2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.6%

Time: 6.7s

Alternatives: 10

Speedup: 1.1×

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: 6.8% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 10.3

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 10.3

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

4. Applied rewrites 10.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

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

   5. metadata-eval 99.5

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1} \]
8. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 1e-5) (* (sqrt (pow x -1.0)) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 1d-5) then
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 1e-5:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = sqrt((x ^ -1.0)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000008e-5

   1. Initial program 5.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 90.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 10^{-5}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (sqrt (pow x -1.0)) 0.5))

C

double code(double x) {
	return sqrt(pow(x, -1.0)) * 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt(math.pow(x, -1.0)) * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.5

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

   \[\leadsto \sqrt{{x}^{-1}} \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 18.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x} + 1\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 10.3

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 10.3

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

4. Applied rewrites 10.3%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 18.8

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

7. Applied rewrites 18.8%

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

8. Final simplification 18.8%

   \[\leadsto {\left(\sqrt{x} + 1\right)}^{-1} \]
9. Add Preprocessing

Alternative 5: 18.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 10.3

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 10.3

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

4. Applied rewrites 10.3%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 18.8

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

7. Applied rewrites 18.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{\frac{1}{x}} \]
9. Step-by-step derivation

10. Applied rewrites 18.7%

    \[\leadsto \sqrt{\frac{1}{x}} \]

11. Final simplification 18.7%

    \[\leadsto \sqrt{{x}^{-1}} \]
12. Add Preprocessing

Alternative 6: 18.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 10.3

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 10.3

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

4. Applied rewrites 10.3%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-sqrt.f64 18.8

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

7. Applied rewrites 18.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 18.8%

    \[\leadsto \frac{\sqrt{x} - 1}{\color{blue}{x}} \]
11. Add Preprocessing

Alternative 7: 4.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 0.5 x (- 1.0 (sqrt x))))

C

double code(double x) {
	return fma(0.5, x, (1.0 - sqrt(x)));
}

Julia

function code(x)
	return fma(0.5, x, Float64(1.0 - sqrt(x)))
end

Wolfram

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

Derivation

1. Initial program 7.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 4.7

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

5. Applied rewrites 4.7%

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

Alternative 8: 4.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. lower-fma.f64 4.7

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

5. Applied rewrites 4.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.6%

   \[\leadsto 0.5 \cdot \color{blue}{x} - \sqrt{x} \]
9. Add Preprocessing

Alternative 9: 1.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.6%

   \[\leadsto \color{blue}{1} - \sqrt{x} \]
6. Add Preprocessing

Alternative 10: 1.1% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* x x) -0.125))

C

double code(double x) {
	return (x * x) * -0.125;
}

Fortran

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

Java

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

Python

def code(x):
	return (x * x) * -0.125

Julia

function code(x)
	return Float64(Float64(x * x) * -0.125)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * -0.125;
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 1.2

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) \]

5. Applied rewrites 1.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{8} \cdot \color{blue}{{x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 1.1%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{-0.125} \]
9. Add Preprocessing

Developer Target 1: 98.0% 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 2024324 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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