2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.1%

Time: 9.6s

Alternatives: 7

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x))))
  (* x (- (/ (- 0.5 (/ 0.125 x)) x) -1.0))))

C

double code(double x) {
	return (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) / (x * (((0.5d0 - (0.125d0 / x)) / x) - (-1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Python

def code(x):
	return (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) / Float64(x * Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) - -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. frac-sub 38.5%

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

   2. *-rgt-identity 38.5%

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

   3. *-un-lft-identity 38.5%

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

   4. +-commutative 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{-\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}} \]

   3. distribute-rgt-neg-in 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left(-x\right)}} \]

7. Simplified 38.5%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)}} \]
8. Step-by-step derivation

   1. flip-- 39.1%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   2. add-sqr-sqrt 22.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   3. add-sqr-sqrt 39.7%

      \[\leadsto \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

9. Applied egg-rr 39.7%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]
10. Step-by-step derivation

    1. associate--l+ 99.5%

       \[\leadsto \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

    2. +-inverses 99.5%

       \[\leadsto \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

    3. metadata-eval 99.5%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

    4. +-commutative 99.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

11. Simplified 99.5%

    \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

12. Final simplification 99.5%

    \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-0.5} - {x}^{-1.5} \cdot 0.375}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (* 0.5 (pow x -0.5)) (* (pow x -1.5) 0.375)) x))

C

double code(double x) {
	return ((0.5 * pow(x, -0.5)) - (pow(x, -1.5) * 0.375)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 * (x ** (-0.5d0))) - ((x ** (-1.5d0)) * 0.375d0)) / x
end function

Java

public static double code(double x) {
	return ((0.5 * Math.pow(x, -0.5)) - (Math.pow(x, -1.5) * 0.375)) / x;
}

Python

def code(x):
	return ((0.5 * math.pow(x, -0.5)) - (math.pow(x, -1.5) * 0.375)) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 * (x ^ -0.5)) - Float64((x ^ -1.5) * 0.375)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 * (x ^ -0.5)) - ((x ^ -1.5) * 0.375)) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -1.5], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around inf 80.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

6. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot 0.5 - {\left({x}^{-0.5}\right)}^{3} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 99.3%

      \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \color{blue}{\left(1 \cdot {\left({x}^{-0.5}\right)}^{3}\right)} \cdot 0.375}{x} \]

   2. pow-pow 99.3%

      \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \left(1 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right) \cdot 0.375}{x} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.375}{x} \]

8. Applied egg-rr 99.3%

   \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.375}{x} \]
9. Step-by-step derivation

   1. *-lft-identity 99.3%

      \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \color{blue}{{x}^{-1.5}} \cdot 0.375}{x} \]

10. Simplified 99.3%

    \[\leadsto \frac{{x}^{-0.5} \cdot 0.5 - \color{blue}{{x}^{-1.5}} \cdot 0.375}{x} \]

11. Final simplification 99.3%

    \[\leadsto \frac{0.5 \cdot {x}^{-0.5} - {x}^{-1.5} \cdot 0.375}{x} \]
12. Add Preprocessing

Alternative 3: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 0.5 (sqrt (/ 1.0 x))) (* x (- (/ (- 0.5 (/ 0.125 x)) x) -1.0))))

C

double code(double x) {
	return (0.5 * sqrt((1.0 / x))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * sqrt((1.0d0 / x))) / (x * (((0.5d0 - (0.125d0 / x)) / x) - (-1.0d0)))
end function

Java

public static double code(double x) {
	return (0.5 * Math.sqrt((1.0 / x))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Python

def code(x):
	return (0.5 * math.sqrt((1.0 / x))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0))

Julia

function code(x)
	return Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / Float64(x * Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) - -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * sqrt((1.0 / x))) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
end

Wolfram

code[x_] := N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. frac-sub 38.5%

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

   2. *-rgt-identity 38.5%

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

   3. *-un-lft-identity 38.5%

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

   4. +-commutative 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{-\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}} \]

   3. distribute-rgt-neg-in 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left(-x\right)}} \]

7. Simplified 38.5%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)}} \]

8. Taylor expanded in x around inf 98.6%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]
9. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

10. Simplified 98.6%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

11. Final simplification 98.6%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \]
12. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-0.5}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 0.5 (pow x -0.5)) (* x (- (/ (- 0.5 (/ 0.125 x)) x) -1.0))))

C

double code(double x) {
	return (0.5 * pow(x, -0.5)) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-0.5d0))) / (x * (((0.5d0 - (0.125d0 / x)) / x) - (-1.0d0)))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -0.5)) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
}

Python

def code(x):
	return (0.5 * math.pow(x, -0.5)) / (x * (((0.5 - (0.125 / x)) / x) - -1.0))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -0.5)) / Float64(x * Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) - -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -0.5)) / (x * (((0.5 - (0.125 / x)) / x) - -1.0));
end

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. frac-sub 38.5%

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

   2. *-rgt-identity 38.5%

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

   3. *-un-lft-identity 38.5%

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

   4. +-commutative 38.5%

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

   5. sqrt-unprod 38.5%

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

   6. +-commutative 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{-x \cdot \left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right)}} \]

   2. *-commutative 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{-\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}} \]

   3. distribute-rgt-neg-in 0.0%

      \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x \cdot {\left(\sqrt{-1}\right)}^{2}}}{x} + {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left(-x\right)}} \]

7. Simplified 38.5%

   \[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)}} \]

8. Taylor expanded in x around inf 98.6%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]
9. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   2. unpow1/2 98.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   3. rem-exp-log 94.4%

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   4. exp-neg 94.5%

      \[\leadsto \frac{{\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   5. exp-prod 94.5%

      \[\leadsto \frac{\color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   6. distribute-lft-neg-out 94.5%

      \[\leadsto \frac{e^{\color{blue}{-\log x \cdot 0.5}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   7. distribute-rgt-neg-in 94.5%

      \[\leadsto \frac{e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   8. metadata-eval 94.5%

      \[\leadsto \frac{e^{\log x \cdot \color{blue}{-0.5}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

   9. exp-to-pow 98.6%

      \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot 0.5}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

10. Simplified 98.6%

    \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot 0.5}}{\left(-1 - \frac{0.5 + \frac{0.125}{-x}}{x}\right) \cdot \left(-x\right)} \]

11. Final simplification 98.6%

    \[\leadsto \frac{0.5 \cdot {x}^{-0.5}}{x \cdot \left(\frac{0.5 - \frac{0.125}{x}}{x} - -1\right)} \]
12. Add Preprocessing

Alternative 5: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around inf 80.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. fma-define 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}}{x} \]

   2. distribute-rgt-out 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.125 + 0.5\right)}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   4. *-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5}\right)}{x} \]

6. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]

7. Taylor expanded in x around inf 98.5%

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

   1. *-commutative 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
11. Add Preprocessing

Alternative 6: 96.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{{x}^{1.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (pow x 1.5)))

C

double code(double x) {
	return 0.5 / pow(x, 1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (x ** 1.5d0)
end function

Java

public static double code(double x) {
	return 0.5 / Math.pow(x, 1.5);
}

Python

def code(x):
	return 0.5 / math.pow(x, 1.5)

Julia

function code(x)
	return Float64(0.5 / (x ^ 1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x ^ 1.5);
end

Wolfram

code[x_] := N[(0.5 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. flip-- 38.4%

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

   2. clear-num 38.4%

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

   3. inv-pow 38.4%

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

   4. sqrt-pow2 38.4%

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

   5. metadata-eval 38.4%

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

   6. pow1/2 38.4%

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

   7. pow-flip 38.4%

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

   8. +-commutative 38.4%

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

   9. metadata-eval 38.4%

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

   10. frac-times 21.2%

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

   11. metadata-eval 21.2%

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

   12. add-sqr-sqrt 21.4%

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

   13. frac-times 24.6%

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

   14. metadata-eval 24.6%

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

   15. add-sqr-sqrt 38.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in x around inf 60.6%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 60.6%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

   2. associate-/r* 60.6%

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

   3. metadata-eval 60.6%

      \[\leadsto 1 \cdot \frac{\color{blue}{0.5}}{\sqrt{{x}^{3}}} \]

   4. sqrt-pow1 96.8%

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

   5. metadata-eval 96.8%

      \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

7. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
8. Step-by-step derivation

   1. *-lft-identity 96.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]

9. Simplified 96.8%

   \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
10. Add Preprocessing

Alternative 7: 5.6% 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 38.5%

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

   1. expm1-log1p-u 38.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-undefine 4.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. inv-pow 4.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. sqrt-pow2 4.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 4.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. log1p-undefine 4.7%

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

   2. rem-exp-log 4.7%

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

   3. +-commutative 4.7%

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

   4. associate--l+ 31.9%

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

   5. metadata-eval 31.9%

      \[\leadsto \left({x}^{-0.5} + \color{blue}{0}\right) - \frac{1}{\sqrt{x + 1}} \]

   6. +-rgt-identity 31.9%

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

6. Simplified 31.9%

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

7. Taylor expanded in x around 0 5.4%

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

   1. unpow1/2 5.4%

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

   2. rem-exp-log 5.4%

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

   3. exp-neg 5.4%

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

   4. exp-prod 5.4%

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

   5. distribute-lft-neg-out 5.4%

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

   6. distribute-rgt-neg-in 5.4%

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

   7. metadata-eval 5.4%

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

   8. exp-to-pow 5.4%

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

9. Simplified 5.4%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024121 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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