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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow-prod-down N/A

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

   9. pow-prod-up N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

       \[\leadsto \sqrt{\frac{1 - {x}^{\color{blue}{4}}}{1 + x \cdot x}} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{\frac{1 - {x}^{4}}{\color{blue}{x \cdot x + 1}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{1 - {x}^{4}}{\color{blue}{x \cdot x} + 1}} \]

   14. lower-fma.f64 100.0

       \[\leadsto \sqrt{\frac{1 - {x}^{4}}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1 - {x}^{4}}{\mathsf{fma}\left(x, x, 1\right)}}} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{1 - {x}^{4}}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{\color{blue}{1 + \left(\mathsf{neg}\left({x}^{4}\right)\right)}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left({x}^{4}\right)\right) + 1}}{\mathsf{fma}\left(x, x, 1\right)}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\mathsf{neg}\left(\color{blue}{{x}^{4}}\right)\right) + 1}{\mathsf{fma}\left(x, x, 1\right)}} \]

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-neg-out N/A

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

   13. lift-neg.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-neg-in N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. lift-*.f64 N/A

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

   10. sqr-neg N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-frac-neg2 N/A

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

   14. lift-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. flip-- N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\left(x + 1\right) \cdot \left(-\left(x - 1\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

      \[\leadsto \sqrt{\left(x + 1\right) \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto \sqrt{\left(x + 1\right) \cdot \left(0 - \color{blue}{\left(x - 1\right)}\right)} \]

   5. associate--r- N/A

      \[\leadsto \sqrt{\left(x + 1\right) \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

   6. neg-sub0 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 100.0

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

   11. lift-+.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 100.0

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 100.0

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

10. Applied rewrites 100.0%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x - -1, -x, x - -1\right)}} \]
11. 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: 99.5% accurate, 1.6× speedup

Math

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

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(Float64(x * x), -0.5, 1.0)
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * -0.5 + 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 4: 98.9% 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. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))