Main:bigenough3 from C

Percentage Accurate: 52.9% → 99.7%

Time: 6.0s

Alternatives: 6

Speedup: 2.0×

Specification

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: 52.9% 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: 99.7% 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 50.0%

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

   1. flip-- 50.5%

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

   2. div-inv 50.5%

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

   3. add-sqr-sqrt 50.3%

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

   4. add-sqr-sqrt 51.1%

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

3. Applied egg-rr 51.1%

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

   1. *-commutative 51.1%

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

   2. associate-/r/ 51.1%

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

   3. +-commutative 51.1%

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

   4. associate--l+ 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-5) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = pow(x, -0.5) * 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 <= 2d-5) then
        tmp = (x ** (-0.5d0)) * 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 <= 2e-5) {
		tmp = Math.pow(x, -0.5) * 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 <= 2e-5:
		tmp = math.pow(x, -0.5) * 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 <= 2e-5)
		tmp = Float64((x ^ -0.5) * 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 <= 2e-5)
		tmp = (x ^ -0.5) * 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, 2e-5], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 2.00000000000000016e-5

   1. Initial program 5.0%

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

      1. flip-- 5.7%

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

      2. div-inv 5.7%

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

      3. add-sqr-sqrt 5.7%

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

      4. add-sqr-sqrt 6.0%

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

   3. Applied egg-rr 6.0%

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

      1. *-commutative 6.0%

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

      2. associate-/r/ 6.0%

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

      3. +-commutative 6.0%

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

      4. associate--l+ 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. /-rgt-identity 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. flip3-+ 67.2%

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

      2. div-inv 67.1%

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

      3. +-commutative 67.1%

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

      4. +-commutative 67.1%

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

      5. sqrt-pow2 67.0%

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

      6. metadata-eval 67.0%

         \[\leadsto \frac{1}{\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      7. sqrt-pow2 66.9%

         \[\leadsto \frac{1}{\left({x}^{1.5} + \color{blue}{{\left(x + 1\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      8. metadata-eval 66.9%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      9. add-sqr-sqrt 67.1%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      10. +-commutative 67.1%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      11. associate-+l+ 67.1%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{x + \left(1 + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)}}} \]

   7. Applied egg-rr 49.3%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}}} \]
   8. Step-by-step derivation

      1. +-commutative 49.3%

         \[\leadsto \frac{1}{\color{blue}{\left({\left(x + 1\right)}^{1.5} + {x}^{1.5}\right)} \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      2. +-commutative 49.3%

         \[\leadsto \frac{1}{\left({\color{blue}{\left(1 + x\right)}}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      3. associate-+r- 49.3%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{\left(\left(1 + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)}}} \]

      4. *-commutative 49.3%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{\color{blue}{x \cdot \left(x + 1\right)}}\right)}} \]

      5. +-commutative 49.3%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \color{blue}{\left(1 + x\right)}}\right)}} \]

   9. Simplified 49.3%

      \[\leadsto \frac{1}{\color{blue}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \left(1 + x\right)}\right)}}} \]

   10. Taylor expanded in x around inf 67.1%

       \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{0.5}}} \]

   11. Taylor expanded in x around inf 99.2%

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

       1. unpow1/2 99.2%

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

       2. unpow-1 99.2%

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

       3. exp-to-pow 91.8%

          \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{\log x \cdot -1}\right)}}^{0.5} \]

       4. *-commutative 91.8%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-1 \cdot \log x}}\right)}^{0.5} \]

       5. neg-mul-1 91.8%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-\log x}}\right)}^{0.5} \]

       6. exp-prod 91.8%

          \[\leadsto 0.5 \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 91.8%

          \[\leadsto 0.5 \cdot e^{\color{blue}{-\log x \cdot 0.5}} \]

       8. distribute-rgt-neg-in 91.8%

          \[\leadsto 0.5 \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \]

       9. metadata-eval 91.8%

          \[\leadsto 0.5 \cdot e^{\log x \cdot \color{blue}{-0.5}} \]

       10. exp-to-pow 99.4%

           \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

       11. *-commutative 99.4%

           \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]

   13. Simplified 99.4%

       \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. inv-pow 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. unpow-prod-down 99.9%

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

      4. pow-prod-up 99.9%

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

      5. +-commutative 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. +-commutative 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. hypot-def 99.9%

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

      10. pow1/2 99.9%

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

      11. sqrt-pow1 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x + 1}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x + 1}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. pow1/2 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({x}^{0.25}, \sqrt{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. sqrt-pow1 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({x}^{0.25}, \color{blue}{{\left(x + 1\right)}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      15. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      16. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({x}^{0.25}, {\left(x + 1\right)}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around 0 99.0%

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

   if 1 < x

   1. Initial program 7.9%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 8.6%

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

      4. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. *-commutative 10.1%

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

      2. associate-/r/ 10.1%

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

      3. +-commutative 10.1%

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

      4. associate--l+ 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. /-rgt-identity 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. flip3-+ 68.5%

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

      2. div-inv 68.5%

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

      3. +-commutative 68.5%

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

      4. +-commutative 68.5%

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

      5. sqrt-pow2 68.4%

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

      6. metadata-eval 68.4%

         \[\leadsto \frac{1}{\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      7. sqrt-pow2 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + \color{blue}{{\left(x + 1\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      8. metadata-eval 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      9. add-sqr-sqrt 68.5%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      10. +-commutative 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      11. associate-+l+ 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{x + \left(1 + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)}}} \]

   7. Applied egg-rr 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}}} \]
   8. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{1}{\color{blue}{\left({\left(x + 1\right)}^{1.5} + {x}^{1.5}\right)} \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      2. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\color{blue}{\left(1 + x\right)}}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      3. associate-+r- 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{\left(\left(1 + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)}}} \]

      4. *-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{\color{blue}{x \cdot \left(x + 1\right)}}\right)}} \]

      5. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \color{blue}{\left(1 + x\right)}}\right)}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \left(1 + x\right)}\right)}}} \]

   10. Taylor expanded in x around inf 67.6%

       \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{0.5}}} \]

   11. Taylor expanded in x around inf 97.0%

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

       1. unpow1/2 97.0%

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

       2. unpow-1 97.0%

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

       3. exp-to-pow 90.0%

          \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{\log x \cdot -1}\right)}}^{0.5} \]

       4. *-commutative 90.0%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-1 \cdot \log x}}\right)}^{0.5} \]

       5. neg-mul-1 90.0%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-\log x}}\right)}^{0.5} \]

       6. exp-prod 90.0%

          \[\leadsto 0.5 \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 90.0%

          \[\leadsto 0.5 \cdot e^{\color{blue}{-\log x \cdot 0.5}} \]

       8. distribute-rgt-neg-in 90.0%

          \[\leadsto 0.5 \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \]

       9. metadata-eval 90.0%

          \[\leadsto 0.5 \cdot e^{\log x \cdot \color{blue}{-0.5}} \]

       10. exp-to-pow 97.2%

           \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

       11. *-commutative 97.2%

           \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]

   13. Simplified 97.2%

       \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 4: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{1} \]

   if 0.25 < x

   1. Initial program 7.9%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 8.6%

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

      4. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. *-commutative 10.1%

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

      2. associate-/r/ 10.1%

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

      3. +-commutative 10.1%

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

      4. associate--l+ 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. /-rgt-identity 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. flip3-+ 68.5%

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

      2. div-inv 68.5%

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

      3. +-commutative 68.5%

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

      4. +-commutative 68.5%

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

      5. sqrt-pow2 68.4%

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

      6. metadata-eval 68.4%

         \[\leadsto \frac{1}{\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      7. sqrt-pow2 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + \color{blue}{{\left(x + 1\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      8. metadata-eval 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      9. add-sqr-sqrt 68.5%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      10. +-commutative 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      11. associate-+l+ 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{x + \left(1 + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)}}} \]

   7. Applied egg-rr 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}}} \]
   8. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{1}{\color{blue}{\left({\left(x + 1\right)}^{1.5} + {x}^{1.5}\right)} \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      2. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\color{blue}{\left(1 + x\right)}}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      3. associate-+r- 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{\left(\left(1 + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)}}} \]

      4. *-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{\color{blue}{x \cdot \left(x + 1\right)}}\right)}} \]

      5. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \color{blue}{\left(1 + x\right)}}\right)}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \left(1 + x\right)}\right)}}} \]

   10. Taylor expanded in x around inf 67.6%

       \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{0.5}}} \]

   11. Taylor expanded in x around inf 97.0%

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

       1. unpow1/2 97.0%

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

       2. unpow-1 97.0%

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

       3. exp-to-pow 90.0%

          \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{\log x \cdot -1}\right)}}^{0.5} \]

       4. *-commutative 90.0%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-1 \cdot \log x}}\right)}^{0.5} \]

       5. neg-mul-1 90.0%

          \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{-\log x}}\right)}^{0.5} \]

       6. exp-prod 90.0%

          \[\leadsto 0.5 \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]

       7. distribute-lft-neg-out 90.0%

          \[\leadsto 0.5 \cdot e^{\color{blue}{-\log x \cdot 0.5}} \]

       8. distribute-rgt-neg-in 90.0%

          \[\leadsto 0.5 \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \]

       9. metadata-eval 90.0%

          \[\leadsto 0.5 \cdot e^{\log x \cdot \color{blue}{-0.5}} \]

       10. exp-to-pow 97.2%

           \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

       11. *-commutative 97.2%

           \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]

   13. Simplified 97.2%

       \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]

Alternative 5: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.1%

      \[\leadsto \color{blue}{1} \]

   if 0.25 < x

   1. Initial program 7.9%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 8.6%

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

      4. add-sqr-sqrt 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. *-commutative 10.1%

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

      2. associate-/r/ 10.1%

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

      3. +-commutative 10.1%

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

      4. associate--l+ 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. /-rgt-identity 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. flip3-+ 68.5%

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

      2. div-inv 68.5%

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

      3. +-commutative 68.5%

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

      4. +-commutative 68.5%

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

      5. sqrt-pow2 68.4%

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

      6. metadata-eval 68.4%

         \[\leadsto \frac{1}{\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      7. sqrt-pow2 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + \color{blue}{{\left(x + 1\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      8. metadata-eval 68.3%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      9. add-sqr-sqrt 68.5%

         \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      10. +-commutative 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}} \]

      11. associate-+l+ 68.5%

          \[\leadsto \frac{1}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{\color{blue}{x + \left(1 + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)}}} \]

   7. Applied egg-rr 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{1.5} + {\left(x + 1\right)}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}}} \]
   8. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{1}{\color{blue}{\left({\left(x + 1\right)}^{1.5} + {x}^{1.5}\right)} \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      2. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\color{blue}{\left(1 + x\right)}}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(1 + \left(x - \sqrt{\left(x + 1\right) \cdot x}\right)\right)}} \]

      3. associate-+r- 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \color{blue}{\left(\left(1 + x\right) - \sqrt{\left(x + 1\right) \cdot x}\right)}}} \]

      4. *-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{\color{blue}{x \cdot \left(x + 1\right)}}\right)}} \]

      5. +-commutative 51.5%

         \[\leadsto \frac{1}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \color{blue}{\left(1 + x\right)}}\right)}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{1}{\color{blue}{\left({\left(1 + x\right)}^{1.5} + {x}^{1.5}\right) \cdot \frac{1}{x + \left(\left(1 + x\right) - \sqrt{x \cdot \left(1 + x\right)}\right)}}} \]

   10. Taylor expanded in x around inf 96.7%

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

       1. add-sqr-sqrt 96.2%

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

       2. sqrt-unprod 96.7%

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

       3. associate-/r* 96.7%

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

       4. metadata-eval 96.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.5}}{\sqrt{x}} \cdot \frac{1}{2 \cdot \sqrt{x}}} \]

       5. associate-/r* 96.7%

          \[\leadsto \sqrt{\frac{0.5}{\sqrt{x}} \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{x}}}} \]

       6. metadata-eval 96.7%

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

       7. frac-times 96.7%

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

       8. metadata-eval 96.7%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{x} \cdot \sqrt{x}}} \]

       9. add-sqr-sqrt 97.0%

          \[\leadsto \sqrt{\frac{0.25}{\color{blue}{x}}} \]

   12. Applied egg-rr 97.0%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \]

Alternative 6: 51.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 50.0%

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

2. Taylor expanded in x around 0 48.1%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 48.1%

   \[\leadsto 1 \]

Developer target: 99.7% 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]

Reproduce

herbie shell --seed 2023290 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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