2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 7.3s

Alternatives: 5

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: 6.7% 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 7.4%

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

   1. flip-- 8.5%

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

   2. div-inv 8.5%

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

   3. add-sqr-sqrt 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 9.8%

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

   5. associate--l+ 9.8%

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

4. Applied egg-rr 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-rgt-identity 9.8%

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

   3. +-commutative 9.8%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
7. Add Preprocessing

Alternative 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]

Derivation

1. Initial program 7.4%

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

   1. flip-- 8.5%

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

   2. div-inv 8.5%

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

   3. add-sqr-sqrt 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 9.8%

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

   5. associate--l+ 9.8%

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

4. Applied egg-rr 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-rgt-identity 9.8%

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

   3. +-commutative 9.8%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around inf 97.4%

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

   1. unpow-1 97.4%

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

   2. metadata-eval 97.4%

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

   3. pow-sqr 97.6%

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

   4. rem-sqrt-square 97.6%

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

   5. rem-square-sqrt 96.7%

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

   6. fabs-sqr 96.7%

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

   7. rem-square-sqrt 97.6%

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

9. Simplified 97.6%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-0.5}} \]
10. Add Preprocessing

Alternative 3: 97.9% 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 7.4%

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

3. Taylor expanded in x around inf 97.4%

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

   1. add-sqr-sqrt 96.8%

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

   2. sqrt-unprod 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. swap-sqr 97.4%

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

   6. add-sqr-sqrt 97.4%

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

   7. metadata-eval 97.4%

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

5. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. metadata-eval 97.4%

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

7. Simplified 97.4%

   \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
8. Add Preprocessing

Alternative 4: 18.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0 1.6%

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

4. Taylor expanded in x around inf 1.6%

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

   1. neg-mul-1 1.6%

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

6. Simplified 1.6%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 5.5%

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

   3. pow1/2 5.5%

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

   4. metadata-eval 5.5%

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

   5. pow-pow 5.5%

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

   6. pow1/3 5.5%

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

   7. pow1/2 5.5%

      \[\leadsto \sqrt{\left(-{\left(\sqrt[3]{x}\right)}^{1.5}\right) \cdot \left(-\color{blue}{{x}^{0.5}}\right)} \]

   8. metadata-eval 5.5%

      \[\leadsto \sqrt{\left(-{\left(\sqrt[3]{x}\right)}^{1.5}\right) \cdot \left(-{x}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}}\right)} \]

   9. pow-pow 5.5%

      \[\leadsto \sqrt{\left(-{\left(\sqrt[3]{x}\right)}^{1.5}\right) \cdot \left(-\color{blue}{{\left({x}^{0.3333333333333333}\right)}^{1.5}}\right)} \]

   10. pow1/3 5.5%

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

   11. sqr-neg 5.5%

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

   12. pow-sqr 5.5%

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

   13. metadata-eval 5.5%

       \[\leadsto \sqrt{{\left(\sqrt[3]{x}\right)}^{\color{blue}{3}}} \]

   14. pow3 5.5%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \]

   15. add-cube-cbrt 5.5%

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

   16. pow1/2 5.5%

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

   17. add-exp-log 5.5%

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

   18. add-sqr-sqrt 5.5%

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

   19. sqrt-unprod 5.5%

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

   20. sqr-neg 5.5%

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

   21. sqrt-unprod 0.0%

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

   22. add-sqr-sqrt 18.7%

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

8. Applied egg-rr 18.7%

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

Alternative 5: 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 7.4%

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

   1. flip-- 8.5%

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

   2. div-inv 8.5%

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

   3. add-sqr-sqrt 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 9.8%

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

   5. associate--l+ 9.8%

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

4. Applied egg-rr 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-rgt-identity 9.8%

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

   3. +-commutative 9.8%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

   1. pow1/2 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. add-cube-cbrt 99.1%

      \[\leadsto \frac{1}{{\left(\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot 1\right)}^{0.5} + \sqrt{1 + x}} \]

   4. associate-*l* 99.1%

      \[\leadsto \frac{1}{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot 1\right)\right)}}^{0.5} + \sqrt{1 + x}} \]

   5. unpow-prod-down 99.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{0.5} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5}} + \sqrt{1 + x}} \]

   6. pow2 99.1%

      \[\leadsto \frac{1}{{\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}}^{0.5} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5} + \sqrt{1 + x}} \]

   7. pow-pow 99.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(2 \cdot 0.5\right)}} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5} + \sqrt{1 + x}} \]

   8. metadata-eval 99.1%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{1}} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5} + \sqrt{1 + x}} \]

   9. pow1 99.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5} + \sqrt{1 + x}} \]

8. Applied egg-rr 99.1%

   \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x} \cdot {\left(\sqrt[3]{x} \cdot 1\right)}^{0.5}} + \sqrt{1 + x}} \]
9. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x} \cdot 1\right)}^{0.5} \cdot \sqrt[3]{x}} + \sqrt{1 + x}} \]

   2. *-rgt-identity 99.1%

      \[\leadsto \frac{1}{{\color{blue}{\left(\sqrt[3]{x}\right)}}^{0.5} \cdot \sqrt[3]{x} + \sqrt{1 + x}} \]

   3. pow-plus 99.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(0.5 + 1\right)}} + \sqrt{1 + x}} \]

   4. metadata-eval 99.1%

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

10. Simplified 99.1%

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

12. Applied egg-rr 0.8%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{-1}{x - \left(x + 1\right)}}\right)} - 1} \]
13. Step-by-step derivation

    1. log1p-undefine 0.8%

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

    2. rem-exp-log 0.8%

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

    3. associate--r+ 7.1%

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

    4. +-inverses 7.1%

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

    5. metadata-eval 7.1%

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

    6. metadata-eval 7.1%

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

    7. metadata-eval 7.1%

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

    8. metadata-eval 7.1%

       \[\leadsto \color{blue}{2} - 1 \]

    9. metadata-eval 7.1%

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

14. Simplified 7.1%

    \[\leadsto \color{blue}{1} \]
15. 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]

Reproduce

herbie shell --seed 2024117 
(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))))

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