std.math

const real E;

e

const real LOG2T;

log₂10

const real LOG2E;

log₂e

const real LOG2;

log₁₀2

const real LOG10E;

log₁₀e

const real LN2;

ln 2

const real LN10;

ln 10

const real PI;

π

const real PI_2;

π / 2

const real PI_4;

π / 4

const real M_1_PI;

1 / π

const real M_2_PI;

2 / π

const real M_2_SQRTPI;

2 / √π

const real SQRT2;

√2

const real SQRT1_2;

√½

real cos(real x);

Returns cosine of x. x is in radians.

Special Values

x	cos( x)	invalid?
NAN	NAN	yes
±∞	NAN	yes

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

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

real acos(real x);

Calculates the arc cosine of x, returning a value ranging from -π/2 to π/2.

Special Values

x	acos( x)	invalid?
>1.0	NAN	yes
<-1.0	NAN	yes
NAN	NAN	yes

real asin(real 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

real atan(real 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

real atan2(real x,real y);

Calculates the arc tangent of y / x, returning a value ranging from -π/2 to π/2.

Special Values

x	y	atan( x, y)
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

real cosh(real x);

Calculates the hyperbolic cosine of x.

Special Values

x	cosh( x)	invalid?
±∞	±0.0	no

real sinh(real x);

Calculates the hyperbolic sine of x.

Special Values

x	sinh( x)	invalid?
±0.0	±0.0	no
±∞	±∞	no

real tanh(real x);

Calculates the hyperbolic tangent of x.

Special Values

x	tanh( x)	invalid?
±0.0	±0.0	no
±∞	±1.0	no

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.

realC 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.

float sqrt(float x);
double sqrt(double x);
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

real exp(real x);

Calculates e ^x.

Special Values

x	exp( x)
+∞	+∞
-∞	+0.0

real exp2(real x);

Calculates 2 ^x.

Special Values

x	exp2( x)
+∞	+∞
-∞	+0.0

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

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*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

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
±∞	+∞	no
NAN	FP_ILOGBNAN	no

real ldexp(real n,int exp);

Compute n * 2 ^exp

References:

frexp

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

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

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

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

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

real modf(real x,inout 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	modf( x, y)	invalid?
±0.0	not 0.0	±0.0	no
±∞	anything	NAN	yes
anything	±0.0	NAN	yes
!=±∞	±∞	x	no

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

real cbrt(real x);

Calculates the cube root x.

Special Values

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

real fabs(real x);

Returns | x|

Special Values

x	fabs( x)
±0.0	+0.0
±∞	+∞

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

real erf(real x);

Returns the error function of x.

real erfc(real x);

Returns the complementary error function of x, which is 1 - erf( x).

real lgamma(real x);

Calculates ln |Γ( x)|

real tgamma(real x);

Calculates the gamma function Γ( x)

real ceil(real x);

Returns the value of x rounded upward to the next integer (toward positive infinity).

real floor(real x);

Returns the value of x rounded downward to the next integer (toward negative infinity).

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.

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.

long lrint(real x);

Rounds x to the nearest integer value, using the current rounding mode.

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.

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.

real trunc(real x);

Returns the integer portion of x, dropping the fractional portion.

This is also know as "chop" rounding.

real remainder(real x,real y);
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 affect 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

int isnan(real e);

Returns !=0 if e is a NaN.

int isfinite(real e);

Returns !=0 if e is finite.

int isnormal(float x);
int isnormal(double d);
int isnormal(real e);

Returns !=0 if x is normalized.

int issubnormal(float f);
int issubnormal(double d);
int issubnormal(real e);

Is number subnormal? (Also called "denormal".) Subnormals have a 0 exponent and a 0 most significant mantissa bit.

int isinf(real e);

Return !=0 if e is ±∞.

int signbit(real e);

Return 1 if sign bit of e is set, 0 if not.

real copysign(real to,real from);

Return a value composed of to with from's sign bit.

real nan(char[] tagp);

Creates a quiet NAN with the information from tagp[] embedded in it.

real nextafter(real x,real 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. 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.

real fdim(real x,real y);

Returns the positive difference between x and y.

Returns:

x, y	fdim( x, y)
x > y	x - y
x <= y	+0.0

real fmax(real x,real y);

Returns the larger of x and y.

real fmin(real x,real y);

Returns the smaller of x and y.

real fma(real x,real y,real z);

Returns ( x * y) + z, rounding only once according to the current rounding mode.

real pow(real x,uint n);
real pow(real x,int n);

Fast integral powers.

real pow(real x,real y);

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

int feqrel(real x,real 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