## Documentation ¶

### Overview ¶

Package hyperdual provides the hyperdual numeric type and functions. Hyperdual numbers are an extension of the real numbers in the form a+bϵ₁+bϵ₂+dϵ₁ϵ₂ where ϵ₁^2=0 and ϵ₂^2=0, but ϵ₁≠0, ϵ₂≠0 and ϵ₁ϵ₂≠0.

See https://doi.org/10.2514/6.2011-886 and http://adl.stanford.edu/hyperdual/ for details of their properties and uses.

### Constants ¶

This section is empty.

### Variables ¶

This section is empty.

### Functions ¶

This section is empty.

### Types ¶

#### type Number ¶

```type Number struct {
Real, E1mag, E2mag, E1E2mag float64
}```

Number is a float64 precision hyperdual number.

Example (Fike)
```Output:

v=(4.4978+4.0534ϵ₁+4.0534ϵ₂+9.4631ϵ₁ϵ₂)
fn(1.5)=4.4978
fn'(1.5)=4.0534
fn''(1.5)=9.4631
```

#### func Abs ¶

`func Abs(d Number) Number`

Abs returns the absolute value of d.

#### func Acos ¶

`func Acos(d Number) Number`

Acos returns the inverse cosine of d.

Special cases are:

```Acos(-1) = (Pi-Infϵ₁-Infϵ₂+Infϵ₁ϵ₂)
Acos(1) = (0-Infϵ₁-Infϵ₂-Infϵ₁ϵ₂)
Acos(x) = NaN if x < -1 or x > 1
```

#### func Acosh ¶

`func Acosh(d Number) Number`

Acosh returns the inverse hyperbolic cosine of d.

Special cases are:

```Acosh(+Inf) = +Inf
Acosh(1) = (0+Infϵ₁+Infϵ₂-Infϵ₁ϵ₂)
Acosh(x) = NaN if x < 1
Acosh(NaN) = NaN
```

`func Add(x, y Number) Number`

Add returns the sum of x and y.

#### func Asin ¶

`func Asin(d Number) Number`

Asin returns the inverse sine of d.

Special cases are:

```Asin(±0) = (±0+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
Asin(±1) = (±Inf+Infϵ₁+Infϵ₂±Infϵ₁ϵ₂)
Asin(x) = NaN if x < -1 or x > 1
```

#### func Asinh ¶

`func Asinh(d Number) Number`

Asinh returns the inverse hyperbolic sine of d.

Special cases are:

```Asinh(±0) = (±0+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Asinh(±Inf) = ±Inf
Asinh(NaN) = NaN
```

#### func Atan ¶

`func Atan(d Number) Number`

Atan returns the inverse tangent of d.

Special cases are:

```Atan(±0) = (±0+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Atan(±Inf) = (±Pi/2+0ϵ₁+0ϵ₂∓0ϵ₁ϵ₂)
```

#### func Atanh ¶

`func Atanh(d Number) Number`

Atanh returns the inverse hyperbolic tangent of d.

Special cases are:

```Atanh(1) = +Inf
Atanh(±0) = (±0+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
Atanh(-1) = -Inf
Atanh(x) = NaN if x < -1 or x > 1
Atanh(NaN) = NaN
```

#### func Cos ¶

`func Cos(d Number) Number`

Cos returns the cosine of d.

Special cases are:

```Cos(±Inf) = NaN
Cos(NaN) = NaN
```

#### func Cosh ¶

`func Cosh(d Number) Number`

Cosh returns the hyperbolic cosine of d.

Special cases are:

```Cosh(±0) = 1
Cosh(±Inf) = +Inf
Cosh(NaN) = NaN
```

#### func Exp ¶

`func Exp(d Number) Number`

Exp returns e**q, the base-e exponential of d.

Special cases are:

```Exp(+Inf) = +Inf
Exp(NaN) = NaN
```

Very large values overflow to 0 or +Inf. Very small values underflow to 1.

#### func Inv ¶

`func Inv(d Number) Number`

Inv returns the hyperdual inverse of d.

Special cases are:

```Inv(±Inf) = ±0-0ϵ₁-0ϵ₂±0ϵ₁ϵ₂
Inv(±0) = ±Inf-Infϵ₁-Infϵ₂±Infϵ₁ϵ₂
```

#### func Log ¶

`func Log(d Number) Number`

Log returns the natural logarithm of d.

Special cases are:

```Log(+Inf) = (+Inf+0ϵ₁+0ϵ₂-0ϵ₁ϵ₂)
Log(0) = (-Inf±Infϵ₁±Infϵ₂-Infϵ₁ϵ₂)
Log(x < 0) = NaN
Log(NaN) = NaN
```

#### func Mul ¶

`func Mul(x, y Number) Number`

Mul returns the hyperdual product of x and y.

#### func Pow ¶

`func Pow(d, p Number) Number`

Pow returns x**p, the base-x exponential of p.

#### func PowReal ¶

`func PowReal(d Number, p float64) Number`

PowReal returns x**p, the base-x exponential of p.

Special cases are (in order):

```PowReal(NaN+xϵ₁+yϵ₂, ±0) = 1+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for any x and y
PowReal(x, ±0) = 1 for any x
PowReal(1+xϵ₁+yϵ₂, z) = 1+xzϵ₁+yzϵ₂+2xyzϵ₁ϵ₂ for any z
PowReal(NaN+xϵ₁+yϵ₂, 1) = NaN+xϵ₁+yϵ₂+NaNϵ₁ϵ₂ for any x
PowReal(x, 1) = x for any x
PowReal(NaN+xϵ₁+xϵ₂, y) = NaN+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂
PowReal(x, NaN) = NaN+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂
PowReal(±0, y) = ±Inf for y an odd integer < 0
PowReal(±0, -Inf) = +Inf
PowReal(±0, +Inf) = +0
PowReal(±0, y) = +Inf for finite y < 0 and not an odd integer
PowReal(±0, y) = ±0 for y an odd integer > 0
PowReal(±0, y) = +0 for finite y > 0 and not an odd integer
PowReal(-1, ±Inf) = 1
PowReal(x+0ϵ₁+0ϵ₂, +Inf) = +Inf+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x+xϵ₁+yϵ₂, +Inf) = +Inf+Infϵ₁+Infϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x, -Inf) = +0+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x+yϵ₁+zϵ₂, +Inf) = +0+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| < 1
PowReal(x+0ϵ₁+0ϵ₂, -Inf) = +Inf+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| < 1
PowReal(x, -Inf) = +Inf-Infϵ₁-Infϵ₂+NaNϵ₁ϵ₂ for |x| < 1
PowReal(+Inf, y) = +Inf for y > 0
PowReal(+Inf, y) = +0 for y < 0
PowReal(-Inf, y) = Pow(-0, -y)
PowReal(x, y) = NaN+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for finite x < 0 and finite non-integer y
```

#### func Scale ¶

`func Scale(f float64, d Number) Number`

Scale returns d scaled by f.

#### func Sin ¶

`func Sin(d Number) Number`

Sin returns the sine of d.

Special cases are:

```Sin(±0) = (±0+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Sin(±Inf) = NaN
Sin(NaN) = NaN
```

#### func Sinh ¶

`func Sinh(d Number) Number`

Sinh returns the hyperbolic sine of d.

Special cases are:

```Sinh(±0) = (±0+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
Sinh(±Inf) = ±Inf
Sinh(NaN) = NaN
```

#### func Sqrt ¶

`func Sqrt(d Number) Number`

Sqrt returns the square root of d.

Special cases are:

```Sqrt(+Inf) = +Inf
Sqrt(±0) = (±0+Infϵ₁+Infϵ₂-Infϵ₁ϵ₂)
Sqrt(x < 0) = NaN
Sqrt(NaN) = NaN
```

#### func Sub ¶

`func Sub(x, y Number) Number`

Sub returns the difference of x and y, x-y.

#### func Tan ¶

`func Tan(d Number) Number`

Tan returns the tangent of d.

Special cases are:

```Tan(±0) = (±0+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
Tan(±Inf) = NaN
Tan(NaN) = NaN
```

#### func Tanh ¶

`func Tanh(d Number) Number`

Tanh returns the hyperbolic tangent of d.

Special cases are:

```Tanh(±0) = (±0+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Tanh(±Inf) = (±1+0ϵ₁+0ϵ₂∓0ϵ₁ϵ₂)
Tanh(NaN) = NaN
```

#### func (Number) Format ¶

`func (d Number) Format(fs fmt.State, c rune)`

Format implements fmt.Formatter.