2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.6%

Time: 12.2s

Alternatives: 4

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -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. inv-pow 33.1%

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

   11. sqrt-pow2 33.1%

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

   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. pow1/2 33.1%

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

   14. pow-flip 33.1%

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

   15. +-commutative 33.1%

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

   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-sub 34.5%

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

   2. div-inv 34.5%

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

   3. *-un-lft-identity 34.5%

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

6. Applied egg-rr 34.5%

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

   1. *-rgt-identity 34.5%

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

   2. associate--l+ 80.5%

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

   3. +-inverses 80.5%

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

   4. metadata-eval 80.5%

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

   5. associate-*r/ 80.5%

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

   6. metadata-eval 80.5%

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

   7. associate-/r* 81.2%

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

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. *-commutative 81.2%

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

   2. frac-times 99.6%

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

   3. *-un-lft-identity 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{2 \cdot \sqrt{x} + \sqrt{\frac{1}{x}} \cdot 1.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 x) (+ (* 2.0 (sqrt x)) (* (sqrt (/ 1.0 x)) 1.5))))

C

double code(double x) {
	return (1.0 / x) / ((2.0 * sqrt(x)) + (sqrt((1.0 / x)) * 1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) / ((2.0d0 * sqrt(x)) + (sqrt((1.0d0 / x)) * 1.5d0))
end function

Java

public static double code(double x) {
	return (1.0 / x) / ((2.0 * Math.sqrt(x)) + (Math.sqrt((1.0 / x)) * 1.5));
}

Python

def code(x):
	return (1.0 / x) / ((2.0 * math.sqrt(x)) + (math.sqrt((1.0 / x)) * 1.5))

Julia

function code(x)
	return Float64(Float64(1.0 / x) / Float64(Float64(2.0 * sqrt(x)) + Float64(sqrt(Float64(1.0 / x)) * 1.5)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) / ((2.0 * sqrt(x)) + (sqrt((1.0 / x)) * 1.5));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $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. inv-pow 33.1%

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

   11. sqrt-pow2 33.1%

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

   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. pow1/2 33.1%

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

   14. pow-flip 33.1%

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

   15. +-commutative 33.1%

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

   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-sub 34.5%

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

   2. div-inv 34.5%

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

   3. *-un-lft-identity 34.5%

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

6. Applied egg-rr 34.5%

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

   1. *-rgt-identity 34.5%

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

   2. associate--l+ 80.5%

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

   3. +-inverses 80.5%

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

   4. metadata-eval 80.5%

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

   5. associate-*r/ 80.5%

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

   6. metadata-eval 80.5%

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

   7. associate-/r* 81.2%

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

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. *-commutative 81.2%

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

   2. frac-times 99.6%

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

   3. *-un-lft-identity 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in x around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. associate-+l+ 99.4%

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

    3. distribute-rgt-out 99.4%

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

    4. metadata-eval 99.4%

       \[\leadsto \frac{\frac{1}{x}}{2 \cdot \sqrt{x} + \sqrt{\frac{1}{x}} \cdot \color{blue}{1.5}} \]

13. Simplified 99.4%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \sqrt{x} + \sqrt{\frac{1}{x}} \cdot 1.5}} \]

14. Final simplification 99.4%

    \[\leadsto \frac{\frac{1}{x}}{2 \cdot \sqrt{x} + \sqrt{\frac{1}{x}} \cdot 1.5} \]
15. Add Preprocessing

Alternative 3: 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 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. inv-pow 33.1%

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

   11. sqrt-pow2 33.1%

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

   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. pow1/2 33.1%

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

   14. pow-flip 33.1%

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

   15. +-commutative 33.1%

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

   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-sub 34.5%

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

   2. div-inv 34.5%

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

   3. *-un-lft-identity 34.5%

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

6. Applied egg-rr 34.5%

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

   1. *-rgt-identity 34.5%

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

   2. associate--l+ 80.5%

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

   3. +-inverses 80.5%

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

   4. metadata-eval 80.5%

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

   5. associate-*r/ 80.5%

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

   6. metadata-eval 80.5%

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

   7. associate-/r* 81.2%

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

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. Taylor expanded in x around inf 63.2%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
10. Step-by-step derivation

    1. exp-to-pow 60.5%

       \[\leadsto 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]

    2. *-commutative 60.5%

       \[\leadsto 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]

    3. exp-neg 61.6%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \]

    4. distribute-lft-neg-in 61.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \]

    5. metadata-eval 61.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

    6. *-commutative 61.6%

       \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

    7. exp-to-pow 64.3%

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

    8. metadata-eval 64.3%

       \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

    9. pow-sqr 64.3%

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

    10. rem-sqrt-square 99.0%

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

    11. rem-square-sqrt 98.5%

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

    12. fabs-sqr 98.5%

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

    13. rem-square-sqrt 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

    \[\leadsto 0.5 \cdot {x}^{-1.5} \]
13. Add Preprocessing

Alternative 4: 35.6% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 33.0%

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

3. Taylor expanded in x around inf 31.6%

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

4. Final simplification 31.6%

   \[\leadsto 0 \]
5. 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)))))