2isqrt (example 3.6)

Percentage Accurate: 38.4% → 99.6%

Time: 13.0s

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: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{fma}\left(1 + x, {x}^{-0.5}, \sqrt{1 + x}\right)}}{x} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(1.0 / fma(Float64(1.0 + x), (x ^ -0.5), sqrt(Float64(1.0 + x)))) / x)
end

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

   1. associate-/r/ 40.1%

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

   2. frac-sub 40.8%

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

   3. frac-times 40.8%

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

   4. *-un-lft-identity 40.8%

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

6. Applied egg-rr 40.8%

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

   1. *-lft-identity 40.8%

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

   2. *-rgt-identity 40.8%

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

   3. associate--l+ 85.3%

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

   4. +-inverses 85.3%

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

   5. metadata-eval 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. fma-define 99.6%

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

   5. pow1 99.6%

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

   6. pow-prod-up 99.7%

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

   7. metadata-eval 99.7%

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

   8. pow1/2 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

   1. associate-/r/ 40.1%

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

   2. frac-sub 40.8%

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

   3. frac-times 40.8%

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

   4. *-un-lft-identity 40.8%

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

6. Applied egg-rr 40.8%

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

   1. *-lft-identity 40.8%

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

   2. *-rgt-identity 40.8%

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

   3. associate--l+ 85.3%

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

   4. +-inverses 85.3%

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

   5. metadata-eval 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. fma-define 99.6%

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

   5. pow1 99.6%

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

   6. pow-prod-up 99.7%

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

   7. metadata-eval 99.7%

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

   8. pow1/2 99.7%

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

10. Applied egg-rr 99.7%

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

    1. fma-define 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

   1. associate-/r/ 40.1%

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

   2. frac-sub 40.8%

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

   3. frac-times 40.8%

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

   4. *-un-lft-identity 40.8%

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

6. Applied egg-rr 40.8%

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

   1. *-lft-identity 40.8%

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

   2. *-rgt-identity 40.8%

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

   3. associate--l+ 85.3%

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

   4. +-inverses 85.3%

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

   5. metadata-eval 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. distribute-lft-in 98.8%

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

   2. pow1 98.8%

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

   3. pow-prod-up 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow1/2 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-1.5}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x -1.5) 2.0))

C

double code(double x) {
	return pow(x, -1.5) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-1.5d0)) / 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.5) / 2.0;
}

Python

def code(x):
	return math.pow(x, -1.5) / 2.0

Julia

function code(x)
	return Float64((x ^ -1.5) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -1.5) / 2.0;
end

Wolfram

code[x_] := N[(N[Power[x, -1.5], $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

5. Taylor expanded in x around inf 67.3%

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

   1. *-commutative 67.3%

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

7. Simplified 67.3%

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

   1. associate-/r* 67.3%

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

   2. sqrt-pow1 97.0%

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

   3. pow-flip 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{{x}^{\left(-\color{blue}{1.5}\right)}}{2} \]

   5. metadata-eval 98.0%

      \[\leadsto \frac{{x}^{\color{blue}{-1.5}}}{2} \]

9. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{{x}^{-1.5}}{2}} \]
10. Add Preprocessing

Alternative 5: 96.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

5. Taylor expanded in x around inf 67.3%

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

   1. *-commutative 67.3%

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

7. Simplified 67.3%

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

   1. *-commutative 67.3%

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

   2. associate-/r* 67.3%

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

   3. metadata-eval 67.3%

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

   4. sqrt-pow1 97.0%

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

   5. metadata-eval 97.0%

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

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
10. 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 40.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. *-un-lft-identity 5.6%

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

5. Applied egg-rr 5.6%

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

   1. *-lft-identity 5.6%

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

7. Simplified 5.6%

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

Alternative 7: 2.0% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x \cdot 0}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (* x 0.0)) x))

C

double code(double x) {
	return (1.0 / (x * 0.0)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x * 0.0d0)) / x
end function

Java

public static double code(double x) {
	return (1.0 / (x * 0.0)) / x;
}

Python

def code(x):
	return (1.0 / (x * 0.0)) / x

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x * 0.0)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x * 0.0)) / x;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. inv-pow 40.1%

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

   7. sqrt-pow2 40.1%

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

   8. +-commutative 40.1%

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

   9. metadata-eval 40.1%

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

   10. frac-times 22.4%

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

   11. metadata-eval 22.4%

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

   12. add-sqr-sqrt 18.0%

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

   13. frac-times 25.8%

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

   14. metadata-eval 25.8%

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

   15. add-sqr-sqrt 40.1%

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

   16. +-commutative 40.1%

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

4. Applied egg-rr 40.1%

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

   1. associate-/r/ 40.1%

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

   2. frac-sub 40.8%

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

   3. frac-times 40.8%

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

   4. *-un-lft-identity 40.8%

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

6. Applied egg-rr 40.8%

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

   1. *-lft-identity 40.8%

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

   2. *-rgt-identity 40.8%

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

   3. associate--l+ 85.3%

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

   4. +-inverses 85.3%

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

   5. metadata-eval 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*l* 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. fma-define 99.6%

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

   5. pow1 99.6%

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

   6. pow-prod-up 99.7%

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

   7. metadata-eval 99.7%

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

   8. pow1/2 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in x around -inf 0.0%

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

    1. associate-*r* 0.0%

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

    2. neg-mul-1 0.0%

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

    3. distribute-lft1-in 0.0%

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

    4. metadata-eval 0.0%

       \[\leadsto \frac{\frac{1}{\left(-x\right) \cdot \left(\color{blue}{0} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{x} \]

    5. mul0-lft 1.2%

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

    6. distribute-lft-neg-in 1.2%

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

    7. distribute-rgt-neg-in 1.2%

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

    8. metadata-eval 2.0%

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

13. Simplified 2.0%

    \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot 0}}}{x} \]
14. 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 2024095 
(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)))))