2sqrt (example 3.1)

Percentage Accurate: 6.9% → 99.6%

Time: 8.0s

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: 6.9% 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: 99.6% 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 6.4%

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

   1. flip-- 7.0%

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

   2. div-inv 7.0%

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

   3. add-sqr-sqrt 6.7%

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

   4. add-sqr-sqrt 7.2%

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

   5. associate--l+ 7.2%

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

4. Applied egg-rr 7.2%

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

   1. associate-*r/ 7.2%

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

   2. *-rgt-identity 7.2%

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

   3. associate-+r- 7.2%

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

   4. +-commutative 7.2%

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

   5. associate--l+ 99.6%

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

   6. +-inverses 99.6%

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

   7. metadata-eval 99.6%

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

   8. +-commutative 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 98.0% 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]

Derivation

1. Initial program 6.4%

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

   1. flip-- 7.0%

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

   2. div-inv 7.0%

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

   3. add-sqr-sqrt 6.7%

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

   4. add-sqr-sqrt 7.2%

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

   5. associate--l+ 7.2%

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

4. Applied egg-rr 7.2%

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

   1. associate-*r/ 7.2%

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

   2. *-rgt-identity 7.2%

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

   3. associate-+r- 7.2%

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

   4. +-commutative 7.2%

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

   5. associate--l+ 99.6%

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

   6. +-inverses 99.6%

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

   7. metadata-eval 99.6%

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

   8. +-commutative 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 97.8%

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

   1. rem-exp-log 90.6%

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

   2. exp-neg 90.6%

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

   3. unpow1/2 90.6%

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

   4. exp-prod 90.6%

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

   5. distribute-lft-neg-out 90.6%

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

   6. distribute-rgt-neg-in 90.6%

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

   7. metadata-eval 90.6%

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

   8. exp-to-pow 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 3: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.25 x)))

C

double code(double x) {
	return sqrt((0.25 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.25d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.25 / x));
}

Python

def code(x):
	return math.sqrt((0.25 / x))

Julia

function code(x)
	return sqrt(Float64(0.25 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.25 / x));
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. flip-- 7.0%

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

   2. div-inv 7.0%

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

   3. add-sqr-sqrt 6.7%

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

   4. add-sqr-sqrt 7.2%

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

   5. associate--l+ 7.2%

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

4. Applied egg-rr 7.2%

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

   1. associate-*r/ 7.2%

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

   2. *-rgt-identity 7.2%

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

   3. associate-+r- 7.2%

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

   4. +-commutative 7.2%

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

   5. associate--l+ 99.6%

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

   6. +-inverses 99.6%

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

   7. metadata-eval 99.6%

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

   8. +-commutative 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 97.7%

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

   1. *-commutative 97.7%

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

9. Simplified 97.7%

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

    1. frac-2neg 97.7%

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

    2. metadata-eval 97.7%

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

    3. div-inv 97.7%

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

    4. distribute-rgt-neg-in 97.7%

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

    5. metadata-eval 97.7%

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

11. Applied egg-rr 97.7%

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

    1. associate-*r/ 97.7%

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

    2. metadata-eval 97.7%

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

    3. *-commutative 97.7%

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

    4. associate-/r* 97.7%

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

    5. metadata-eval 97.7%

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

13. Simplified 97.7%

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

    1. *-un-lft-identity 97.7%

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

    2. *-commutative 97.7%

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

    3. add-sqr-sqrt 97.2%

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

    4. sqrt-unprod 97.7%

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

    5. frac-times 97.7%

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

    6. metadata-eval 97.7%

       \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{x} \cdot \sqrt{x}}} \cdot 1 \]

    7. add-sqr-sqrt 97.8%

       \[\leadsto \sqrt{\frac{0.25}{\color{blue}{x}}} \cdot 1 \]

15. Applied egg-rr 97.8%

    \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}} \cdot 1} \]
16. Step-by-step derivation

    1. *-rgt-identity 97.8%

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

17. Simplified 97.8%

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

18. Final simplification 97.8%

    \[\leadsto \sqrt{\frac{0.25}{x}} \]
19. Add Preprocessing

Alternative 4: 6.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 6.4%

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

   1. flip-- 7.0%

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

   2. div-inv 7.0%

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

   3. add-sqr-sqrt 6.7%

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

   4. add-sqr-sqrt 7.2%

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

   5. associate--l+ 7.2%

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

4. Applied egg-rr 7.2%

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

   1. associate-*r/ 7.2%

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

   2. *-rgt-identity 7.2%

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

   3. associate-+r- 7.2%

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

   4. +-commutative 7.2%

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

   5. associate--l+ 99.6%

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

   6. +-inverses 99.6%

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

   7. metadata-eval 99.6%

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

   8. +-commutative 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 97.8%

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

   1. rem-exp-log 90.6%

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

   2. exp-neg 90.6%

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

   3. unpow1/2 90.6%

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

   4. exp-prod 90.6%

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

   5. distribute-lft-neg-out 90.6%

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

   6. distribute-rgt-neg-in 90.6%

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

   7. metadata-eval 90.6%

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

   8. exp-to-pow 98.1%

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

9. Simplified 98.1%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-0.5}} \]
10. Step-by-step derivation

    1. metadata-eval 98.1%

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

    2. pow-flip 97.7%

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

    3. pow1/2 97.7%

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

    4. add-sqr-sqrt 97.2%

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

    5. associate-/r* 97.2%

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

    6. metadata-eval 97.2%

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

    7. sqrt-div 97.5%

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

    8. pow1/2 97.5%

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

    9. pow-flip 97.6%

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

    10. metadata-eval 97.6%

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

    11. sqrt-pow1 97.5%

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

    12. metadata-eval 97.5%

        \[\leadsto 0.5 \cdot \frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\sqrt{x}}} \]

    13. pow1/2 97.5%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{\sqrt{\color{blue}{{x}^{0.5}}}} \]

    14. sqrt-pow1 97.5%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}} \]

    15. add-exp-log 92.0%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\color{blue}{\left(e^{\log x}\right)}}^{\left(\frac{0.5}{2}\right)}} \]

    16. add-sqr-sqrt 90.4%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\left(e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}\right)}^{\left(\frac{0.5}{2}\right)}} \]

    17. sqrt-unprod 92.0%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\left(e^{\color{blue}{\sqrt{\log x \cdot \log x}}}\right)}^{\left(\frac{0.5}{2}\right)}} \]

    18. sqr-neg 92.0%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\left(e^{\sqrt{\color{blue}{\left(-\log x\right) \cdot \left(-\log x\right)}}}\right)}^{\left(\frac{0.5}{2}\right)}} \]

    19. sqrt-unprod 0.0%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\left(e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}\right)}^{\left(\frac{0.5}{2}\right)}} \]

    20. add-sqr-sqrt 6.9%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\left(e^{\color{blue}{-\log x}}\right)}^{\left(\frac{0.5}{2}\right)}} \]

    21. rec-exp 6.9%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{{\color{blue}{\left(\frac{1}{e^{\log x}}\right)}}^{\left(\frac{0.5}{2}\right)}} \]

    22. add-exp-log 6.9%

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

    23. sqrt-pow1 6.9%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{\color{blue}{\sqrt{{\left(\frac{1}{x}\right)}^{0.5}}}} \]

    24. inv-pow 6.9%

        \[\leadsto 0.5 \cdot \frac{{x}^{-0.25}}{\sqrt{{\color{blue}{\left({x}^{-1}\right)}}^{0.5}}} \]

11. Applied egg-rr 6.9%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-0.25}}{{x}^{-0.25}}} \]
12. Step-by-step derivation

    1. *-inverses 6.9%

       \[\leadsto 0.5 \cdot \color{blue}{1} \]

13. Simplified 6.9%

    \[\leadsto 0.5 \cdot \color{blue}{1} \]

14. Final simplification 6.9%

    \[\leadsto 0.5 \]
15. 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 2024095 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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