Package com.github.tommyettinger.digital
Class TrigTools
java.lang.Object
com.github.tommyettinger.digital.TrigTools
Various trigonometric approximations, using a lookup table for sin() and cos(), a non-tabular approximation for
sinSmooth() and cosSmooth(), a Padé approximant for tan(), and Taylor series for the inverses of sin(), cos(), and
tan(). This supplies variants for radians, degrees, and turns. This also has an atan2() approximation defined with
output in radians, degrees, and turns. The lookup-table-based sin() and cos() can be extraordinarily fast if the 64KB
table can stay in a processor cache, while the "smooth" approximations may have higher quality but perform less
quickly compared to an in-cache lookup table.
This is primarily derived from libGDX's MathUtils class. The main new functionalities are the variants that take or return measurements in turns, the now-available
MathUtils had its sin and cos methods created by Riven on JavaGaming.org . The asin(), acos(), and atan() methods all use Taylor series approximations from the 1955 research study "Approximations for Digital Computers," by RAND Corporation; though one might think such code would be obsolete over 60 years later, the approximations from that study seem to have higher accuracy and speed than most attempts in later decades, often those aimed at DSP usage. Even older is the basis for sinSmooth() and cosSmooth(); the versions here are updated to be more precise, but are closely related to a 7th-century sine approximation by Bhaskara I. The update was given in this Stack Exchange answer by WimC. Also from Stack Exchange, this Stack Exchange answer by Soonts provided the tan() approximation used here.
This is primarily derived from libGDX's MathUtils class. The main new functionalities are the variants that take or return measurements in turns, the now-available
SIN_TABLE
and SIN_TABLE_D
, and double variants in
general. Using the sin table directly has other uses mentioned in its docs (in particular, uniform random unit
vectors). Because using a lookup table for sine and cosine has very small "jumps" between what it returns for
smoothly increasing inputs, it may be unsuitable for some usage, such as calculating tan() or some statistical code.
TrigTools provides sinSmooth(), cosSmooth(), and degree/turn variants of those for when the precision should be high
but it is most important to have a smoothly-curving graph of returns. A different smooth approximation is used for
tan().
MathUtils had its sin and cos methods created by Riven on JavaGaming.org . The asin(), acos(), and atan() methods all use Taylor series approximations from the 1955 research study "Approximations for Digital Computers," by RAND Corporation; though one might think such code would be obsolete over 60 years later, the approximations from that study seem to have higher accuracy and speed than most attempts in later decades, often those aimed at DSP usage. Even older is the basis for sinSmooth() and cosSmooth(); the versions here are updated to be more precise, but are closely related to a 7th-century sine approximation by Bhaskara I. The update was given in this Stack Exchange answer by WimC. Also from Stack Exchange, this Stack Exchange answer by Soonts provided the tan() approximation used here.
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Field Summary
FieldsModifier and TypeFieldDescriptionstatic final float
Multiply by this to convert from degrees to radians.static final double
Multiply by this to convert from degrees to radians.static final float
static final double
static final float
static final double
static final float
Thefloat
value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.static final double
Thedouble
value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.static final float
1.0f divided byPI
.static final double
1.0 divided byPI
.static final float
static final double
static final float
PI
divided by 4f.static final double
Math.PI
divided by 4.0.static final float
Multiply by this to convert from radians to degrees.static final double
Multiply by this to convert from radians to degrees.static final float[]
A precalculated table of 16384 floats, corresponding to the y-value of points on the unit circle, ordered by increasing angle.static final double[]
A precalculated table of 16384 doubles, corresponding to the y-value of points on the unit circle, ordered by increasing angle.static final int
If you add this to an index used inSIN_TABLE
, you get the result of the cosine instead of the sine.static final int
The bitmask that can be used to confine any int to wrap withinTABLE_SIZE
.static final int
The size ofSIN_TABLE
, available separately from the table's length for convenience.static final float
static final double
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Method Summary
Modifier and TypeMethodDescriptionstatic double
acos
(double a) Returns arccosine in radians; less accurate than Math.acos but may be faster.static float
acos
(float a) Returns arccosine in radians; less accurate than Math.acos but may be faster.static double
acosDeg
(double a) Returns arccosine in degrees.static float
acosDeg
(float a) Returns arccosine in degrees.static double
acosTurns
(double a) Returns arccosine in turns.static float
acosTurns
(float a) Returns arccosine in turns.static double
asin
(double a) Returns arcsine in radians; less accurate than Math.asin but may be faster.static float
asin
(float a) Returns arcsine in radians; less accurate than Math.asin but may be faster.static double
asinDeg
(double a) Returns arcsine in degrees.static float
asinDeg
(float a) Returns arcsine in degrees.static double
asinTurns
(double a) Returns arcsine in turns.static float
asinTurns
(float a) Returns arcsine in turns.static double
atan
(double i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static float
atan
(float i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static double
atan2
(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using radians.static float
atan2
(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using radians.static double
atan2Deg
(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees.static float
atan2Deg
(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees.static double
atan2Deg360
(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only.static float
atan2Deg360
(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only.static double
atan2Turns
(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only.static float
atan2Turns
(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only.static double
atanDeg
(double i) Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static float
atanDeg
(float i) Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static double
atanTurns
(double i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static float
atanTurns
(float i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).static double
atanUnchecked
(double i) A variant onatan(float)
that does not tolerate infinite inputs for speed reasons.static double
atanUncheckedDeg
(double i) A variant onatanDeg(float)
that does not tolerate infinite inputs for speed reasons.static double
atanUncheckedTurns
(double i) A variant onatanTurns(float)
that does not tolerate infinite inputs for speed reasons.static double
cos
(double radians) Returns the cosine in radians from a lookup table.static float
cos
(float radians) Returns the cosine in radians from a lookup table.static double
cosDeg
(double degrees) Returns the cosine in degrees from a lookup table.static float
cosDeg
(float degrees) Returns the cosine in degrees from a lookup table.static double
cosSmooth
(double radians) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
cosSmooth
(float radians) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
cosSmoothDeg
(double degrees) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
cosSmoothDeg
(float degrees) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
cosSmoothTurns
(double turns) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
cosSmoothTurns
(float turns) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
cosTurns
(double turns) Returns the cosine in turns from a lookup table.static float
cosTurns
(float turns) Returns the cosine in turns from a lookup table.static double
sin
(double radians) Returns the sine in radians from a lookup table.static float
sin
(float radians) Returns the sine in radians from a lookup table.static double
sinDeg
(double degrees) Returns the sine in degrees from a lookup table.static float
sinDeg
(float degrees) Returns the sine in degrees from a lookup table.static double
sinSmooth
(double radians) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
sinSmooth
(float radians) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
sinSmoothDeg
(double degrees) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
sinSmoothDeg
(float degrees) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
sinSmoothTurns
(double turns) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static float
sinSmoothTurns
(float turns) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century.static double
sinTurns
(double turns) Returns the sine in turns from a lookup table.static float
sinTurns
(float turns) Returns the sine in turns from a lookup table.static double
tan
(double radians) Returns the tangent in radians, using a Padé approximant.static float
tan
(float radians) Returns the tangent in radians, using a Padé approximant.static double
tanDeg
(double degrees) Returns the tangent in degrees, using a Padé approximant.static float
tanDeg
(float degrees) Returns the tangent in degrees, using a Padé approximant.static double
tanTurns
(double turns) Returns the tangent in turns, using a Padé approximant.static float
tanTurns
(float turns) Returns the tangent in turns, using a Padé approximant.
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Field Details
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PI
public static final float PIThefloat
value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.- See Also:
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PI_D
public static final double PI_DThedouble
value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.- See Also:
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PI_INVERSE
public static final float PI_INVERSE1.0f divided byPI
.- See Also:
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PI2
public static final float PI2- See Also:
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TAU
public static final float TAU- See Also:
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HALF_PI
public static final float HALF_PI- See Also:
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ETA
public static final float ETA- See Also:
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PI_INVERSE_D
public static final double PI_INVERSE_D1.0 divided byPI
.- See Also:
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PI2_D
public static final double PI2_D- See Also:
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TAU_D
public static final double TAU_D- See Also:
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HALF_PI_D
public static final double HALF_PI_D- See Also:
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ETA_D
public static final double ETA_D- See Also:
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QUARTER_PI
public static final float QUARTER_PIPI
divided by 4f.- See Also:
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QUARTER_PI_D
public static final double QUARTER_PI_DMath.PI
divided by 4.0.- See Also:
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TABLE_SIZE
public static final int TABLE_SIZEThe size ofSIN_TABLE
, available separately from the table's length for convenience.- See Also:
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SIN_TO_COS
public static final int SIN_TO_COSIf you add this to an index used inSIN_TABLE
, you get the result of the cosine instead of the sine.- See Also:
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TABLE_MASK
public static final int TABLE_MASKThe bitmask that can be used to confine any int to wrap withinTABLE_SIZE
. Any accesses toSIN_TABLE
with an index that could be out of bounds should probably be wrapped using this, as withSIN_TABLE[index & TABLE_MASK]
.- See Also:
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radiansToDegrees
public static final float radiansToDegreesMultiply by this to convert from radians to degrees.- See Also:
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degreesToRadians
public static final float degreesToRadiansMultiply by this to convert from degrees to radians.- See Also:
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radiansToDegreesD
public static final double radiansToDegreesDMultiply by this to convert from radians to degrees.- See Also:
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degreesToRadiansD
public static final double degreesToRadiansDMultiply by this to convert from degrees to radians.- See Also:
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SIN_TABLE
public static final float[] SIN_TABLEA precalculated table of 16384 floats, corresponding to the y-value of points on the unit circle, ordered by increasing angle. This should not be mutated, but it can be accessed directly for things like getting random unit vectors, or implementing the "sincos" method (which assigns sin() to one item and cos() to another).
A quick way to get a random unit vector is to get a random 14-bit number, as withint angle = random.nextInt() >>> 18;
, look up angle in this table to get y, then look up(angle + TrigTools.SIN_TO_COS) & TrigTools.TABLE_MASK
(or(angle + 4096) & 16383
) to get x. -
SIN_TABLE_D
public static final double[] SIN_TABLE_DA precalculated table of 16384 doubles, corresponding to the y-value of points on the unit circle, ordered by increasing angle. This should not be mutated, but it can be accessed directly for things like getting random unit vectors, or implementing the "sincos" method (which assigns sin() to one item and cos() to another).
A quick way to get a random unit vector is to get a random 14-bit number, as withint angle = random.nextInt() >>> 18;
, look up angle in this table to get y, then look up(angle + TrigTools.SIN_TO_COS) & TrigTools.TABLE_MASK
(or(angle + 4096) & 16383
) to get x.
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Method Details
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sin
public static float sin(float radians) Returns the sine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).- Parameters:
radians
- an angle in radians, where 0 toPI2
is one rotation
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cos
public static float cos(float radians) Returns the cosine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).- Parameters:
radians
- an angle in radians, where 0 toPI2
is one rotation
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tan
public static float tan(float radians) Returns the tangent in radians, using a Padé approximant. Padé approximants tend to be most accurate when they aren't producing results of extreme magnitude; in the tan() function, those results occur on and near odd multiples ofPI/2
, and this method is least accurate when given inputs near those multiples.
For inputs between -1.57 to 1.57 (just inside half-pi), separated by 0x1p-20f, absolute error is 0.00890192, relative error is 0.00000090, and the maximum error is 17.98901367 when given 1.56999838. The maximum error might seem concerning, but it's the difference between the correct 1253.22167969 and the 1235.23266602 this returns, so for many purposes the difference won't be noticeable.
For inputs between -1.55 to 1.55 (getting less close to half-pi), separated by 0x1p-20f, absolute error is 0.00023368, relative error is -0.00000009, and the maximum error is 0.02355957 when given -1.54996467. The maximum error is the difference between the correct -47.99691010 and the -47.97335052 this returns.
While you don't have to use a dedicated method for tan(), and you can usesin(x)/cos(x)
, approximating tan() in this way is very susceptible to error building up from any of sin(), cos() or the division. Where this tan() has a maximum error in the -1.55 to 1.55 range of 0.02355957, the simpler division technique on the same range has a maximum error of 1.25724030 (about 50 times worse), as well as larger absolute and relative errors. Casting the double result ofMath.tan(double)
to float will get the highest precision, but can be anywhere from 2.5x to nearly 4x slower than this, depending on JVM.
Based on this Stack Exchange answer by Soonts.- Parameters:
radians
- a float angle in radians, where 0 toPI2
is one rotation- Returns:
- a float approximation of tan()
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sinDeg
public static float sinDeg(float degrees) Returns the sine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation
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cosDeg
public static float cosDeg(float degrees) Returns the cosine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation
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tanDeg
public static float tanDeg(float degrees) Returns the tangent in degrees, using a Padé approximant. Based on this Stack Exchange answer.- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation- Returns:
- a float approximation of tan()
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sinTurns
public static float sinTurns(float turns) Returns the sine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation
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cosTurns
public static float cosTurns(float turns) Returns the cosine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation
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tanTurns
public static float tanTurns(float turns) Returns the tangent in turns, using a Padé approximant. Based on this Stack Exchange answer.- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation- Returns:
- a float approximation of tan()
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sin
public static double sin(double radians) Returns the sine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).- Parameters:
radians
- an angle in radians, where 0 toPI2_D
is one rotation
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cos
public static double cos(double radians) Returns the cosine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).- Parameters:
radians
- an angle in radians, where 0 toPI2_D
is one rotation
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tan
public static double tan(double radians) Returns the tangent in radians, using a Padé approximant. Based on this Stack Exchange answer.- Parameters:
radians
- a double angle in radians, where 0 toPI2
is one rotation- Returns:
- a double approximation of tan()
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sinDeg
public static double sinDeg(double degrees) Returns the sine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation
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cosDeg
public static double cosDeg(double degrees) Returns the cosine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation
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tanDeg
public static double tanDeg(double degrees) Returns the tangent in degrees, using a Padé approximant. Based on this Stack Exchange answer.- Parameters:
degrees
- an angle in degrees, where 0 to 360 is one rotation- Returns:
- a double approximation of tan()
-
sinTurns
public static double sinTurns(double turns) Returns the sine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation
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cosTurns
public static double cosTurns(double turns) Returns the cosine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation
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tanTurns
public static double tanTurns(double turns) Returns the tangent in turns, using a Padé approximant. Based on this Stack Exchange answer.- Parameters:
turns
- an angle in turns, where 0 to 1 is one rotation- Returns:
- a double approximation of tan()
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sinSmooth
public static float sinSmooth(float radians) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in radians, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sin(); in particular, only 16384 outputs are possible fromsin(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sin() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
radians
- an angle in radians; most precise between -PI2 and PI2- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmooth
public static float cosSmooth(float radians) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in radians, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cos(); in particular, only 16384 outputs are possible fromcos(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cos() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
radians
- an angle in radians; most precise between -PI2 and PI2- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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sinSmooth
public static double sinSmooth(double radians) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in radians, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sin(); in particular, only 16384 outputs are possible fromsin(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sin() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
radians
- an angle in radians; most precise between -PI2 and PI2- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmooth
public static double cosSmooth(double radians) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in radians, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cos(); in particular, only 16384 outputs are possible fromcos(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cos() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
radians
- an angle in radians; most precise between -PI2 and PI2- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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sinSmoothDeg
public static float sinSmoothDeg(float degrees) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in degrees, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sinDeg(); in particular, only 16384 outputs are possible fromsinDeg(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sinDeg() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
degrees
- an angle in degrees; most precise between -360 and 360- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmoothDeg
public static float cosSmoothDeg(float degrees) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in degrees, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cosDeg(); in particular, only 16384 outputs are possible fromcosDeg(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cosDeg() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
degrees
- an angle in degrees; most precise between -360 and 360- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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sinSmoothDeg
public static double sinSmoothDeg(double degrees) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in degrees, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sinDeg(); in particular, only 16384 outputs are possible fromsinDeg(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sinDeg() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
degrees
- an angle in degrees; most precise between -360 and 360- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmoothDeg
public static double cosSmoothDeg(double degrees) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in degrees, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cosDeg(); in particular, only 16384 outputs are possible fromcosDeg(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cosDeg() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
degrees
- an angle in degrees; most precise between -360 and 360- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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sinSmoothTurns
public static float sinSmoothTurns(float turns) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in turns, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sinTurns(); in particular, only 16384 outputs are possible fromsinTurns(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sinTurns() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
turns
- an angle in turns; most precise between -1 and 1- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmoothTurns
public static float cosSmoothTurns(float turns) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in turns, and takes and returns floats. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cosTurns(); in particular, only 16384 outputs are possible fromcosTurns(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cosTurns() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
turns
- an angle in turns; most precise between -1 and 1- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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sinSmoothTurns
public static double sinSmoothTurns(double turns) A smooth sine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in turns, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of sinTurns(); in particular, only 16384 outputs are possible fromsinTurns(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible sinTurns() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
turns
- an angle in turns; most precise between -1 and 1- Returns:
- the approximate sine of the given angle, from -1 to 1 inclusive
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cosSmoothTurns
public static double cosSmoothTurns(double turns) A smooth cosine approximation (not table-based) built around Bhaskara I's sine approximation from the 7th century. This takes an input in turns, and takes and returns doubles. This was updated more recently than the 7th century, and has better precision than the original. You may want to use this if you notice statistical issues with the tabular approximation of cosTurns(); in particular, only 16384 outputs are possible fromcosTurns(float)
, and about half of those are duplicates, so if you need more possible results in-between the roughly 8192 possible cosTurns() returns, you can use this.
Credit to This Stack Exchange answer by WimC.- Parameters:
turns
- an angle in turns; most precise between -1 and 1- Returns:
- the approximate cosine of the given angle, from -1 to 1 inclusive
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atanUnchecked
public static double atanUnchecked(double i) A variant onatan(float)
that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atan(float), and returns a float like that method. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used insideatan2(float, float)
, but it may be a tiny bit faster than atan(float) in other code.- Parameters:
i
- any finite double or float, but more commonly a float- Returns:
- an output from the inverse tangent function in radians, from
-HALF_PI
toHALF_PI
inclusive
-
atanUncheckedTurns
public static double atanUncheckedTurns(double i) A variant onatanTurns(float)
that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atanTurns(float), but returns a double in case external code needs higher precision. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used insideatan2Turns(float, float)
, but it may be a tiny bit faster than atanTurns(float) in other code.- Parameters:
i
- any finite double or float, but more commonly a float- Returns:
- an output from the inverse tangent function in turns, from
-0.25
to0.25
inclusive
-
atanUncheckedDeg
public static double atanUncheckedDeg(double i) A variant onatanDeg(float)
that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atanDeg(float), and returns a float like that method. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used insideatan2(float, float)
, but it may be a tiny bit faster than atanDeg(float) in other code.- Parameters:
i
- any finite double or float, but more commonly a float- Returns:
- an output from the inverse tangent function in degrees, from
-90
to90
inclusive
-
atan2
public static float atan2(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using radians. Average error is 1.057E-6 radians; maximum error is 1.922E-6. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in radians. It is about 4 times faster thanMath.atan2(double, double)
(roughly 15 ns instead of roughly 60 ns for Math, on Java 8 HotSpot).
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(float)
method, and that cleanly translates to atan2().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in radians as a float; ranges from
-PI
toPI
-
atan2Deg
public static float atan2Deg(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees. Average absolute error is 0.00006037 degrees; relative error is 0 degrees, maximum error is 0.00010396 degrees. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in degrees.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(float)
method, and that cleanly translates to atan2Deg().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in degrees as a float; ranges from
-180
to180
-
atan2Deg360
public static float atan2Deg360(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only. Average absolute error is 0.00006045 degrees; relative error is 0 degrees; maximum error is 0.00011178 degrees. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in degrees.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(float)
method, and that cleanly translates to atan2Deg360().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in degrees as a float; ranges from
0
to360
-
atan2Turns
public static float atan2Turns(float y, float x) Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only. Average absolute error is 0.00000030 turns; relative error is 0 turns; maximum error is 0.00000017 turns. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in turns. Because this always returns a float between 0.0 (inclusive) and 1.0 (exclusive), it can be useful for various kinds of calculations that must store angles as a small fraction, such as packing a hue angle into a byte.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(float)
method, and that cleanly translates to atan2Turns().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in turns as a float; ranges from
0.0f
to1.0f
-
atan2
public static double atan2(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using radians. Average error is 1.057E-6 radians; maximum error is 1.922E-6. Takes y and x (in that unusual order) as doubles, and returns the angle from the origin to that point in radians. It is about 4 times faster thanMath.atan2(double, double)
(roughly 15 ns instead of roughly 60 ns for Math, on Java 8 HotSpot).
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(double)
method, and that cleanly translates to atan2().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in radians as a double; ranges from
-PI
toPI
-
atan2Deg
public static double atan2Deg(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees. Average absolute error is 0.00006037 degrees; relative error is 0 degrees, maximum error is 0.00010396 degrees. Takes y and x (in that unusual order) as doubles, and returns the angle from the origin to that point in degrees.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(double)
method, and that cleanly translates to atan2Deg().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in degrees as a double; ranges from
-180
to180
-
atan2Deg360
public static double atan2Deg360(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only. Average absolute error is 0.00006045 degrees; relative error is 0 degrees; maximum error is 0.00011178 degrees. Takes y and x (in that unusual order) as doubles, and returns the angle from the origin to that point in degrees.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(double)
method, and that cleanly translates to atan2Deg360().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in degrees as a double; ranges from
0
to360
-
atan2Turns
public static double atan2Turns(double y, double x) Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only. Average absolute error is 0.00000030 turns; relative error is 0 turns; maximum error is 0.00000017 turns. Takes y and x (in that unusual order) as doubles, and returns the angle from the origin to that point in turns. Because this always returns a double between 0.0 (inclusive) and 1.0 (exclusive), it can be useful for various kinds of calculations that must store angles as a small fraction, such as packing a hue angle into a byte.
Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides anatan(double)
method, and that cleanly translates to atan2Turns().- Parameters:
y
- y-component of the point to find the angle towards; note the parameter order is unusual by conventionx
- x-component of the point to find the angle towards; note the parameter order is unusual by convention- Returns:
- the angle to the given point, in turns as a double; ranges from
0.0
to1.0
-
asin
public static float asin(float a) Returns arcsine in radians; less accurate than Math.asin but may be faster. Average error of 0.000028447 radians (0.0016298931 degrees), largest error of 0.000067592 radians (0.0038727364 degrees). This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.- Parameters:
a
- asin is defined only when a is between -1f and 1f, inclusive- Returns:
- between
-HALF_PI
andHALF_PI
when a is in the defined range
-
asinDeg
public static float asinDeg(float a) Returns arcsine in degrees. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.- Parameters:
a
- asin is defined only when a is between -1f and 1f, inclusive- Returns:
- between
-90
and90
when a is in the defined range
-
asinTurns
public static float asinTurns(float a) Returns arcsine in turns. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN. Note that unlikeatan2Turns(float, float)
, this can return negative turn values.- Parameters:
a
- asin is defined only when a is between -1f and 1f, inclusive- Returns:
- between
-0.25
and0.25
when a is in the defined range
-
acos
public static float acos(float a) Returns arccosine in radians; less accurate than Math.acos but may be faster. Average error of 0.00002845 radians (0.0016300649 degrees), largest error of 0.000067548 radians (0.0038702153 degrees). This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1f and 1f, inclusive- Returns:
- between
0
andPI
when a is in the defined range
-
acosDeg
public static float acosDeg(float a) Returns arccosine in degrees. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1f and 1f, inclusive- Returns:
- between
0
and180
when a is in the defined range
-
acosTurns
public static float acosTurns(float a) Returns arccosine in turns. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1f and 1f, inclusive- Returns:
- between
0
and0.5
when a is in the defined range
-
asin
public static double asin(double a) Returns arcsine in radians; less accurate than Math.asin but may be faster. Average error of 0.000028447 radians (0.0016298931 degrees), largest error of 0.000067592 radians (0.0038727364 degrees). This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.- Parameters:
a
- asin is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
-HALF_PI
andHALF_PI
when a is in the defined range
-
asinDeg
public static double asinDeg(double a) Returns arcsine in degrees. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.- Parameters:
a
- asin is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
-90
and90
when a is in the defined range
-
asinTurns
public static double asinTurns(double a) Returns arcsine in turns. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN. Note that unlikeatan2Turns(double, double)
, this can return negative turn values.- Parameters:
a
- asin is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
-0.25
and0.25
when a is in the defined range
-
acos
public static double acos(double a) Returns arccosine in radians; less accurate than Math.acos but may be faster. Average error of 0.00002845 radians (0.0016300649 degrees), largest error of 0.000067548 radians (0.0038702153 degrees). This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
0
andPI
when a is in the defined range
-
acosDeg
public static double acosDeg(double a) Returns arccosine in degrees. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
0
and180
when a is in the defined range
-
acosTurns
public static double acosTurns(double a) Returns arccosine in turns. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.- Parameters:
a
- acos is defined only when a is between -1.0 and 1.0, inclusive- Returns:
- between
0
and0.5
when a is in the defined range
-
atan
public static float atan(float i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). This method is usually about 4x faster thanMath.atan(double)
, but is somewhat less precise than Math's implementation. For finite inputs only, you may get a tiny speedup by usingatanUnchecked(double)
, but this method will be correct enough for infinite inputs, and atanUnchecked() will not be.- Parameters:
i
- an input to the inverse tangent function; any float is accepted- Returns:
- an output from the inverse tangent function in radians, from
-HALF_PI
toHALF_PI
inclusive - See Also:
-
atanDeg
public static float atanDeg(float i) Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by usingatanUncheckedDeg(double)
, but this method will be correct enough for infinite inputs, and atanUncheckedDeg() will not be.- Parameters:
i
- an input to the inverse tangent function; any float is accepted- Returns:
- an output from the inverse tangent function in degrees, from
-90
to90
inclusive - See Also:
-
atanTurns
public static float atanTurns(float i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by usingatanUncheckedTurns(double)
, but this method will be correct enough for infinite inputs, and atanUncheckedTurns() will not be.- Parameters:
i
- an input to the inverse tangent function; any float is accepted- Returns:
- an output from the inverse tangent function in turns, from
-0.25
to0.25
inclusive - See Also:
-
atan
public static double atan(double i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). This method is usually about 4x faster thanMath.atan(double)
, but is somewhat less precise than Math's implementation. For finite inputs only, you may get a tiny speedup by usingatanUnchecked(double)
, but this method will be correct enough for infinite inputs, and atanUnchecked() will not be.- Parameters:
i
- an input to the inverse tangent function; any double is accepted- Returns:
- an output from the inverse tangent function in radians, from
-HALF_PI
toHALF_PI
inclusive - See Also:
-
atanDeg
public static double atanDeg(double i) Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by usingatanUncheckedDeg(double)
, but this method will be correct enough for infinite inputs, and atanUncheckedDeg() will not be.- Parameters:
i
- an input to the inverse tangent function; any double is accepted- Returns:
- an output from the inverse tangent function in degrees, from
-90
to90
inclusive - See Also:
-
atanTurns
public static double atanTurns(double i) Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by usingatanUncheckedTurns(double)
, but this method will be correct enough for infinite inputs, and atanUncheckedTurns() will not be.- Parameters:
i
- an input to the inverse tangent function; any double is accepted- Returns:
- an output from the inverse tangent function in turns, from
-0.25
to0.25
inclusive - See Also:
-