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Last update Wed Aug 1 09:43:26 2012

# std.math

Elementary mathematical functions

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

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.

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.

Source:
std/math.d

License:
Boost License 1.0.

Authors:
Walter Bright, Don Clugston

Source:
std/math.d

real E;
  e = 2.718281...

real LOG2T;
  log210 = 3.321928...

real LOG2E;
  log2e = 1.442695...

real LOG2;
  log102 = 0.301029...

real LOG10E;
  log10e = 0.434294...

real LN2;
  ln 2 = 0.693147...

real LN10;
  ln 10 = 2.302585...

real PI;
  π = 3.141592...

real PI_2;
  π / 2 = 1.570796...

real PI_4;
  π / 4 = 0.785398...

real M_1_PI;
  1 / π = 0.318309...

real M_2_PI;
  2 / π = 0.636619...

real M_2_SQRTPI;
  2 / √π = 1.128379...

real SQRT2;
  √2 = 1.414213...

real SQRT1_2;
  √½ = 0.707106...

pure nothrow @safe Num abs(Num)(Num x);
pure nothrow @safe real abs(Num)(Num y);
  Calculates the absolute value

  For complex numbers, abs(z) = sqrt( z.re2 + z.im2 ) = hypot(z.re, z.im).

pure nothrow @safe creal conj(creal z);
pure nothrow @safe ireal conj(ireal y);
  Complex conjugate

  conj(x + iy) = x - iy

  Note that z * conj(z) = z.re2 - z.im2 is always a real number

pure nothrow @safe real cos(real 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 @safe real sin(real 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

  BUGS:
  Results are undefined if |x| >= 264.

pure nothrow @safe creal sin(creal z);
pure nothrow @safe ireal sin(ireal y);
  sine, complex and imaginary

  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).

pure nothrow @safe creal cos(creal z);
pure nothrow @safe real cos(ireal y);
  cosine, complex and imaginary

  cos(z) = cos(z.re)*cosh(z.im) - sin(z.re)*sinh(z.im)i

pure nothrow @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 @safe real acos(real x);
pure nothrow @safe double acos(double x);
pure nothrow @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 @safe real asin(real x);
pure nothrow @safe double asin(double x);
pure nothrow @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 @safe real atan(real x);
pure nothrow @safe double atan(double x);
pure nothrow @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 @trusted real atan2(real y, real x);
pure nothrow @safe double atan2(double y, double x);
pure nothrow @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 @safe real cosh(real x);
pure nothrow @safe double cosh(double x);
pure nothrow @safe float cosh(float x);
  Calculates the hyperbolic cosine of x.

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

pure nothrow @safe real sinh(real x);
pure nothrow @safe double sinh(double x);
pure nothrow @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 @safe real tanh(real x);
pure nothrow @safe double tanh(double x);
pure nothrow @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 @safe real acosh(real x);
pure nothrow @safe double acosh(double x);
pure nothrow @safe float acosh(float x);
  Calculates the inverse hyperbolic cosine of x.

  Mathematically, acosh(x) = log(x + sqrt( x*x - 1))

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

pure nothrow @safe real asinh(real x);
pure nothrow @safe double asinh(double x);
pure nothrow @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 @safe real atanh(real x);
pure nothrow @safe double atanh(double x);
pure nothrow @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

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

pure nothrow @safe long rndtol(real 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 @safe float sqrt(float x);
pure nothrow @safe double sqrt(double x);
pure nothrow @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 @trusted real exp(real x);
pure nothrow @safe double exp(double x);
pure nothrow @safe float exp(float x);
  Calculates ex.

  Special Values
  x | ex
  --- | ---
  +∞ | +∞
  -∞ | +0.0
  NAN | NAN

pure nothrow @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 | ex-1
  --- | ---
  ±0.0 | ±0.0
  +∞ | +∞
  -∞ | -1.0
  NAN | NAN

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

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

pure nothrow @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.

pure nothrow @trusted real frexp(real value, out int exp);
  Separate floating point value into significand and exponent.

  Returns:
  Calculate and return x and exp such that value =x*2exp 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

nothrow @trusted int ilogb(real x);
  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 @safe real ldexp(real n, int exp);
  Compute n * 2exp

  References:
  frexp

pure nothrow @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

pure nothrow @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

pure nothrow @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 @safe real log2(real x);
  Calculates the base-2 logarithm of x: log2x

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

nothrow @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 @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 @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 @trusted real scalbn(real x, int n);
  Efficiently calculates x * 2n.

  scalbn handles underflow and overflow in the same fashion as the basic arithmetic operators.

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

nothrow @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 @safe real fabs(real x);
  Returns |x|

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

pure nothrow @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(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

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

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

nothrow @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 @safe real rint(real 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 @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.

nothrow @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 to the even integer.

nothrow @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.

nothrow @trusted real trunc(real x);
  Returns the integer portion of x, dropping the fractional portion.

  This is also known as "chop" rounding.

nothrow @trusted real remainder(real x, real y);
nothrow @trusted real remquo(real x, real y, out int n);
  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

  Note:
  remquo 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.

  Example:

          real a=3.5;
          // Set all the flags to zero
          resetIeeeFlags();
          assert(!ieeeFlags.divByZero);
          // Perform a division by zero.
          a/=0.0L;
          assert(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();
          assert(ieeeFlags == f);

  @property bool inexact();
    The result cannot be represented exactly, so rounding occured. (example: x = sin(0.1); )

  @property bool underflow();
    A zero was generated by underflow (example: x = real.min*real.epsilon/2;)

  @property bool overflow();
    An infinity was generated by overflow (example: x = real.max*2;)

  @property bool divByZero();
    An infinity was generated by division by zero (example: x = 3/0.0; )

  @property bool invalid();
    A machine NaN was generated. (example: x = real.infinity * 0.0; )

void resetIeeeFlags();
  Set all of the floating-point status flags to false.

@property IeeeFlags ieeeFlags();
  Return a 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:

        {
          // Enable hardware exceptions for division by zero, overflow to infinity,
          // invalid operations, and uninitialized floating-point variables.

          FloatingPointControl fpctrl;
          fpctrl.enableExceptions(FloatingPointControl.severeExceptions);

          double y = x*3.0; // will generate a hardware exception, if x is uninitialized.
          //
          fpctrl.rounding = FloatingPointControl.roundUp;

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

  severeExceptions
    Severe = The overflow, division by zero, and invalid exceptions.

  void enableExceptions(uint exceptions);
    Enable (unmask) specific hardware exceptions. Multiple exceptions may be ORed together.

  void disableExceptions(uint exceptions);
    Disable (mask) specific hardware exceptions. Multiple exceptions may be ORed together.

  @property void rounding(RoundingMode newMode);
    Change the floating-point hardware rounding mode

  static @property uint enabledExceptions();
    Return the exceptions which are currently enabled (unmasked)

  static @property RoundingMode rounding();
    Return the currently active rounding mode

pure nothrow @trusted bool isNaN(real x);
  Returns !=0 if e is a NaN.

pure nothrow @trusted int isFinite(real e);
  Returns !=0 if e is finite (not infinite or NAN).

pure nothrow @trusted int isNormal(X)(X x);
  Returns !=0 if x is normalized (not zero, subnormal, infinite, or NAN).

pure nothrow @trusted int isSubnormal(float f);
pure nothrow @trusted int isSubnormal(double d);
pure nothrow @trusted int isSubnormal(real x);
  Is number subnormal? (Also called "denormal".) Subnormals have a 0 exponent and a 0 most significant mantissa bit.

pure nothrow @trusted bool isInfinity(real x);
  Return !=0 if e is ±∞.

pure nothrow @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 @trusted int signbit(real x);
  Return 1 if sign bit of e is set, 0 if not.

pure nothrow @trusted real copysign(real to, real from);
  Return a value composed of to with from's sign bit.

pure nothrow @safe F sgn(F)(F x);
  Returns -1 if x < 0, x if x == 0, 1 if x > 0, and NAN if x==NAN.

pure nothrow @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 @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 @trusted real nextUp(real x);
pure nothrow @trusted double nextUp(double x);
pure nothrow @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 @safe real nextDown(real x);
pure nothrow @safe double nextDown(double x);
pure nothrow @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

pure nothrow @safe T nextafter(T)(T x, T y);
  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.

pure nothrow @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 @safe real fmax(real x, real y);
  Returns the larger of x and y.

pure nothrow @safe real fmin(real x, real y);
  Returns the smaller of x and y.

pure nothrow @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 @trusted Unqual!(F) pow(F, G)(F x, G n);
  Compute the value of x n, where n is an integer

pure nothrow @trusted typeof(Unqual!(F).init * Unqual!(G).init) pow(F, G)(F x, G n);
  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.

pure nothrow @trusted real pow(I, F)(I x, F y);
  Computes integer to floating point powers.

pure nothrow @trusted Unqual!(Largest!(F,G)) pow(F, G)(F x, G y);
  Calculates xy.

  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

pure nothrow @trusted int feqrel(X)(X x, X y);
  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 @trusted real poly(real x, const real[] A);
  Evaluate polynomial A(x) = a0 + a1x + a2x2 + a3x3; ...

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

  Parameters:

  real[] A | array of coefficients a0, a1, etc.

bool approxEqual(T, U, V)(T lhs, U rhs, V maxRelDiff, V maxAbsDiff = 1e-05);
  Computes whether lhs is approximately equal to rhs admitting a maximum relative difference maxRelDiff and a maximum absolute difference maxAbsDiff.

  If the two inputs are ranges, 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).

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