2isqrt (example 3.6)

Percentage Accurate: 38.3% → 98.8%

Time: 9.3s

Alternatives: 6

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: 38.3% 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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around inf 85.2%

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

   1. distribute-rgt-in 85.2%

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

   2. *-un-lft-identity 85.2%

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

   3. distribute-lft-in 85.2%

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

   4. pow-flip 85.2%

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

   5. sqrt-pow1 85.2%

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

   6. metadata-eval 85.2%

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

   7. metadata-eval 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{\color{blue}{-2.5}} + -0.5 \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   8. *-commutative 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\color{blue}{\left(x \cdot 0.5\right)} \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   9. pow-flip 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{{x}^{\left(-5\right)}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   10. sqrt-pow1 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{x}^{\left(\frac{-5}{2}\right)}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   11. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\left(\frac{\color{blue}{-5}}{2}\right)}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   12. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\color{blue}{-2.5}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.2%

   \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{-2.5}\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. associate-*r* 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + \color{blue}{\left(-0.5 \cdot \left(x \cdot 0.5\right)\right) \cdot {x}^{-2.5}}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. distribute-rgt-out 85.2%

      \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \left(x \cdot 0.5\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. associate-*r* 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{\left(-0.5 \cdot 0.5\right) \cdot x}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{-0.25} \cdot x\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{x \cdot -0.25}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.2%

   \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + x \cdot -0.25\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

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

Alternative 2: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around inf 85.2%

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

   1. distribute-rgt-in 85.2%

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

   2. *-un-lft-identity 85.2%

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

   3. distribute-lft-in 85.2%

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

   4. pow-flip 85.2%

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

   5. sqrt-pow1 85.2%

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

   6. metadata-eval 85.2%

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

   7. metadata-eval 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{\color{blue}{-2.5}} + -0.5 \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   8. *-commutative 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\color{blue}{\left(x \cdot 0.5\right)} \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   9. pow-flip 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{{x}^{\left(-5\right)}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   10. sqrt-pow1 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{x}^{\left(\frac{-5}{2}\right)}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   11. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\left(\frac{\color{blue}{-5}}{2}\right)}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   12. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\color{blue}{-2.5}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.2%

   \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{-2.5}\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. associate-*r* 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + \color{blue}{\left(-0.5 \cdot \left(x \cdot 0.5\right)\right) \cdot {x}^{-2.5}}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. distribute-rgt-out 85.2%

      \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \left(x \cdot 0.5\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. associate-*r* 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{\left(-0.5 \cdot 0.5\right) \cdot x}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{-0.25} \cdot x\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{x \cdot -0.25}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.2%

   \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + x \cdot -0.25\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

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

    1. div-inv 98.7%

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

    2. *-commutative 98.7%

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

    3. distribute-lft-out-- 98.7%

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

12. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.8%

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

    2. *-rgt-identity 98.8%

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

14. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x}} \]
15. Add Preprocessing

Alternative 3: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around inf 85.2%

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

   1. distribute-rgt-in 85.2%

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

   2. *-un-lft-identity 85.2%

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

   3. distribute-lft-in 85.2%

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

   4. pow-flip 85.2%

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

   5. sqrt-pow1 85.2%

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

   6. metadata-eval 85.2%

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

   7. metadata-eval 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{\color{blue}{-2.5}} + -0.5 \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   8. *-commutative 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\color{blue}{\left(x \cdot 0.5\right)} \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   9. pow-flip 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{{x}^{\left(-5\right)}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   10. sqrt-pow1 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{x}^{\left(\frac{-5}{2}\right)}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   11. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\left(\frac{\color{blue}{-5}}{2}\right)}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   12. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\color{blue}{-2.5}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.2%

   \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{-2.5}\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. associate-*r* 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + \color{blue}{\left(-0.5 \cdot \left(x \cdot 0.5\right)\right) \cdot {x}^{-2.5}}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. distribute-rgt-out 85.2%

      \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \left(x \cdot 0.5\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. associate-*r* 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{\left(-0.5 \cdot 0.5\right) \cdot x}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{-0.25} \cdot x\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{x \cdot -0.25}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.2%

   \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + x \cdot -0.25\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

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

    1. div-inv 98.7%

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

    2. *-commutative 98.7%

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

    3. distribute-lft-out-- 98.7%

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

12. Applied egg-rr 98.7%

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

    1. associate-*l* 98.7%

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

    2. associate-*r/ 98.7%

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

    3. *-rgt-identity 98.7%

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

    4. sub-neg 98.7%

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

    5. distribute-neg-frac 98.7%

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

    6. metadata-eval 98.7%

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

14. Simplified 98.7%

    \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{0.5 + \frac{-0.375}{x}}{x}} \]
15. Add Preprocessing

Alternative 4: 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 38.6%

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

3. Taylor expanded in x around inf 85.2%

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

   1. distribute-rgt-in 85.2%

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

   2. *-un-lft-identity 85.2%

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

   3. distribute-lft-in 85.2%

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

   4. pow-flip 85.2%

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

   5. sqrt-pow1 85.2%

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

   6. metadata-eval 85.2%

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

   7. metadata-eval 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{\color{blue}{-2.5}} + -0.5 \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   8. *-commutative 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\color{blue}{\left(x \cdot 0.5\right)} \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   9. pow-flip 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{{x}^{\left(-5\right)}}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   10. sqrt-pow1 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot \color{blue}{{x}^{\left(\frac{-5}{2}\right)}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   11. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\left(\frac{\color{blue}{-5}}{2}\right)}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   12. metadata-eval 85.2%

       \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{\color{blue}{-2.5}}\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

5. Applied egg-rr 85.2%

   \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot {x}^{-2.5} + -0.5 \cdot \left(\left(x \cdot 0.5\right) \cdot {x}^{-2.5}\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. associate-*r* 85.2%

      \[\leadsto \frac{\left(-0.5 \cdot {x}^{-2.5} + \color{blue}{\left(-0.5 \cdot \left(x \cdot 0.5\right)\right) \cdot {x}^{-2.5}}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   2. distribute-rgt-out 85.2%

      \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \left(x \cdot 0.5\right)\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   3. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + -0.5 \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   4. associate-*r* 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{\left(-0.5 \cdot 0.5\right) \cdot x}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   5. metadata-eval 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{-0.25} \cdot x\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

   6. *-commutative 85.2%

      \[\leadsto \frac{{x}^{-2.5} \cdot \left(-0.5 + \color{blue}{x \cdot -0.25}\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

7. Simplified 85.2%

   \[\leadsto \frac{\color{blue}{{x}^{-2.5} \cdot \left(-0.5 + x \cdot -0.25\right)} - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}} \]

8. Taylor expanded in x around inf 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in x around inf 97.9%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]
12. Add Preprocessing

Alternative 5: 36.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot x\right)}^{-0.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* x x) -0.25))

C

double code(double x) {
	return pow((x * x), -0.25);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) ** (-0.25d0)
end function

Java

public static double code(double x) {
	return Math.pow((x * x), -0.25);
}

Python

def code(x):
	return math.pow((x * x), -0.25)

Julia

function code(x)
	return Float64(x * x) ^ -0.25
end

MATLAB

function tmp = code(x)
	tmp = (x * x) ^ -0.25;
end

Wolfram

code[x_] := N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0 5.6%

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

   1. inv-pow 5.6%

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

   2. sqrt-pow1 5.6%

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

   3. metadata-eval 5.6%

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

   4. sqr-pow 5.6%

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

   5. pow-prod-down 37.2%

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

   6. pow2 37.2%

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

   7. metadata-eval 37.2%

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

5. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Applied egg-rr 37.2%

   \[\leadsto {\color{blue}{\left(x \cdot x\right)}}^{-0.25} \]
8. Add Preprocessing

Alternative 6: 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 38.6%

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

   1. sub-neg 38.6%

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

   2. inv-pow 38.6%

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

   3. sqrt-pow2 27.4%

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

   4. metadata-eval 27.4%

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

   5. distribute-neg-frac 27.4%

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

   6. metadata-eval 27.4%

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

   7. +-commutative 27.4%

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

4. Applied egg-rr 27.4%

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

   1. *-rgt-identity 27.4%

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

   2. metadata-eval 27.4%

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

   3. distribute-rgt-neg-in 27.4%

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

   4. sub-neg 27.4%

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

   5. associate-*l/ 27.4%

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

   6. metadata-eval 27.4%

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

   7. unpow1/2 27.4%

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

   8. exp-to-pow 6.9%

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

   9. log1p-undefine 6.9%

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

   10. *-commutative 6.9%

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

   11. exp-neg 6.9%

       \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]

   12. *-commutative 6.9%

       \[\leadsto {x}^{-0.5} - e^{-\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.5}} \]

   13. distribute-rgt-neg-in 6.9%

       \[\leadsto {x}^{-0.5} - e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(-0.5\right)}} \]

   14. log1p-undefine 6.9%

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

   15. metadata-eval 6.9%

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

   16. exp-to-pow 38.6%

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

6. Simplified 38.6%

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

7. Taylor expanded in x around 0 5.6%

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

   1. unpow-1 5.6%

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

   2. metadata-eval 5.6%

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

   3. pow-sqr 5.6%

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

   4. rem-sqrt-square 5.6%

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

   5. metadata-eval 5.6%

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

   6. pow-sqr 5.6%

      \[\leadsto \left|\color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}}\right| \]

   7. fabs-sqr 5.6%

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

   8. pow-sqr 5.6%

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

   9. metadata-eval 5.6%

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

9. Simplified 5.6%

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

Developer Target 1: 98.3% 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]

Developer Target 2: 38.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
end

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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