sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 10.2s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 99.4%

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

   1. +-commutative N/A

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

   2. associate-+l- N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f64 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto -0.5 + \left(x - \frac{0.125 + \frac{0.0625}{x}}{x}\right) \]
9. 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 99.2%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
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 99.4%

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

Alternative 3: 99.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 99.2%

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

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

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

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

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

   5. /-lowering-/.f64 99.2%

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

7. Applied egg-rr 99.2%

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

Alternative 4: 99.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x - N[(N[(0.125 / x), $MachinePrecision] + 0.5), $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. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 99.1% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -0.5 + x

Julia

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

MATLAB

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

Wolfram

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

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

   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. *-rgt-identity N/A

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

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

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

   5. metadata-eval N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. rem-square-sqrt N/A

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

   10. unpow2 N/A

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

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

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. unpow2 N/A

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

   15. rem-square-sqrt N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

5. Simplified 98.8%

   \[\leadsto \color{blue}{x + -0.5} \]

6. Final simplification 98.8%

   \[\leadsto -0.5 + x \]
7. Add Preprocessing

Alternative 6: 98.1% 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 99.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 97.5%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))