2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 8.6s

Alternatives: 5

Speedup: 1.2×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

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: 6.7% accurate, 1.0× speedup

Math

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

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, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-+.f64 N/A

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

   13. associate--l+ N/A

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. lower-+.f64 N/A

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

   17. metadata-eval N/A

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

   18. *-rgt-identity N/A

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

   19. lower--.f64 N/A

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

   20. lower-+.f64 8.0

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

4. Applied rewrites 8.0%

   \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}} \]
5. Step-by-step derivation

   1. associate-+r- N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   5. metadata-eval 99.6

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]

6. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
8. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (sqrt (/ 1.0 x))))

C

double code(double x) {
	return 0.5 * sqrt((1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * sqrt((1.0d0 / x))
end function

Java

public static double code(double x) {
	return 0.5 * Math.sqrt((1.0 / x));
}

Python

def code(x):
	return 0.5 * math.sqrt((1.0 / x))

Julia

function code(x)
	return Float64(0.5 * sqrt(Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * sqrt((1.0 / x));
end

Wolfram

code[x_] := N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.6

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
6. Add Preprocessing

Alternative 3: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (sqrt x)))

C

double code(double x) {
	return 0.5 / sqrt(x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 / Math.sqrt(x);
}

Python

def code(x):
	return 0.5 / math.sqrt(x)

Julia

function code(x)
	return Float64(0.5 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / sqrt(x);
end

Wolfram

code[x_] := N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.6

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{x}}} \]

   5. lower-/.f64 98.4

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]
8. Add Preprocessing

Alternative 4: 5.4% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt x))

C

double code(double x) {
	return sqrt(x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt(x)

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 4.3

      \[\leadsto \mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) \]

5. Applied rewrites 4.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{-1 \cdot \sqrt{x}}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)}\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\sqrt{x}\right)}\right) \]

   3. lower-sqrt.f64 4.3

      \[\leadsto \mathsf{fma}\left(x, 0.5, -\color{blue}{\sqrt{x}}\right) \]

8. Applied rewrites 4.3%

   \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{-\sqrt{x}}\right) \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. neg-sub0 N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(0 - \sqrt{x}\right)} \]

   3. flip3-- N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{{0}^{3} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)}} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{0} - {\left(\sqrt{x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   5. neg-sub0 N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{\mathsf{neg}\left({\left(\sqrt{x}\right)}^{3}\right)}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   6. neg-mul-1 N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{-1 \cdot {\left(\sqrt{x}\right)}^{3}}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{{\left(\sqrt{x}\right)}^{3} \cdot -1}}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{{\color{blue}{\left(\sqrt{x}\right)}}^{3} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   9. sqrt-pow2 N/A

      \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)}} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   10. rem-square-sqrt N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\left(\color{blue}{\sqrt{x}} \cdot \sqrt{x}\right)}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   12. lift-sqrt.f64 N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\left(\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right)}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   13. sqr-neg N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)}}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   14. lift-neg.f64 N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   15. lift-neg.f64 N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{{\left(\left(\mathsf{neg}\left(\sqrt{x}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}\right)}^{\left(\frac{3}{2}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   16. pow-prod-down N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{\left({\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

   17. sqr-pow N/A

       \[\leadsto x \cdot \frac{1}{2} + \frac{\color{blue}{{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)}^{3}} \cdot -1}{0 \cdot 0 + \left(\sqrt{x} \cdot \sqrt{x} + 0 \cdot \sqrt{x}\right)} \]

10. Applied rewrites 4.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \sqrt{x}\right)} \]

11. Taylor expanded in x around 0

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

    1. lower-sqrt.f64 5.2

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

13. Applied rewrites 5.2%

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

Alternative 5: 3.8% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 3.8

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

5. Applied rewrites 3.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. +-inverses 3.8

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))
end

MATLAB

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

Wolfram

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

Developer Target 2: 98.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

double code(double x) {
	return 0.5 * pow(x, -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -0.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -0.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -0.5);
end

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

  :alt
  (! :herbie-platform default (* 1/2 (pow x -1/2)))

  (- (sqrt (+ x 1.0)) (sqrt x)))