2isqrt (example 3.6)

Percentage Accurate: 38.8% → 98.7%

Time: 9.2s

Alternatives: 4

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: 38.8% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - 0.375 \cdot {x}^{-1.5}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (* (sqrt (/ 1.0 x)) 0.5) (* 0.375 (pow x -1.5))) x))

C

double code(double x) {
	return ((sqrt((1.0 / x)) * 0.5) - (0.375 * pow(x, -1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt((1.0d0 / x)) * 0.5d0) - (0.375d0 * (x ** (-1.5d0)))) / x
end function

Java

public static double code(double x) {
	return ((Math.sqrt((1.0 / x)) * 0.5) - (0.375 * Math.pow(x, -1.5))) / x;
}

Python

def code(x):
	return ((math.sqrt((1.0 / x)) * 0.5) - (0.375 * math.pow(x, -1.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) - Float64(0.375 * (x ^ -1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((1.0 / x)) * 0.5) - (0.375 * (x ^ -1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.375 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 37.5%

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

3. Taylor expanded in x around inf 80.2%

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

4. Taylor expanded in x around inf 98.6%

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

   1. +-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

   4. *-commutative 98.6%

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

   5. distribute-rgt-out 98.6%

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

   6. metadata-eval 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

6. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}}{x}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}}}{x} \]

   2. div-inv 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l* 98.6%

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

   5. add-sqr-sqrt 98.6%

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

   6. cube-mult 98.6%

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

   7. inv-pow 98.6%

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

   8. sqrt-pow1 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - 1 \cdot \left(0.375 \cdot {\color{blue}{\left({x}^{\left(\frac{-1}{2}\right)}\right)}}^{3}\right)}{x} \]

   9. metadata-eval 98.6%

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

   10. pow-pow 98.6%

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

   11. metadata-eval 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{0.375 \cdot {x}^{-1.5}}}{x} \]

10. Simplified 98.6%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{0.375 \cdot {x}^{-1.5}}}{x} \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

3. Taylor expanded in x around inf 80.2%

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

4. Taylor expanded in x around inf 98.6%

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

   1. +-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

   4. *-commutative 98.6%

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

   5. distribute-rgt-out 98.6%

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

   6. metadata-eval 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

6. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}}{x}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{x}}}{x} \]

   2. div-inv 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l* 98.6%

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

   5. add-sqr-sqrt 98.6%

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

   6. cube-mult 98.6%

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

   7. inv-pow 98.6%

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

   8. sqrt-pow1 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - 1 \cdot \left(0.375 \cdot {\color{blue}{\left({x}^{\left(\frac{-1}{2}\right)}\right)}}^{3}\right)}{x} \]

   9. metadata-eval 98.6%

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

   10. pow-pow 98.6%

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

   11. metadata-eval 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{0.375 \cdot {x}^{-1.5}}}{x} \]

10. Simplified 98.6%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{0.375 \cdot {x}^{-1.5}}}{x} \]
11. Step-by-step derivation

    1. sqrt-div 98.5%

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \cdot 0.5 - 0.375 \cdot {x}^{-1.5}}{x} \]

    2. metadata-eval 98.5%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt{x}} \cdot 0.5 - 0.375 \cdot {x}^{-1.5}}{x} \]

    3. associate-*l/ 98.5%

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot 0.5}{\sqrt{x}}} - 0.375 \cdot {x}^{-1.5}}{x} \]

    4. metadata-eval 98.5%

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

12. Applied egg-rr 98.5%

    \[\leadsto \frac{\color{blue}{\frac{0.5}{\sqrt{x}}} - 0.375 \cdot {x}^{-1.5}}{x} \]
13. Add Preprocessing

Alternative 3: 98.0% accurate, 2.0× speedup

Math

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

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 37.5%

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

3. Taylor expanded in x around inf 66.9%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 56.3%

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

   2. pow1/3 54.8%

      \[\leadsto \color{blue}{{\left(\left(\left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \cdot \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right) \cdot \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right)}^{0.3333333333333333}} \]

   3. pow3 54.8%

      \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. *-commutative 54.8%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. pow-flip 54.8%

      \[\leadsto {\left({\left(\sqrt{\color{blue}{{x}^{\left(-3\right)}}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   6. sqrt-pow1 54.8%

      \[\leadsto {\left({\left(\color{blue}{{x}^{\left(\frac{-3}{2}\right)}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   7. metadata-eval 54.8%

      \[\leadsto {\left({\left({x}^{\left(\frac{\color{blue}{-3}}{2}\right)} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

   8. metadata-eval 54.8%

      \[\leadsto {\left({\left({x}^{\color{blue}{-1.5}} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333} \]

5. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{{\left({\left({x}^{-1.5} \cdot 0.5\right)}^{3}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 56.4%

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

   2. rem-cbrt-cube 97.4%

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

   3. *-commutative 97.4%

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

7. Simplified 97.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]
8. Add Preprocessing

Alternative 4: 36.0% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. add-cube-cbrt 13.6%

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

   4. distribute-lft-neg-in 13.6%

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

   5. fma-define 8.5%

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

4. Applied egg-rr 8.8%

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

5. Taylor expanded in x around inf 33.9%

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

   1. distribute-rgt1-in 33.9%

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

   2. metadata-eval 33.9%

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

   3. mul0-lft 33.9%

      \[\leadsto \color{blue}{0} \]

7. Simplified 33.9%

   \[\leadsto \color{blue}{0} \]
8. 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 2024117 
(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)))))