sqrt times

Percentage Accurate: 99.2% → 99.2%

Time: 2.9s

Alternatives: 4

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 99.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ x - \left(0.5 + \frac{0.125}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x (+ 0.5 (/ 0.125 x))))

C

double code(double x) {
	return x - (0.5 + (0.125 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - (0.5d0 + (0.125d0 / x))
end function

Java

public static double code(double x) {
	return x - (0.5 + (0.125 / x));
}

Python

def code(x):
	return x - (0.5 + (0.125 / x))

Julia

function code(x)
	return Float64(x - Float64(0.5 + Float64(0.125 / x)))
end

MATLAB

function tmp = code(x)
	tmp = x - (0.5 + (0.125 / x));
end

Wolfram

code[x_] := N[(x - N[(0.5 + N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x}}{x}\right)} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125}{x}\right)} \]
6. Add Preprocessing

Alternative 3: 99.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ x + -0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x + -0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. metadata-eval N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. rem-square-sqrt N/A

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

   10. unpow2 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. unpow2 N/A

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

   15. rem-square-sqrt N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{x + -0.5} \]
6. Add Preprocessing

Alternative 4: 98.0% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity 97.7

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

5. Applied rewrites 97.7%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))