2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.6%

Time: 16.4s

Alternatives: 8

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. flip-- 33.0%

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

   2. div-inv 33.0%

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

   3. frac-times 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-sqr-sqrt 17.0%

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

   6. frac-times 24.5%

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

   7. metadata-eval 24.5%

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

   8. add-sqr-sqrt 33.1%

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

   9. +-commutative 33.1%

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

   10. pow1/2 33.1%

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

   11. pow-flip 33.1%

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

   12. metadata-eval 33.1%

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

   13. inv-pow 33.1%

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

   14. sqrt-pow2 33.1%

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

   15. +-commutative 33.1%

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

   16. metadata-eval 33.1%

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

4. Applied egg-rr 33.1%

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

   1. frac-2neg 33.1%

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

   2. metadata-eval 33.1%

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

   3. frac-sub 34.5%

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

   4. *-un-lft-identity 34.5%

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

   5. distribute-neg-in 34.5%

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

   6. metadata-eval 34.5%

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

   7. distribute-neg-in 34.5%

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

   8. metadata-eval 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r* 34.6%

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

   2. associate--l+ 81.2%

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

   3. *-commutative 81.2%

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

   4. neg-mul-1 81.2%

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

   5. +-inverses 81.2%

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

   6. metadata-eval 81.2%

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

   7. unsub-neg 81.2%

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

8. Simplified 81.2%

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

   1. associate-/l/ 80.5%

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

   2. associate-*l/ 80.5%

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

   3. sub-neg 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-neg-in 80.5%

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

   6. distribute-lft-neg-in 80.5%

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

   7. associate-*l/ 80.5%

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

   8. metadata-eval 80.5%

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

   9. frac-2neg 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-/r* 81.2%

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

   12. frac-times 99.6%

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

   13. *-un-lft-identity 99.6%

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

   14. frac-2neg 99.6%

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

   15. metadata-eval 99.6%

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

   16. distribute-neg-in 99.6%

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

   17. metadata-eval 99.6%

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

   18. sub-neg 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. flip-- 33.0%

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

   2. div-inv 33.0%

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

   3. frac-times 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-sqr-sqrt 17.0%

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

   6. frac-times 24.5%

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

   7. metadata-eval 24.5%

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

   8. add-sqr-sqrt 33.1%

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

   9. +-commutative 33.1%

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

   10. pow1/2 33.1%

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

   11. pow-flip 33.1%

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

   12. metadata-eval 33.1%

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

   13. inv-pow 33.1%

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

   14. sqrt-pow2 33.1%

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

   15. +-commutative 33.1%

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

   16. metadata-eval 33.1%

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

4. Applied egg-rr 33.1%

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

   1. frac-2neg 33.1%

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

   2. metadata-eval 33.1%

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

   3. frac-sub 34.5%

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

   4. *-un-lft-identity 34.5%

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

   5. distribute-neg-in 34.5%

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

   6. metadata-eval 34.5%

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

   7. distribute-neg-in 34.5%

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

   8. metadata-eval 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r* 34.6%

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

   2. associate--l+ 81.2%

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

   3. *-commutative 81.2%

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

   4. neg-mul-1 81.2%

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

   5. +-inverses 81.2%

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

   6. metadata-eval 81.2%

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

   7. unsub-neg 81.2%

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

8. Simplified 81.2%

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

   1. associate-/l/ 80.5%

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

   2. associate-*l/ 80.5%

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

   3. sub-neg 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-neg-in 80.5%

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

   6. distribute-lft-neg-in 80.5%

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

   7. associate-*l/ 80.5%

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

   8. metadata-eval 80.5%

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

   9. frac-2neg 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-/r* 81.2%

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

   12. frac-times 99.6%

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

   13. *-un-lft-identity 99.6%

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

   14. frac-2neg 99.6%

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

   15. metadata-eval 99.6%

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

   16. distribute-neg-in 99.6%

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

   17. metadata-eval 99.6%

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

   18. sub-neg 99.6%

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

10. Applied egg-rr 99.6%

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

    1. *-commutative 99.6%

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

    2. distribute-rgt-in 99.6%

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

    3. pow-plus 99.6%

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

    4. metadata-eval 99.6%

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

    5. pow1/2 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 3: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{-1 - x}}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ -1.0 (- -1.0 x)) (* x (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	return (-1.0 / (-1.0 - x)) / (x * (2.0 * sqrt((1.0 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / ((-1.0d0) - x)) / (x * (2.0d0 * sqrt((1.0d0 / x))))
end function

Java

public static double code(double x) {
	return (-1.0 / (-1.0 - x)) / (x * (2.0 * Math.sqrt((1.0 / x))));
}

Python

def code(x):
	return (-1.0 / (-1.0 - x)) / (x * (2.0 * math.sqrt((1.0 / x))))

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(-1.0 - x)) / Float64(x * Float64(2.0 * sqrt(Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / (-1.0 - x)) / (x * (2.0 * sqrt((1.0 / x))));
end

Wolfram

code[x_] := N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

   1. flip-- 33.0%

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

   2. div-inv 33.0%

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

   3. frac-times 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-sqr-sqrt 17.0%

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

   6. frac-times 24.5%

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

   7. metadata-eval 24.5%

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

   8. add-sqr-sqrt 33.1%

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

   9. +-commutative 33.1%

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

   10. pow1/2 33.1%

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

   11. pow-flip 33.1%

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

   12. metadata-eval 33.1%

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

   13. inv-pow 33.1%

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

   14. sqrt-pow2 33.1%

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

   15. +-commutative 33.1%

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

   16. metadata-eval 33.1%

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

4. Applied egg-rr 33.1%

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

   1. frac-2neg 33.1%

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

   2. metadata-eval 33.1%

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

   3. frac-sub 34.5%

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

   4. *-un-lft-identity 34.5%

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

   5. distribute-neg-in 34.5%

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

   6. metadata-eval 34.5%

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

   7. distribute-neg-in 34.5%

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

   8. metadata-eval 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r* 34.6%

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

   2. associate--l+ 81.2%

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

   3. *-commutative 81.2%

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

   4. neg-mul-1 81.2%

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

   5. +-inverses 81.2%

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

   6. metadata-eval 81.2%

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

   7. unsub-neg 81.2%

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

8. Simplified 81.2%

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

   1. associate-/l/ 80.5%

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

   2. associate-*l/ 80.5%

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

   3. sub-neg 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-neg-in 80.5%

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

   6. distribute-lft-neg-in 80.5%

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

   7. associate-*l/ 80.5%

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

   8. metadata-eval 80.5%

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

   9. frac-2neg 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-/r* 81.2%

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

   12. frac-times 99.6%

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

   13. *-un-lft-identity 99.6%

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

   14. frac-2neg 99.6%

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

   15. metadata-eval 99.6%

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

   16. distribute-neg-in 99.6%

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

   17. metadata-eval 99.6%

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

   18. sub-neg 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in x around inf 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \frac{\frac{-1}{-1 - x}}{x \cdot \left(2 \cdot \sqrt{\frac{1}{x}}\right)} \]
13. Add Preprocessing

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{-1 - x}}{\sqrt{x} \cdot 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ -1.0 (- -1.0 x)) (* (sqrt x) 2.0)))

C

double code(double x) {
	return (-1.0 / (-1.0 - x)) / (sqrt(x) * 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / ((-1.0d0) - x)) / (sqrt(x) * 2.0d0)
end function

Java

public static double code(double x) {
	return (-1.0 / (-1.0 - x)) / (Math.sqrt(x) * 2.0);
}

Python

def code(x):
	return (-1.0 / (-1.0 - x)) / (math.sqrt(x) * 2.0)

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(-1.0 - x)) / Float64(sqrt(x) * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / (-1.0 - x)) / (sqrt(x) * 2.0);
end

Wolfram

code[x_] := N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

   1. flip-- 33.0%

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

   2. div-inv 33.0%

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

   3. frac-times 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-sqr-sqrt 17.0%

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

   6. frac-times 24.5%

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

   7. metadata-eval 24.5%

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

   8. add-sqr-sqrt 33.1%

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

   9. +-commutative 33.1%

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

   10. pow1/2 33.1%

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

   11. pow-flip 33.1%

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

   12. metadata-eval 33.1%

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

   13. inv-pow 33.1%

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

   14. sqrt-pow2 33.1%

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

   15. +-commutative 33.1%

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

   16. metadata-eval 33.1%

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

4. Applied egg-rr 33.1%

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

   1. frac-2neg 33.1%

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

   2. metadata-eval 33.1%

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

   3. frac-sub 34.5%

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

   4. *-un-lft-identity 34.5%

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

   5. distribute-neg-in 34.5%

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

   6. metadata-eval 34.5%

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

   7. distribute-neg-in 34.5%

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

   8. metadata-eval 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r* 34.6%

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

   2. associate--l+ 81.2%

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

   3. *-commutative 81.2%

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

   4. neg-mul-1 81.2%

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

   5. +-inverses 81.2%

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

   6. metadata-eval 81.2%

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

   7. unsub-neg 81.2%

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

8. Simplified 81.2%

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

   1. associate-/l/ 80.5%

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

   2. associate-*l/ 80.5%

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

   3. sub-neg 80.5%

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

   4. metadata-eval 80.5%

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

   5. distribute-neg-in 80.5%

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

   6. distribute-lft-neg-in 80.5%

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

   7. associate-*l/ 80.5%

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

   8. metadata-eval 80.5%

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

   9. frac-2neg 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-/r* 81.2%

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

   12. frac-times 99.6%

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

   13. *-un-lft-identity 99.6%

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

   14. frac-2neg 99.6%

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

   15. metadata-eval 99.6%

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

   16. distribute-neg-in 99.6%

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

   17. metadata-eval 99.6%

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

   18. sub-neg 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in x around inf 98.7%

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

12. Final simplification 98.7%

    \[\leadsto \frac{\frac{-1}{-1 - x}}{\sqrt{x} \cdot 2} \]
13. Add Preprocessing

Alternative 5: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. frac-sub 33.1%

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

   2. *-un-lft-identity 33.1%

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

   3. +-commutative 33.1%

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

   4. *-rgt-identity 33.1%

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

   5. sqrt-unprod 33.1%

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

   6. +-commutative 33.1%

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

4. Applied egg-rr 33.1%

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

5. Taylor expanded in x around 0 33.5%

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

6. Final simplification 33.5%

   \[\leadsto \frac{1}{\sqrt{x \cdot \left(x + 1\right)}} \]
7. Add Preprocessing

Alternative 6: 7.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. flip-- 33.0%

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

   2. div-inv 33.0%

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

   3. frac-times 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-sqr-sqrt 17.0%

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

   6. frac-times 24.5%

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

   7. metadata-eval 24.5%

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

   8. add-sqr-sqrt 33.1%

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

   9. +-commutative 33.1%

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

   10. pow1/2 33.1%

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

   11. pow-flip 33.1%

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

   12. metadata-eval 33.1%

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

   13. inv-pow 33.1%

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

   14. sqrt-pow2 33.1%

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

   15. +-commutative 33.1%

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

   16. metadata-eval 33.1%

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

4. Applied egg-rr 33.1%

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

5. Taylor expanded in x around 0 7.8%

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

   1. distribute-rgt-in 7.8%

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

   2. *-un-lft-identity 7.8%

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

   3. pow-plus 7.8%

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

   4. metadata-eval 7.8%

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

   5. pow1/2 7.8%

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

   6. +-commutative 7.8%

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

7. Applied egg-rr 7.8%

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

8. Final simplification 7.8%

   \[\leadsto \frac{1}{x + \sqrt{x}} \]
9. Add Preprocessing

Alternative 7: 3.2% 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 Float64(-(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 33.0%

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

   1. expm1-log1p-u 33.0%

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

   2. expm1-udef 6.0%

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

   3. inv-pow 6.0%

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

   4. sqrt-pow2 6.0%

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

   5. +-commutative 6.0%

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

   6. metadata-eval 6.0%

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

4. Applied egg-rr 6.0%

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

   1. expm1-def 26.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)\right)} \]

   2. expm1-log1p 26.1%

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

6. Simplified 26.1%

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

7. Taylor expanded in x around inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. rem-exp-log 3.1%

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

   3. exp-neg 3.1%

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

   4. unpow1/2 3.1%

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

   5. exp-prod 3.1%

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

   6. distribute-lft-neg-out 3.1%

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

   7. distribute-rgt-neg-in 3.1%

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

   8. metadata-eval 3.1%

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

   9. exp-to-pow 3.1%

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

9. Simplified 3.1%

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

10. Final simplification 3.1%

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

Alternative 8: 5.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. *-un-lft-identity 33.0%

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

   2. inv-pow 33.0%

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

   3. add-cube-cbrt 12.2%

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

   4. unpow-prod-down 12.8%

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

   5. prod-diff 6.2%

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

4. Applied egg-rr 6.2%

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

   1. +-commutative 6.2%

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

   2. fma-udef 6.2%

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

   3. *-lft-identity 6.2%

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

6. Simplified 15.4%

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

7. Taylor expanded in x around inf 5.5%

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

8. Final simplification 5.5%

   \[\leadsto \sqrt{\frac{1}{x}} \]
9. 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 2024039 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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