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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(0 - x, x, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (fma (- 0.0 x) x 1.0)))

C

double code(double x) {
	return sqrt(fma((0.0 - x), x, 1.0));
}

Julia

function code(x)
	return sqrt(fma(Float64(0.0 - x), x, 1.0))
end

Wolfram

code[x_] := N[Sqrt[N[(N[(0.0 - x), $MachinePrecision] * x + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right) + 1}} \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot x} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), x, 1\right)}} \]

   5. neg-sub0 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{0 - x}, x, 1\right)} \]

   6. --lowering--.f64 100.0

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{0 - x}, x, 1\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, x, 1\right)} \]

   2. neg-lowering-neg.f64 100.0

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{-x}, x, 1\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{-x}, x, 1\right)} \]

7. Final simplification 100.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(0 - x, x, 1\right)} \]
8. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 3: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sqrt{1 - x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \]

   3. --rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{{x}^{2} - 0}, 1\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(0\right)\right)}, 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right), 1\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, x \cdot x + \color{blue}{0}, 1\right) \]

   7. accelerator-lowering-fma.f64 99.1

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 1\right) \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(x, x, 0\right), 1\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1 \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{2}} + 1 \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{-1}{2}\right)} + 1 \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right)} \]

   5. *-lowering-*.f64 99.1

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

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 19.0× 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 98.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))