2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.5%

Time: 17.7s

Alternatives: 5

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: 38.6% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. flip-- 37.6%

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

   2. clear-num 37.6%

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

   3. inv-pow 37.6%

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

   4. sqrt-pow2 37.6%

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

   5. metadata-eval 37.6%

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

   6. inv-pow 37.6%

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

   7. sqrt-pow2 37.6%

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

   8. +-commutative 37.6%

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

   9. metadata-eval 37.6%

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

   10. frac-times 20.8%

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

   11. metadata-eval 20.8%

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

   12. add-sqr-sqrt 22.4%

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

   13. frac-times 24.7%

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

   14. metadata-eval 24.7%

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

   15. add-sqr-sqrt 37.8%

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

   16. +-commutative 37.8%

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

4. Applied egg-rr 37.8%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-un-lft-identity 40.6%

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

   4. *-rgt-identity 40.6%

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

   5. metadata-eval 40.6%

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

   6. frac-times 40.6%

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

   7. un-div-inv 40.6%

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

6. Applied egg-rr 40.6%

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

   1. clear-num 40.6%

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

   2. div-inv 40.6%

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

   3. associate--l+ 83.2%

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

   4. +-inverses 83.2%

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

   5. metadata-eval 83.2%

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

   6. *-un-lft-identity 83.2%

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

   7. +-commutative 83.2%

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

   8. *-un-lft-identity 83.2%

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

   9. *-un-lft-identity 83.2%

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

   10. +-commutative 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 99.4%

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

   2. div-inv 99.5%

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

   3. clear-num 99.5%

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

   4. associate-/r/ 99.6%

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

   5. /-rgt-identity 99.6%

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

   6. +-commutative 99.6%

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

   7. +-commutative 99.6%

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

   8. +-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. flip-- 37.6%

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

   2. clear-num 37.6%

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

   3. inv-pow 37.6%

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

   4. sqrt-pow2 37.6%

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

   5. metadata-eval 37.6%

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

   6. inv-pow 37.6%

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

   7. sqrt-pow2 37.6%

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

   8. +-commutative 37.6%

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

   9. metadata-eval 37.6%

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

   10. frac-times 20.8%

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

   11. metadata-eval 20.8%

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

   12. add-sqr-sqrt 22.4%

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

   13. frac-times 24.7%

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

   14. metadata-eval 24.7%

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

   15. add-sqr-sqrt 37.8%

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

   16. +-commutative 37.8%

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

4. Applied egg-rr 37.8%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-un-lft-identity 40.6%

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

   4. *-rgt-identity 40.6%

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

   5. metadata-eval 40.6%

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

   6. frac-times 40.6%

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

   7. un-div-inv 40.6%

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

6. Applied egg-rr 40.6%

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

   1. clear-num 40.6%

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

   2. div-inv 40.6%

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

   3. associate--l+ 83.2%

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

   4. +-inverses 83.2%

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

   5. metadata-eval 83.2%

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

   6. *-un-lft-identity 83.2%

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

   7. +-commutative 83.2%

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

   8. *-un-lft-identity 83.2%

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

   9. *-un-lft-identity 83.2%

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

   10. +-commutative 83.2%

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

8. Applied egg-rr 83.2%

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

   1. associate-*l/ 99.4%

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

   2. div-inv 99.5%

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

   3. clear-num 99.5%

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

   4. associate-/r* 98.8%

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

   5. *-commutative 98.8%

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

   6. +-commutative 98.8%

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

   7. associate-/r/ 98.8%

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

   8. /-rgt-identity 98.8%

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

   9. +-commutative 98.8%

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

   10. +-commutative 98.8%

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

10. Applied egg-rr 98.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. flip-- 37.6%

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

   2. clear-num 37.6%

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

   3. inv-pow 37.6%

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

   4. sqrt-pow2 37.6%

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

   5. metadata-eval 37.6%

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

   6. inv-pow 37.6%

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

   7. sqrt-pow2 37.6%

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

   8. +-commutative 37.6%

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

   9. metadata-eval 37.6%

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

   10. frac-times 20.8%

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

   11. metadata-eval 20.8%

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

   12. add-sqr-sqrt 22.4%

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

   13. frac-times 24.7%

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

   14. metadata-eval 24.7%

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

   15. add-sqr-sqrt 37.8%

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

   16. +-commutative 37.8%

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

4. Applied egg-rr 37.8%

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

   1. frac-sub 40.6%

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

   2. div-inv 40.6%

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

   3. *-un-lft-identity 40.6%

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

   4. *-rgt-identity 40.6%

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

   5. metadata-eval 40.6%

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

   6. frac-times 40.6%

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

   7. un-div-inv 40.6%

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

6. Applied egg-rr 40.6%

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

   1. clear-num 40.6%

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

   2. div-inv 40.6%

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

   3. associate--l+ 83.2%

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

   4. +-inverses 83.2%

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

   5. metadata-eval 83.2%

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

   6. *-un-lft-identity 83.2%

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

   7. +-commutative 83.2%

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

   8. *-un-lft-identity 83.2%

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

   9. *-un-lft-identity 83.2%

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

   10. +-commutative 83.2%

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

8. Applied egg-rr 83.2%

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

   1. *-commutative 83.2%

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

   2. frac-times 99.5%

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

   3. *-un-lft-identity 99.5%

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

   4. +-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

10. Applied egg-rr 99.5%

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

Alternative 4: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. add-exp-log 37.6%

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

   2. inv-pow 37.2%

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

   3. sqrt-pow2 30.8%

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

   4. metadata-eval 30.8%

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

   5. inv-pow 31.2%

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

   6. sqrt-pow2 37.7%

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

   7. +-commutative 37.7%

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

   8. metadata-eval 37.7%

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

4. Applied egg-rr 37.7%

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

5. Taylor expanded in x around inf 90.6%

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

   1. exp-sum 76.5%

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

   2. add-exp-log 76.3%

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

   3. distribute-lft-neg-in 76.3%

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

   4. metadata-eval 76.3%

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

   5. associate-*l* 76.3%

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

   6. *-commutative 76.3%

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

   7. exp-to-pow 80.2%

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

   8. pow2 80.2%

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

   9. inv-pow 80.2%

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

   10. inv-pow 80.2%

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

   11. pow-prod-up 80.4%

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

   12. metadata-eval 80.4%

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

7. Applied egg-rr 80.4%

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

   1. pow1/2 80.4%

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

   2. pow-prod-up 97.0%

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

   3. metadata-eval 97.0%

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

9. Applied egg-rr 97.0%

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

Alternative 5: 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 37.6%

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

3. Taylor expanded in x around 0 5.7%

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

   1. inv-pow 5.7%

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

   2. sqrt-pow1 5.7%

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

   3. metadata-eval 5.7%

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

5. Applied egg-rr 5.7%

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

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(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)))))