2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 8.3s

Alternatives: 4

Speedup: 1.9×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. flip-- 8.3%

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

   2. div-inv 8.3%

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

   3. add-sqr-sqrt 9.0%

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

   4. add-sqr-sqrt 11.0%

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

   5. associate--l+ 11.0%

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

4. Applied egg-rr 11.0%

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

   1. associate-*r/ 11.0%

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

   2. *-rgt-identity 11.0%

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

   3. associate-+r- 11.0%

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

   4. +-commutative 11.0%

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

   5. associate--l+ 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. +-inverses 99.7%

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

   2. metadata-eval 99.7%

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

   3. inv-pow 99.7%

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

8. Applied egg-rr 99.7%

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

   1. unpow-1 99.7%

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

   2. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0) (/ 1.0 (+ (sqrt x) (+ 1.0 (* x 0.5)))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / (sqrt(x) + (1.0 + (x * 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((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 / (sqrt(x) + (1.0d0 + (x * 0.5d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / (Math.sqrt(x) + (1.0 + (x * 0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 / (math.sqrt(x) + (1.0 + (x * 0.5)))
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < 0.0

   1. Initial program 3.9%

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

      1. flip-- 3.9%

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

      2. div-inv 3.9%

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

      3. add-sqr-sqrt 3.8%

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

      4. add-sqr-sqrt 3.9%

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

      5. associate--l+ 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. associate-*r/ 3.9%

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

      2. *-rgt-identity 3.9%

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

      3. associate-+r- 3.9%

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

      4. +-commutative 3.9%

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

      5. associate--l+ 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. +-inverses 99.7%

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

      2. metadata-eval 99.7%

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

      3. inv-pow 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around 0 6.6%

       \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\left(1 + 0.5 \cdot x\right)}} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 58.6%

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

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 7.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. flip-- 8.3%

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

   2. div-inv 8.3%

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

   3. add-sqr-sqrt 9.0%

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

   4. add-sqr-sqrt 11.0%

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

   5. associate--l+ 11.0%

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

4. Applied egg-rr 11.0%

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

   1. associate-*r/ 11.0%

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

   2. *-rgt-identity 11.0%

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

   3. associate-+r- 11.0%

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

   4. +-commutative 11.0%

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

   5. associate--l+ 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. +-inverses 99.7%

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

   2. metadata-eval 99.7%

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

   3. inv-pow 99.7%

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

8. Applied egg-rr 99.7%

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

   1. unpow-1 99.7%

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

   2. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in x around 0 7.1%

    \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{\left(1 + 0.5 \cdot x\right)}} \]

12. Final simplification 7.1%

    \[\leadsto \frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)} \]
13. Add Preprocessing

Alternative 4: 4.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot 0.5\right) - \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0 4.6%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot x\right) - \sqrt{x}} \]

4. Final simplification 4.6%

   \[\leadsto \left(1 + x \cdot 0.5\right) - \sqrt{x} \]
5. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.0× 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, 2.0× 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 2024179 
(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)))