sqrt times

Percentage Accurate: 99.2% → 99.4%

Time: 2.4s

Alternatives: 3

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: 99.4% 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 99.3%

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

Alternative 2: 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-rgt-neg-in N/A

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

   3. metadata-eval N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-in N/A

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

   12. rem-square-sqrt N/A

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

   13. unpow2 N/A

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

   14. +-commutative N/A

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

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

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

   16. distribute-lft-in N/A

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

5. Applied rewrites 99.0%

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

Alternative 3: 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 97.5

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

5. Applied rewrites 97.5%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))