2isqrt (example 3.6)

Percentage Accurate: 38.5% → 98.7%

Time: 13.9s

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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left({x}^{-0.5} + {x}^{-2.5}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (* 0.5 (+ (pow x -0.5) (pow x -2.5))) (* (pow x -0.5) (/ 0.375 x))) x))

C

double code(double x) {
	return ((0.5 * (pow(x, -0.5) + pow(x, -2.5))) - (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 ** (-2.5d0)))) - ((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, -2.5))) - (Math.pow(x, -0.5) * (0.375 / x))) / x;
}

Python

def code(x):
	return ((0.5 * (math.pow(x, -0.5) + math.pow(x, -2.5))) - (math.pow(x, -0.5) * (0.375 / x))) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.5 * Float64((x ^ -0.5) + (x ^ -2.5))) - Float64((x ^ -0.5) * Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 * ((x ^ -0.5) + (x ^ -2.5))) - ((x ^ -0.5) * (0.375 / x))) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[x, -2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf 79.3%

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

4. Taylor expanded in x around 0 79.2%

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

   1. unpow-1 79.2%

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

   2. exp-to-pow 79.2%

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

   3. *-commutative 79.2%

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

   4. exp-prod 79.2%

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

   5. *-commutative 79.2%

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

   6. associate-*r* 79.2%

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

   7. metadata-eval 79.2%

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

   8. *-commutative 79.2%

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

   9. exp-to-pow 79.2%

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

   10. metadata-eval 79.2%

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

   11. pow-sqr 79.2%

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

   12. rem-sqrt-square 79.2%

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

   13. rem-square-sqrt 79.2%

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

   14. fabs-sqr 79.2%

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

   15. rem-square-sqrt 79.2%

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

6. Simplified 79.2%

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

7. Taylor expanded in x around inf 98.5%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + \left(0.5 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)}{x}} \]
8. Step-by-step derivation

9. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left({x}^{-0.5} + \sqrt{\frac{1}{{x}^{5}}}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x}} \]
10. Step-by-step derivation

    1. distribute-lft-in 98.5%

       \[\leadsto \frac{\color{blue}{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

    2. pow-flip 98.5%

       \[\leadsto \frac{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{\color{blue}{{x}^{\left(-5\right)}}}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

    3. sqrt-pow1 98.5%

       \[\leadsto \frac{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot \color{blue}{{x}^{\left(\frac{-5}{2}\right)}}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

    4. metadata-eval 98.5%

       \[\leadsto \frac{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot {x}^{\left(\frac{\color{blue}{-5}}{2}\right)}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

    5. metadata-eval 98.5%

       \[\leadsto \frac{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot {x}^{\color{blue}{-2.5}}\right) - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

11. Applied egg-rr 98.5%

    \[\leadsto \frac{\color{blue}{\left(0.5 \cdot {x}^{-0.5} + 0.5 \cdot {x}^{-2.5}\right)} - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]
12. Step-by-step derivation

    1. distribute-lft-in 98.5%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \left({x}^{-0.5} + {x}^{-2.5}\right)} - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]

13. Simplified 98.5%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \left({x}^{-0.5} + {x}^{-2.5}\right)} - {x}^{-0.5} \cdot \frac{0.375}{x}}{x} \]
14. Add Preprocessing

Alternative 2: 98.6% 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 35.5%

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

3. Taylor expanded in x around inf 79.3%

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

4. Taylor expanded in x around 0 79.2%

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

   1. unpow-1 79.2%

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

   2. exp-to-pow 79.2%

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

   3. *-commutative 79.2%

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

   4. exp-prod 79.2%

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

   5. *-commutative 79.2%

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

   6. associate-*r* 79.2%

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

   7. metadata-eval 79.2%

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

   8. *-commutative 79.2%

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

   9. exp-to-pow 79.2%

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

   10. metadata-eval 79.2%

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

   11. pow-sqr 79.2%

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

   12. rem-sqrt-square 79.2%

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

   13. rem-square-sqrt 79.2%

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

   14. fabs-sqr 79.2%

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

   15. rem-square-sqrt 79.2%

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

6. Simplified 79.2%

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

7. Taylor expanded in x around inf 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \left({x}^{-1.5} - {x}^{-0.5}\right)}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf 77.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 77.9%

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

5. Simplified 77.9%

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

   1. *-commutative 77.9%

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

   2. unpow2 77.9%

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

   3. times-frac 97.0%

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

   4. inv-pow 97.0%

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

   5. sqrt-pow1 97.0%

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

   6. metadata-eval 97.0%

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

7. Applied egg-rr 97.0%

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

   1. associate-*r/ 97.1%

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

   2. div-sub 97.1%

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

   3. pow1 97.1%

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

   4. pow-div 97.1%

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

   5. metadata-eval 97.1%

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

   6. pow1/2 97.1%

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

   7. pow1 97.1%

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

   8. pow-div 97.2%

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

   9. metadata-eval 97.2%

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

9. Applied egg-rr 97.2%

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

10. Final simplification 97.2%

    \[\leadsto \frac{-0.5 \cdot \left({x}^{-1.5} - {x}^{-0.5}\right)}{x} \]
11. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({x}^{-1.5} - {x}^{-0.5}\right) \cdot \frac{-0.5}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf 77.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 77.9%

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

5. Simplified 77.9%

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

   1. *-commutative 77.9%

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

   2. unpow2 77.9%

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

   3. times-frac 97.0%

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

   4. inv-pow 97.0%

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

   5. sqrt-pow1 97.0%

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

   6. metadata-eval 97.0%

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

7. Applied egg-rr 97.0%

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

   1. div-sub 97.0%

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

   2. pow1 97.0%

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

   3. pow-div 97.0%

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

   4. metadata-eval 97.0%

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

   5. pow1/2 97.0%

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

   6. pow1 97.0%

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

   7. pow-div 97.2%

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

   8. metadata-eval 97.2%

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

9. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\left({x}^{-1.5} - {x}^{-0.5}\right)} \cdot \frac{-0.5}{x} \]
10. Add Preprocessing

Alternative 5: 97.5% 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 35.5%

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

3. Taylor expanded in x around inf 77.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 77.9%

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

5. Simplified 77.9%

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

   1. *-commutative 77.9%

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

   2. unpow2 77.9%

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

   3. times-frac 97.0%

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

   4. inv-pow 97.0%

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

   5. sqrt-pow1 97.0%

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

   6. metadata-eval 97.0%

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

7. Applied egg-rr 97.0%

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

   1. associate-*l/ 97.0%

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

9. Applied egg-rr 97.0%

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

10. Taylor expanded in x around inf 97.0%

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

Alternative 6: 97.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf 77.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 77.9%

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

5. Simplified 77.9%

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

   1. *-commutative 77.9%

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

   2. unpow2 77.9%

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

   3. times-frac 97.0%

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

   4. inv-pow 97.0%

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

   5. sqrt-pow1 97.0%

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

   6. metadata-eval 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around inf 96.9%

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

   1. mul-1-neg 96.9%

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

   2. unpow1/2 96.9%

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

   3. exp-to-pow 92.8%

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

   4. log-rec 92.8%

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

   5. distribute-lft-neg-out 92.8%

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

   6. distribute-rgt-neg-in 92.8%

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

   7. metadata-eval 92.8%

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

   8. exp-to-pow 97.0%

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

10. Simplified 97.0%

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

11. Final simplification 97.0%

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

Alternative 7: 35.5% 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 35.5%

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

   1. sub-neg 35.5%

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

   2. +-commutative 35.5%

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

   3. add-cube-cbrt 13.5%

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

   4. distribute-lft-neg-in 13.5%

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

   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 8.9%

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

5. Taylor expanded in x around inf 31.7%

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

   1. distribute-rgt1-in 31.7%

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

   2. metadata-eval 31.7%

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

   3. unpow-1 31.7%

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

   4. metadata-eval 31.7%

      \[\leadsto 0 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   5. pow-sqr 31.7%

      \[\leadsto 0 \cdot \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \]

   6. rem-sqrt-square 31.7%

      \[\leadsto 0 \cdot \color{blue}{\left|{x}^{-0.5}\right|} \]

   7. metadata-eval 31.7%

      \[\leadsto 0 \cdot \left|{x}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

   8. pow-sqr 31.7%

      \[\leadsto 0 \cdot \left|\color{blue}{{x}^{-0.25} \cdot {x}^{-0.25}}\right| \]

   9. fabs-sqr 31.7%

      \[\leadsto 0 \cdot \color{blue}{\left({x}^{-0.25} \cdot {x}^{-0.25}\right)} \]

   10. pow-sqr 31.7%

       \[\leadsto 0 \cdot \color{blue}{{x}^{\left(2 \cdot -0.25\right)}} \]

   11. metadata-eval 31.7%

       \[\leadsto 0 \cdot {x}^{\color{blue}{-0.5}} \]

   12. mul0-lft 31.7%

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

7. Simplified 31.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer target: 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 2024114 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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