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

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

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

\[\begin{array}{l} \\ {\left(\sqrt[3]{1 - x \cdot x}\right)}^{1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (cbrt (- 1.0 (* x x))) 1.5))

C

double code(double x) {
	return pow(cbrt((1.0 - (x * x))), 1.5);
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt((1.0 - (x * x))), 1.5);
}

Julia

function code(x)
	return cbrt(Float64(1.0 - Float64(x * x))) ^ 1.5
end

Wolfram

code[x_] := N[Power[N[Power[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-cube-cbrt 100.0%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{1 - x \cdot x} \cdot \sqrt[3]{1 - x \cdot x}\right) \cdot \sqrt[3]{1 - x \cdot x}}} \]

   2. sqrt-prod 100.0%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{1 - x \cdot x} \cdot \sqrt[3]{1 - x \cdot x}} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}}} \]

   3. pow2 100.0%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{2}} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}}} \]
4. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{1 - x \cdot x} \cdot \sqrt[3]{1 - x \cdot x}}} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   2. rem-sqrt-square 100.0%

      \[\leadsto \color{blue}{\left|\sqrt[3]{1 - x \cdot x}\right|} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   3. unpow1 100.0%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{1}}\right| \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   4. metadata-eval 100.0%

      \[\leadsto \left|{\left(\sqrt[3]{1 - x \cdot x}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right| \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   5. pow-sqr 100.0%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5}}\right| \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \color{blue}{\left({\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5}\right)} \cdot \sqrt{\sqrt[3]{1 - x \cdot x}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left({\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5}\right) \cdot \color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5}} \]

   8. unpow1/2 100.0%

      \[\leadsto \left(\color{blue}{\sqrt{\sqrt[3]{1 - x \cdot x}}} \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5}\right) \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \]

   9. unpow1/2 100.0%

      \[\leadsto \left(\sqrt{\sqrt[3]{1 - x \cdot x}} \cdot \color{blue}{\sqrt{\sqrt[3]{1 - x \cdot x}}}\right) \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \]

   10. rem-square-sqrt 100.0%

       \[\leadsto \color{blue}{\sqrt[3]{1 - x \cdot x}} \cdot {\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \]

   11. *-commutative 100.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{0.5} \cdot \sqrt[3]{1 - x \cdot x}} \]

   12. pow-plus 100.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{\left(0.5 + 1\right)}} \]

   13. metadata-eval 100.0%

       \[\leadsto {\left(\sqrt[3]{1 - x \cdot x}\right)}^{\color{blue}{1.5}} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - x \cdot x}\right)}^{1.5}} \]

6. Final simplification 100.0%

   \[\leadsto {\left(\sqrt[3]{1 - x \cdot x}\right)}^{1.5} \]

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: 99.6% 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: 99.1% 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 2023200 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))