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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sqr-neg 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. +-commutative 100.0%

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

   4. +-lft-identity 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. fma-neg 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sqrt{-\mathsf{fma}\left(x, x, -1\right)} \]
6. 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. Final simplification 100.0%

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

Alternative 3: 1.6% 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. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. +-commutative 100.0%

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

   4. +-lft-identity 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. fma-neg 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 0.0%

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

   1. unpow2 0.0%

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

   2. rem-square-sqrt 1.6%

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

7. Simplified 1.6%

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

8. Final simplification 1.6%

   \[\leadsto -1 \]
9. Add Preprocessing

Alternative 4: 99.0% 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. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. +-commutative 100.0%

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

   4. +-lft-identity 100.0%

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

   5. cancel-sign-sub 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. fma-neg 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. distribute-neg-in 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. sub-neg 100.0%

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

   6. flip-- 100.0%

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

   7. sqrt-div 100.0%

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

   8. metadata-eval 100.0%

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

   9. pow2 100.0%

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

   10. pow2 100.0%

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

   11. pow-prod-up 100.0%

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

   12. metadata-eval 100.0%

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

   13. hypot-1-def 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 98.1%

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

8. Final simplification 98.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))