Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))

C

double code(double x) {
	return sqrt((1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 - (x * x)));
}

Python

def code(x):
	return math.sqrt((1.0 - (x * x)))

Julia

function code(x)
	return sqrt(Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 - (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))

C

double code(double x) {
	return sqrt((1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 - (x * x)));
}

Python

def code(x):
	return math.sqrt((1.0 - (x * x)))

Julia

function code(x)
	return sqrt(Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 - (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(-x \cdot x\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (* (log1p (- (* x x))) 0.5)))

C

double code(double x) {
	return exp((log1p(-(x * x)) * 0.5));
}

Java

public static double code(double x) {
	return Math.exp((Math.log1p(-(x * x)) * 0.5));
}

Python

def code(x):
	return math.exp((math.log1p(-(x * x)) * 0.5))

Julia

function code(x)
	return exp(Float64(log1p(Float64(-Float64(x * x))) * 0.5))
end

Wolfram

code[x_] := N[Exp[N[(N[Log[1 + (-N[(x * x), $MachinePrecision])], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]
2. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto \color{blue}{{\left(1 - x \cdot x\right)}^{0.5}} \]

   2. pow-to-exp 100.0%

      \[\leadsto \color{blue}{e^{\log \left(1 - x \cdot x\right) \cdot 0.5}} \]

   3. sub-neg 100.0%

      \[\leadsto e^{\log \color{blue}{\left(1 + \left(-x \cdot x\right)\right)} \cdot 0.5} \]

   4. log1p-def 100.0%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-x \cdot x\right)} \cdot 0.5} \]

   5. distribute-rgt-neg-in 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(-x\right)}\right) \cdot 0.5} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) \cdot 0.5}} \]

4. Final simplification 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(-x \cdot x\right) \cdot 0.5} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))

C

double code(double x) {
	return sqrt((1.0 - (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 - (x * x)));
}

Python

def code(x):
	return math.sqrt((1.0 - (x * x)))

Julia

function code(x)
	return sqrt(Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 - (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]

2. Final simplification 100.0%

   \[\leadsto \sqrt{1 - x \cdot x} \]

Alternative 3: 99.5% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (* x x) -0.5)))

C

double code(double x) {
	return 1.0 + ((x * x) * -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (-0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * -0.5);
}

Python

def code(x):
	return 1.0 + ((x * x) * -0.5)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * -0.5);
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]

2. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

4. Simplified 99.5%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]

5. Final simplification 99.5%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot -0.5 \]

Alternative 4: 98.9% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]

2. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 99.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))