2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.6%

Time: 12.8s

Alternatives: 4

Speedup: 205.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. flip-- 8.2%

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

   2. div-inv 8.2%

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

   3. add-sqr-sqrt 8.1%

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

   4. add-sqr-sqrt 9.0%

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

   5. associate--l+ 9.0%

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

4. Applied egg-rr 9.0%

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

   1. associate-*r/ 9.0%

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

   2. *-rgt-identity 9.0%

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

   3. associate-+r- 9.0%

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

   4. remove-double-neg 9.0%

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

   5. sub-neg 9.0%

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

   6. div-sub 7.3%

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

   7. rem-square-sqrt 7.1%

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

   8. sqr-neg 7.1%

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

   9. div-sub 8.1%

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

   10. +-commutative 8.1%

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

   11. sqr-neg 8.1%

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

   12. rem-square-sqrt 9.0%

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

   13. associate--l+ 99.6%

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

   14. +-inverses 99.6%

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

   15. metadata-eval 99.6%

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

   16. sub-neg 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. flip3-- 5.9%

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

   2. div-inv 5.9%

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

   3. sqrt-pow2 6.1%

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

   4. metadata-eval 6.1%

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

   5. sqrt-pow2 6.0%

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

   6. metadata-eval 6.0%

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

   7. add-sqr-sqrt 6.0%

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

   8. associate-+l+ 6.0%

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

   9. add-sqr-sqrt 6.0%

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

   10. sqrt-unprod 6.0%

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

4. Applied egg-rr 6.0%

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

5. Taylor expanded in x around inf 97.5%

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

   1. *-commutative 97.5%

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

7. Simplified 97.5%

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

   1. *-un-lft-identity 97.5%

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

   2. inv-pow 97.5%

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

   3. sqrt-pow1 97.7%

      \[\leadsto \left(1 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

   4. metadata-eval 97.7%

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

9. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot 0.5 \]
10. Step-by-step derivation

    1. *-lft-identity 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

    \[\leadsto {x}^{-0.5} \cdot 0.5 \]
13. Add Preprocessing

Alternative 3: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. flip3-- 5.9%

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

   2. div-inv 5.9%

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

   3. sqrt-pow2 6.1%

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

   4. metadata-eval 6.1%

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

   5. sqrt-pow2 6.0%

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

   6. metadata-eval 6.0%

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

   7. add-sqr-sqrt 6.0%

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

   8. associate-+l+ 6.0%

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

   9. add-sqr-sqrt 6.0%

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

   10. sqrt-unprod 6.0%

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

4. Applied egg-rr 6.0%

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

5. Taylor expanded in x around inf 54.2%

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

6. Taylor expanded in x around inf 96.9%

   \[\leadsto \left(1.5 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{0.3333333333333333}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 96.9%

      \[\leadsto \color{blue}{\frac{\left(1.5 \cdot \sqrt{x}\right) \cdot 0.3333333333333333}{x}} \]

   2. clear-num 96.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\left(1.5 \cdot \sqrt{x}\right) \cdot 0.3333333333333333}}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\left(\sqrt{x} \cdot 1.5\right)} \cdot 0.3333333333333333}} \]

   4. associate-*l* 97.1%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\sqrt{x} \cdot \left(1.5 \cdot 0.3333333333333333\right)}}} \]

   5. metadata-eval 97.1%

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

8. Applied egg-rr 97.1%

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

   1. associate-/r/ 97.1%

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

   2. associate-*l/ 97.3%

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

   3. *-lft-identity 97.3%

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

   4. rem-square-sqrt 97.2%

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

   5. associate-/l/ 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-/l* 97.3%

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

   8. *-inverses 97.3%

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

   9. metadata-eval 97.3%

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

10. Simplified 97.3%

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

11. Final simplification 97.3%

    \[\leadsto \frac{0.5}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 4: 6.9% 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 7.2%

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

3. Taylor expanded in x around 0 7.0%

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

4. Final simplification 7.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 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]

Reproduce

herbie shell --seed 2024052 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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