow @nogc @safe double `fabs`(double `x`);
pure nothrow @nogc @safe float `fabs`(float `x`);
  Returns |`x`|

  Special Values
  `x` | `fabs`(`x`)
  --- | ---
  ±0.0 | +0.0
  ±∞ | +∞

pure nothrow @nogc @safe real `hypot`(real `x`, real `y`);
  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(`x`2 + `y`2)
  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

pure nothrow @nogc @trusted real `ceil`(real `x`);
  Returns the value of `x` rounded upward to the next integer (toward positive infinity).
  Examples:

      writeln(ceil(+123.456L)); // +124
      writeln(ceil(-123.456L)); // -123
      writeln(ceil(-1.234L)); // -1
      writeln(ceil(-0.123L)); // 0
      writeln(ceil(0.0L)); // 0
      writeln(ceil(+0.123L)); // 1
      writeln(ceil(+1.234L)); // 2
      writeln(ceil(real.infinity)); // real.infinity
      assert(isNaN(ceil(real.nan)));
      assert(isNaN(ceil(real.init)));

pure nothrow @nogc @trusted real `floor`(real `x`);
  Returns the value of `x` rounded downward to the next integer (toward negative infinity).
  Examples:

      writeln(floor(+123.456L)); // +123
      writeln(floor(-123.456L)); // -124
      writeln(floor(-1.234L)); // -2
      writeln(floor(-0.123L)); // -1
      writeln(floor(0.0L)); // 0
      writeln(floor(+0.123L)); // 0
      writeln(floor(+1.234L)); // 1
      writeln(floor(real.infinity)); // real.infinity
      assert(isNaN(floor(real.nan)));
      assert(isNaN(floor(real.init)));

Unqual!F `quantize`(alias rfunc = rint, F)(const F `val`, const F `unit`)
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F);
  Round `val` to a multiple of `unit`. rfunc specifies the rounding function to use; by default this is rint, which uses the current rounding mode.
  Examples:

      writeln(12345.6789L.quantize(0.01L)); // 12345.68L
      writeln(12345.6789L.quantize!floor(0.01L)); // 12345.67L
      writeln(12345.6789L.quantize(22.0L)); // 12342.0L

  Examples:

      writeln(12345.6789L.quantize(0)); // 12345.6789L
      assert(12345.6789L.quantize(real.infinity).isNaN);
      assert(12345.6789L.quantize(real.nan).isNaN);
      writeln(real.infinity.quantize(0.01L)); // real.infinity
      assert(real.infinity.quantize(real.nan).isNaN);
      assert(real.nan.quantize(0.01L).isNaN);
      assert(real.nan.quantize(real.infinity).isNaN);
      assert(real.nan.quantize(real.nan).isNaN);

Unqual!F `quantize`(real base, alias rfunc = rint, F, E)(const F `val`, const E `exp`)
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F && isIntegral!E);
Unqual!F `quantize`(real base, long exp = 1, alias rfunc = rint, F)(const F `val`)
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F);
  Round `val` to a multiple of pow(base, `exp`). rfunc specifies the rounding function to use; by default this is rint, which uses the current rounding mode.
  Examples:

      writeln(12345.6789L.quantize!10(-2)); // 12345.68L
      writeln(12345.6789L.quantize!(10, -2)); // 12345.68L
      writeln(12345.6789L.quantize!(10, floor)(-2)); // 12345.67L
      writeln(12345.6789L.quantize!(10, -2, floor)); // 12345.67L

      writeln(12345.6789L.quantize!22(1)); // 12342.0L
      writeln(12345.6789L.quantize!22); // 12342.0L

nothrow @nogc @trusted real `nearbyint`(real `x`);
  Rounds `x` to the nearest integer value, using the current rounding mode.
  Unlike the rint functions, `nearbyint` does not raise the FE_INEXACT exception.
pure nothrow @nogc @safe real `rint`(real `x`);
pure nothrow @nogc @safe double `rint`(double `x`);
pure nothrow @nogc @safe float `rint`(float `x`);
  Rounds `x` to the nearest integer value, using the current rounding mode. If the return value is not equal to `x`, the FE_INEXACT exception is raised. nearbyint performs the same operation, but does not set the FE_INEXACT exception.
pure nothrow @nogc @trusted long `lrint`(real `x`);
  Rounds `x` to the nearest integer value, using the current rounding mode.
  This is generally the fastest method to convert a floating-point number to an integer. Note that the results from this function depend on the rounding mode, if the fractional part of `x` is exactly 0.5. If using the default rounding mode (ties round to even integers) `lrint`(4.5) == 4, `lrint`(5.5)==6.
  Examples:

      writeln(lrint(4.5)); // 4
      writeln(lrint(5.5)); // 6
      writeln(lrint(-4.5)); // -4
      writeln(lrint(-5.5)); // -6

      writeln(lrint(int.max - 0.5)); // 2147483646L
      writeln(lrint(int.max + 0.5)); // 2147483648L
      writeln(lrint(int.min - 0.5)); // -2147483648L
      writeln(lrint(int.min + 0.5)); // -2147483648L

nothrow @nogc @trusted real `round`(real `x`);
  Return the value of `x` rounded to the nearest integer. If the fractional part of `x` is exactly 0.5, the return value is rounded away from zero.
nothrow @nogc @trusted long `lround`(real `x`);
  Return the value of `x` rounded to the nearest integer.
  If the fractional part of `x` is exactly 0.5, the return value is rounded away from zero.
  This function is Posix-Only.
nothrow @nogc @trusted real `trunc`(real `x`);
  Returns the integer portion of `x`, dropping the fractional portion.
  This is also known as "chop" rounding.
nothrow @nogc @trusted real `remainder`(real `x`, real `y`);
  Calculate the `remainder` `x` REM `y`, following IEC 60559.
  REM is the value of `x` - `y` * n, where n is the integer nearest the exact value of `x` / `y`. If |n - `x` / `y`| == 0.5, n is even. If the result is zero, it has the same sign as `x`. Otherwise, the sign of the result is the sign of `x` / `y`. Precision mode has no effect on the `remainder` functions.
  remquo returns n in the parameter n.

  Special Values
  `x` | `y` | `remainder`(`x`, `y`) | n | invalid?
  --- | --- | --- | --- | ---
  ±0.0 | not 0.0 | ±0.0 | 0.0 | no
  ±∞ | anything | NAN | ? | yes
  anything | ±0.0 | NAN | ? | yes
  != ±∞ | ±∞ | `x` | ? | no

  remquo and `remainder` not supported on Windows.
struct `IeeeFlags`;
  IEEE exception status flags ('sticky bits')
  These flags indicate that an exceptional floating-point condition has occurred. They indicate that a NaN or an infinity has been generated, that a result is inexact, or that a signalling NaN has been encountered. If floating-point exceptions are enabled (unmasked), a hardware exception will be generated instead of setting these flags.
  Examples:

      static void func() {
          int a = 10 * 10;
      }

      real a=3.5;
      // Set all the flags to zero
      resetIeeeFlags();
      assert(!ieeeFlags.divByZero);
      // Perform a division by zero.
      a/=0.0L;
      writeln(a); // real.infinity
      assert(ieeeFlags.divByZero);
      // Create a NaN
      a*=0.0L;
      assert(ieeeFlags.invalid);
      assert(isNaN(a));

      // Check that calling func() has no effect on the
      // status flags.
      IeeeFlags f = ieeeFlags;
      func();
      writeln(ieeeFlags); // f

  const @property bool `inexact`();
    The result cannot be represented exactly, so rounding occurred.

    Example x = sin(0.1);

  const @property bool `underflow`();
    A zero was generated by `underflow`

    Example x = real.min*real.epsilon/2;

  const @property bool `overflow`();
    An infinity was generated by `overflow`

    Example x = real.max*2;

  const @property bool `divByZero`();
    An infinity was generated by division by zero

    Example x = 3/0.0;

  const @property bool `invalid`();
    A machine NaN was generated.

    Example x = real.infinity * 0.0;

@nogc void `resetIeeeFlags`();
  Set all of the floating-point status flags to `false`.
@property IeeeFlags `ieeeFlags`();
  Returns:
  snapshot of the current state of the floating-point status flags
struct `FloatingPointControl`;
  Control the Floating point hardware
  Change the IEEE754 floating-point rounding mode and the floating-point hardware exceptions.
  By default, the rounding mode is roundToNearest and all hardware exceptions are disabled. For most applications, debugging is easier if the division by zero, overflow, and invalid operation exceptions are enabled. These three are combined into a severeExceptions value for convenience. Note in particular that if invalidException is enabled, a hardware trap will be generated whenever an uninitialized floating-point variable is used.
  All changes are temporary. The previous state is restored at the end of the scope.

  Example

      {
          FloatingPointControl fpctrl;

          // Enable hardware exceptions for division by zero, overflow to infinity,
          // invalid operations, and uninitialized floating-point variables.
          fpctrl.enableExceptions(FloatingPointControl.severeExceptions);

          // This will generate a hardware exception, if x is a
          // default-initialized floating point variable:
          real x; // Add `= 0` or even `= real.nan` to not throw the exception.
          real y = x * 3.0;

          // The exception is only thrown for default-uninitialized NaN-s.
          // NaN-s with other payload are valid:
          real z = y * real.nan; // ok

          // Changing the rounding mode:
          fpctrl.rounding = FloatingPointControl.roundUp;
          assert(rint(1.1) == 2);

          // The set hardware exceptions will be disabled when leaving this scope.
          // The original rounding mode will also be restored.
      }

      // Ensure previous values are returned:
      assert(!FloatingPointControl.enabledExceptions);
      assert(FloatingPointControl.rounding == FloatingPointControl.roundToNearest);
      assert(rint(1.1) == 1);

  alias `RoundingMode` = uint;
  `roundToNearest`
  `roundDown`
  `roundUp`
  `roundToZero`
  `roundingMask`
    IEEE rounding modes. The default mode is `roundToNearest`.
    `roundingMask` = A mask of all rounding modes.
  @nogc @property void `rounding`(RoundingMode `newMode`);
    Change the floating-point hardware `rounding` mode
  static @nogc @property RoundingMode `rounding`();
    Returns:
    the currently active `rounding` mode
  alias `ExceptionMask` = uint;
  `subnormalException`
  `inexactException`
  `underflowException`
  `overflowException`
  `divByZeroException`
  `invalidException`
  `severeExceptions`
  `allExceptions`
    IEEE hardware exceptions. By default, all exceptions are masked (disabled).
    `severeExceptions` = The overflow, division by zero, and invalid exceptions.
  static nothrow @nogc @property @safe bool `hasExceptionTraps`();
    Returns:
    `true` if the current FPU supports exception trapping
  @nogc void `enableExceptions`(ExceptionMask `exceptions`);
    Enable (unmask) specific hardware `exceptions`. Multiple `exceptions` may be ORed together.
  @nogc void `disableExceptions`(ExceptionMask `exceptions`);
    Disable (mask) specific hardware `exceptions`. Multiple `exceptions` may be ORed together.
  static @nogc @property ExceptionMask `enabledExceptions`();
    Returns:
    the exceptions which are currently enabled (unmasked)
pure nothrow @nogc @trusted bool `isNaN`(X)(X `x`)
if (isFloatingPoint!X);
  Determines if `x` is NaN.
  Parameters:

  X `x` | a floating point number.

  Returns:
  `true` if `x` is Nan.
  Examples:

      assert( isNaN(float.init));
      assert( isNaN(-double.init));
      assert( isNaN(real.nan));
      assert( isNaN(-real.nan));
      assert(!isNaN(cast(float) 53.6));
      assert(!isNaN(cast(real)-53.6));

pure nothrow @nogc @trusted bool `isFinite`(X)(X `x`)
if (isFloatingPoint!X);
  Determines if `x` is finite.
  Parameters:

  X `x` | a floating point number.

  Returns:
  `true` if `x` is finite.
  Examples:

      assert( isFinite(1.23f));
      assert( isFinite(float.max));
      assert( isFinite(float.min_normal));
      assert(!isFinite(float.nan));
      assert(!isFinite(float.infinity));

pure nothrow @nogc @trusted bool `isNormal`(X)(X `x`)
if (isFloatingPoint!X);
  Determines if `x` is normalized.
  A normalized number must not be zero, subnormal, infinite nor NAN.
  Parameters:

  X `x` | a floating point number.

  Returns:
  `true` if `x` is normalized.
  Examples:

      float f = 3;
      double d = 500;
      real e = 10e+48;

      assert(isNormal(f));
      assert(isNormal(d));
      assert(isNormal(e));
      f = d = e = 0;
      assert(!isNormal(f));
      assert(!isNormal(d));
      assert(!isNormal(e));
      assert(!isNormal(real.infinity));
      assert(isNormal(-real.max));
      assert(!isNormal(real.min_normal/4));

pure nothrow @nogc @trusted bool `isSubnormal`(X)(X `x`)
if (isFloatingPoint!X);
  Determines if `x` is subnormal.
  Subnormals (also known as "denormal number"), have a 0 exponent and a 0 most significant mantissa bit.
  Parameters:

  X `x` | a floating point number.

  Returns:
  `true` if `x` is a denormal number.
  Examples:

      import std.meta : AliasSeq;

      foreach (T; AliasSeq!(float, double, real))
      {
          T f;
          for (f = 1.0; !isSubnormal(f); f /= 2)
              assert(f != 0);
      }

pure nothrow @nogc @trusted bool `isInfinity`(X)(X `x`)
if (isFloatingPoint!X);
  Determines if `x` is ±∞.
  Parameters:

  X `x` | a floating point number.

  Returns:
  `true` if `x` is ±∞.
  Examples:

      assert(!isInfinity(float.init));
      assert(!isInfinity(-float.init));
      assert(!isInfinity(float.nan));
      assert(!isInfinity(-float.nan));
      assert(isInfinity(float.infinity));
      assert(isInfinity(-float.infinity));
      assert(isInfinity(-1.0f / 0.0f));

pure nothrow @nogc @trusted bool `isIdentical`(real `x`, real `y`);
  Is the binary representation of `x` identical to `y`?
  Same as ==, except that positive and negative zero are not identical, and two NANs are identical if they have the same 'payload'.
pure nothrow @nogc @trusted int `signbit`(X)(X `x`)
if (isFloatingPoint!X);
  Return 1 if sign bit of e is set, 0 if not.
  Examples:

      assert(!signbit(float.nan));
      assert(signbit(-float.nan));
      assert(!signbit(168.1234f));
      assert(signbit(-168.1234f));
      assert(!signbit(0.0f));
      assert(signbit(-0.0f));
      assert(signbit(-float.max));
      assert(!signbit(float.max));

      assert(!signbit(double.nan));
      assert(signbit(-double.nan));
      assert(!signbit(168.1234));
      assert(signbit(-168.1234));
      assert(!signbit(0.0));
      assert(signbit(-0.0));
      assert(signbit(-double.max));
      assert(!signbit(double.max));

      assert(!signbit(real.nan));
      assert(signbit(-real.nan));
      assert(!signbit(168.1234L));
      assert(signbit(-168.1234L));
      assert(!signbit(0.0L));
      assert(signbit(-0.0L));
      assert(signbit(-real.max));
      assert(!signbit(real.max));

pure nothrow @nogc @trusted R `copysign`(R, X)(R `to`, X `from`)
if (isFloatingPoint!R && isFloatingPoint!X);
  Return a value composed of `to` with `from`'s sign bit.
pure nothrow @nogc @safe F `sgn`(F)(F `x`)
if (isNumeric!F);
  Returns -1 if `x` < 0, `x` if `x` == 0, 1 if `x` > 0, and NAN if `x`==NAN.
  Examples:

      writeln(sgn(168.1234)); // 1
      writeln(sgn(-168.1234)); // -1
      writeln(sgn(0.0)); // 0
      writeln(sgn(-0.0)); // 0

pure nothrow @nogc @trusted real `NaN`(ulong `payload`);
  Create a quiet NAN, storing an integer inside the `payload`.
  For floats, the largest possible `payload` is 0x3F_FFFF. For doubles, it is 0x3_FFFF_FFFF_FFFF. For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
pure nothrow @nogc @trusted ulong `getNaNPayload`(real `x`);
  Extract an integral payload from a NAN.
  Returns:
  the integer payload as a ulong.
  For floats, the largest possible payload is 0x3F_FFFF. For doubles, it is 0x3_FFFF_FFFF_FFFF. For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
pure nothrow @nogc @trusted real `nextUp`(real `x`);
pure nothrow @nogc @trusted double `nextUp`(double `x`);
pure nothrow @nogc @trusted float `nextUp`(float `x`);
  Calculate the next largest floating point value after `x`.
  Return the least number greater than `x` that is representable as a real; thus, it gives the next point on the IEEE number line.

  Special Values
  `x` | `nextUp`(`x`)
  --- | ---
  -∞ | -real.max
  ±0.0 | real.min_normal*real.epsilon
  real.max | ∞
  ∞ | ∞
  NAN | NAN

pure nothrow @nogc @safe real `nextDown`(real `x`);
pure nothrow @nogc @safe double `nextDown`(double `x`);
pure nothrow @nogc @safe float `nextDown`(float `x`);
  Calculate the next smallest floating point value before `x`.
  Return the greatest number less than `x` that is representable as a real; thus, it gives the previous point on the IEEE number line.

  Special Values
  `x` | `nextDown`(`x`)
  --- | ---
  ∞ | real.max
  ±0.0 | -real.min_normal*real.epsilon
  -real.max | -∞
  -∞ | -∞
  NAN | NAN

  Examples:

      writeln(nextDown(1.0 + real.epsilon)); // 1.0

pure nothrow @nogc @safe T `nextafter`(T)(const T `x`, const T `y`)
if (isFloatingPoint!T);
  Calculates the next representable value after `x` in the direction of `y`.
  If `y` > `x`, the result will be the next largest floating-point value; if `y` < `x`, the result will be the next smallest value. If `x` == `y`, the result is `y`.

  Remarks This function is not generally very useful; it's almost always better to use the faster functions nextUp() or nextDown() instead.

  The FE_INEXACT and FE_OVERFLOW exceptions will be raised if `x` is finite and the function result is infinite. The FE_INEXACT and FE_UNDERFLOW exceptions will be raised if the function value is subnormal, and `x` is not equal to `y`.
  Examples:

      float a = 1;
      assert(is(typeof(nextafter(a, a)) == float));
      assert(nextafter(a, a.infinity) > a);

      double b = 2;
      assert(is(typeof(nextafter(b, b)) == double));
      assert(nextafter(b, b.infinity) > b);

      real c = 3;
      assert(is(typeof(nextafter(c, c)) == real));
      assert(nextafter(c, c.infinity) > c);

pure nothrow @nogc @safe real `fdim`(real `x`, real `y`);
  Returns the positive difference between `x` and `y`.
  Returns:

  Special Values
  `x`, `y` | `fdim`(`x`, `y`)
  --- | ---
  `x` > `y` | `x` - `y`
  `x` <= `y` | +0.0

pure nothrow @nogc @safe real `fmax`(real `x`, real `y`);
  Returns the larger of `x` and `y`.
pure nothrow @nogc @safe real `fmin`(real `x`, real `y`);
  Returns the smaller of `x` and `y`.
pure nothrow @nogc @safe real `fma`(real `x`, real `y`, real `z`);
  Returns (`x` * `y`) + `z`, rounding only once according to the current rounding mode.
  Bugs:
  Not currently implemented - rounds twice.
pure nothrow @nogc @trusted Unqual!F `pow`(F, G)(F `x`, G `n`)
if (isFloatingPoint!F && isIntegral!G);
  Compute the value of `x` `n`, where `n` is an integer
pure nothrow @nogc @trusted typeof(Unqual!F.init * Unqual!G.init) `pow`(F, G)(F `x`, G `n`)
if (isIntegral!F && isIntegral!G);
  Compute the value of an integer `x`, raised to the power of a positive integer `n`.
  If both `x` and `n` are 0, the result is 1. If `n` is negative, an integer divide error will occur at runtime, regardless of the value of `x`.
  Examples:

      immutable int one = 1;
      immutable byte two = 2;
      immutable ubyte three = 3;
      immutable short four = 4;
      immutable long ten = 10;

      writeln(pow(two, three)); // 8
      writeln(pow(two, ten)); // 1024
      writeln(pow(one, ten)); // 1
      writeln(pow(ten, four)); // 10_000
      writeln(pow(four, 10)); // 1_048_576
      writeln(pow(three, four)); // 81

pure nothrow @nogc @trusted real `pow`(I, F)(I `x`, F `y`)
if (isIntegral!I && isFloatingPoint!F);
  Computes integer to floating point powers.
pure nothrow @nogc @trusted Unqual!(Largest!(F, G)) `pow`(F, G)(F `x`, G `y`)
if (isFloatingPoint!F && isFloatingPoint!G);
  Calculates `x``y`.

  Special Values
  `x` | `y` | `pow`(`x`, `y`) | div 0 | invalid?
  --- | --- | --- | --- | ---
  anything | ±0.0 | 1.0 | no | no
  |`x`| > 1 | +∞ | +∞ | no | no
  |`x`| < 1 | +∞ | +0.0 | no | no
  |`x`| > 1 | -∞ | +0.0 | no | no
  |`x`| < 1 | -∞ | +∞ | no | no
  +∞ | > 0.0 | +∞ | no | no
  +∞ | < 0.0 | +0.0 | no | no
  -∞ | odd integer > 0.0 | -∞ | no | no
  -∞ | > 0.0, not odd integer | +∞ | no | no
  -∞ | odd integer < 0.0 | -0.0 | no | no
  -∞ | < 0.0, not odd integer | +0.0 | no | no
  ±1.0 | ±∞ | NAN | no | yes
  < 0.0 | finite, nonintegral | NAN | no | yes
  ±0.0 | odd integer < 0.0 | ±∞ | yes | no
  ±0.0 | < 0.0, not odd integer | +∞ | yes | no
  ±0.0 | odd integer > 0.0 | ±0.0 | no | no
  ±0.0 | > 0.0, not odd integer | +0.0 | no | no

Unqual!(Largest!(F, H)) `powmod`(F, G, H)(F `x`, G `n`, H `m`)
if (isUnsigned!F && isUnsigned!G && isUnsigned!H);
  Computes the value of a positive integer `x`, raised to the power `n`, modulo `m`.
  Parameters:

  F `x` | base
  G `n` | exponent
  H `m` | modulus

  Returns:
  `x` to the power `n`, modulo `m`. The return type is the largest of `x`'s and `m`'s type.
  The function requires that all values have unsigned types.
pure nothrow @nogc @trusted int `feqrel`(X)(const X `x`, const X `y`)
if (isFloatingPoint!X);
  To what precision is `x` equal to `y`?
  Returns:
  the number of mantissa bits which are equal in `x` and `y`. eg, 0x1.F8p+60 and 0x1.F1p+60 are equal to 5 bits of precision.

  Special Values
  `x` | `y` | `feqrel`(`x`, `y`)
  --- | --- | ---
  `x` | `x` | real.mant_dig
  `x` | >= 2*`x` | 0
  `x` | <= `x`/2 | 0
  NAN | any | 0
  any | NAN | 0

pure nothrow @nogc @trusted Unqual!(CommonType!(T1, T2)) `poly`(T1, T2)(T1 `x`, in T2[] `A`)
if (isFloatingPoint!T1 && isFloatingPoint!T2);
  Evaluate polynomial `A`(`x`) = a0 + a1`x` + a2`x`2 + a3`x`3; ...
  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 = 3.1;
      static real[] pp = [56.1, 32.7, 6];

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

bool `approxEqual`(T, U, V)(T `lhs`, U `rhs`, V `maxRelDiff`, V `maxAbsDiff` = 1e-05)
if ((isNumeric!T || isInputRange!T && isNumeric!(ElementType!T)) && (isNumeric!U || isInputRange!U && isNumeric!(ElementType!U)) && isNumeric!V);
  Computes whether two values are approximately equal, admitting a maximum relative difference, and a maximum absolute difference.
  Parameters:

  T `lhs` | First item to compare.
  U `rhs` | Second item to compare.
  V `maxRelDiff` | Maximum allowable difference relative to `rhs`.
  V `maxAbsDiff` | Maximum absolute difference.

  Returns:
  true if the two items are approximately equal under either criterium. If one item is a range, and the other is a single value, then the result is the logical and-ing of calling `approxEqual` on each element of the ranged item against the single item. If both items are ranges, then `approxEqual` returns true if and only if the ranges have the same number of elements and if `approxEqual` evaluates to true for each pair of elements.
bool `approxEqual`(T, U)(T `lhs`, U `rhs`);
  Returns `approxEqual`(`lhs`, `rhs`, 1e-2, 1e-5).
  Examples:

      assert(approxEqual(1.0, 1.0099));
      assert(!approxEqual(1.0, 1.011));
      float[] arr1 = [ 1.0, 2.0, 3.0 ];
      double[] arr2 = [ 1.001, 1.999, 3 ];
      assert(approxEqual(arr1, arr2));

      real num = real.infinity;
      assert(num == real.infinity);  // Passes.
      assert(approxEqual(num, real.infinity));  // Fails.
      num = -real.infinity;
      assert(num == -real.infinity);  // Passes.
      assert(approxEqual(num, -real.infinity));  // Fails.

      assert(!approxEqual(3, 0));
      assert(approxEqual(3, 3));
      assert(approxEqual(3.0, 3));
      assert(approxEqual([3, 3, 3], 3.0));
      assert(approxEqual([3.0, 3.0, 3.0], 3));
      int a = 10;
      assert(approxEqual(10, a));

pure nothrow @nogc @trusted int `cmp`(T)(const(T) `x`, const(T) `y`)
if (isFloatingPoint!T);
  Defines a total order on all floating-point numbers.
  The order is defined as follows:
  - All numbers in [-∞, +∞] are ordered the same way as by built-in comparison, with the exception of -0.0, which is less than +0.0;
  - If the sign bit is set (that is, it's 'negative'), NAN is less than any number; if the sign bit is not set (it is 'positive'), NAN is greater than any number;
  - NANs of the same sign are ordered by the payload ('negative' ones - in reverse order).
  Returns:
  negative value if `x` precedes `y` in the order specified above; 0 if `x` and `y` are identical, and positive value otherwise.
  See Also:
  isIdentical
  Standards:
  Conforms to IEEE 754-2008
  Examples:
  Most numbers are ordered naturally.

      assert(cmp(-double.infinity, -double.max) < 0);
      assert(cmp(-double.max, -100.0) < 0);
      assert(cmp(-100.0, -0.5) < 0);
      assert(cmp(-0.5, 0.0) < 0);
      assert(cmp(0.0, 0.5) < 0);
      assert(cmp(0.5, 100.0) < 0);
      assert(cmp(100.0, double.max) < 0);
      assert(cmp(double.max, double.infinity) < 0);

      writeln(cmp(1.0, 1.0)); // 0

  Examples:
  Positive and negative zeroes are distinct.

      assert(cmp(-0.0, +0.0) < 0);
      assert(cmp(+0.0, -0.0) > 0);

  Examples:
  Depending on the sign, NANs go to either end of the spectrum.

      assert(cmp(-double.nan, -double.infinity) < 0);
      assert(cmp(double.infinity, double.nan) < 0);
      assert(cmp(-double.nan, double.nan) < 0);

  Examples:
  NANs of the same sign are ordered by the payload.

      assert(cmp(NaN(10), NaN(20)) < 0);
      assert(cmp(-NaN(20), -NaN(10)) < 0);

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

pure nothrow @nogc @safe bool `isPowerOf2`(X)(const X `x`)
if (isNumeric!X);
  Check whether a number is an integer power of two.
  Note that only positive numbers can be integer powers of two. This function always return false if `x` is negative or zero.
  Parameters:

  X `x` | the number to test

  Returns:
  true if `x` is an integer power of two.
  Examples:

      assert( isPowerOf2(1.0L));
      assert( isPowerOf2(2.0L));
      assert( isPowerOf2(0.5L));
      assert( isPowerOf2(pow(2.0L, 96)));
      assert( isPowerOf2(pow(2.0L, -77)));

      assert(!isPowerOf2(-2.0L));
      assert(!isPowerOf2(-0.5L));
      assert(!isPowerOf2(0.0L));
      assert(!isPowerOf2(4.315));
      assert(!isPowerOf2(1.0L / 3.0L));

      assert(!isPowerOf2(real.nan));
      assert(!isPowerOf2(real.infinity));

  Examples:

      assert( isPowerOf2(1));
      assert( isPowerOf2(2));
      assert( isPowerOf2(1uL << 63));

      assert(!isPowerOf2(-4));
      assert(!isPowerOf2(0));
      assert(!isPowerOf2(1337u));

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