sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 4.5s

Alternatives: 5

Speedup: 29.3×

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: 99.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \left(x + -0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (+ x -0.5) (/ (+ 0.125 (/ 0.0625 x)) x)))

C

double code(double x) {
	return (x + -0.5) - ((0.125 + (0.0625 / x)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (-0.5d0)) - ((0.125d0 + (0.0625d0 / x)) / x)
end function

Java

public static double code(double x) {
	return (x + -0.5) - ((0.125 + (0.0625 / x)) / x);
}

Python

def code(x):
	return (x + -0.5) - ((0.125 + (0.0625 / x)) / x)

Julia

function code(x)
	return Float64(Float64(x + -0.5) - Float64(Float64(0.125 + Float64(0.0625 / x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = (x + -0.5) - ((0.125 + (0.0625 / x)) / x);
end

Wolfram

code[x_] := N[(N[(x + -0.5), $MachinePrecision] - N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{x}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{x + \left(-0.5 - \frac{0.125 + \frac{0.0625}{x}}{x}\right)} \]
6. Step-by-step derivation

   1. associate-+r- N/A

      \[\leadsto \left(x + \frac{-1}{2}\right) - \color{blue}{\frac{\frac{1}{8} + \frac{\frac{1}{16}}{x}}{x}} \]

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

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \frac{-1}{2}\right), \color{blue}{\left(\frac{\frac{1}{8} + \frac{\frac{1}{16}}{x}}{x}\right)}\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \left(\frac{\color{blue}{\frac{1}{8} + \frac{\frac{1}{16}}{x}}}{x}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\frac{1}{8} + \frac{\frac{1}{16}}{x}\right), \color{blue}{x}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{\frac{1}{16}}{x}\right)\right), x\right)\right) \]

   6. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{/.f64}\left(\frac{1}{16}, x\right)\right), x\right)\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(x + -0.5\right) - \frac{0.125 + \frac{0.0625}{x}}{x}} \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x + \left(-0.5 - \frac{0.125 + \frac{0.0625}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (- -0.5 (/ (+ 0.125 (/ 0.0625 x)) x))))

C

double code(double x) {
	return x + (-0.5 - ((0.125 + (0.0625 / x)) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-0.5d0) - ((0.125d0 + (0.0625d0 / x)) / x))
end function

Java

public static double code(double x) {
	return x + (-0.5 - ((0.125 + (0.0625 / x)) / x));
}

Python

def code(x):
	return x + (-0.5 - ((0.125 + (0.0625 / x)) / x))

Julia

function code(x)
	return Float64(x + Float64(-0.5 - Float64(Float64(0.125 + Float64(0.0625 / x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x + (-0.5 - ((0.125 + (0.0625 / x)) / x));
end

Wolfram

code[x_] := N[(x + N[(-0.5 - N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{x}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{x + \left(-0.5 - \frac{0.125 + \frac{0.0625}{x}}{x}\right)} \]
6. Add Preprocessing

Alternative 3: 99.4% accurate, 29.3× speedup

Math

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

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(Float64(x + -0.5) + Float64(-0.125 / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
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. Simplified 99.7%

   \[\leadsto \color{blue}{x + \left(-0.5 + \frac{-0.125}{x}\right) \cdot 1} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{8}}{x}\right), \color{blue}{\left(x + \frac{-1}{2}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, x\right), \left(\color{blue}{x} + \frac{-1}{2}\right)\right) \]

   8. +-lowering-+.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, x\right), \mathsf{+.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.0% accurate, 68.3× 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 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
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

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. rem-square-sqrt N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. rem-square-sqrt N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac2 N/A

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

   17. associate-/l* N/A

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

   18. *-rgt-identity N/A

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

   19. associate-*r/ N/A

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

   20. rgt-mult-inverse N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

5. Simplified 99.5%

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

Alternative 5: 97.9% accurate, 205.0× 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 99.3%

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

3. Taylor expanded in x around inf

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

5. Simplified 98.5%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))