2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.6%

Time: 14.6s

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: 38.6% 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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

   1. flip-- 39.1%

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

   2. div-inv 39.1%

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

   3. frac-times 22.9%

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

   4. metadata-eval 22.9%

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

   5. add-sqr-sqrt 20.6%

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

   6. frac-times 27.0%

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

   7. metadata-eval 27.0%

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

   8. add-sqr-sqrt 39.3%

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

   9. +-commutative 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. metadata-eval 39.3%

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

   13. pow1/2 39.3%

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

   14. pow-flip 39.3%

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

   15. +-commutative 39.3%

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

   16. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. frac-sub 41.2%

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

   2. associate-/r* 41.2%

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

   3. *-un-lft-identity 41.2%

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

   4. *-rgt-identity 41.2%

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

   5. associate--l+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. clear-num 81.1%

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

   2. frac-times 81.1%

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

   3. metadata-eval 81.1%

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

   4. div-inv 81.1%

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

   5. +-inverses 81.1%

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

   6. metadata-eval 81.1%

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

   7. clear-num 81.1%

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

   8. /-rgt-identity 81.1%

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

   9. *-commutative 81.1%

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

   10. +-commutative 81.1%

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

   11. +-commutative 81.1%

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

8. Applied egg-rr 81.1%

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

   1. associate-*l* 97.8%

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

   2. associate-/r* 99.5%

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

   3. distribute-rgt-in 99.5%

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

   4. fma-define 99.5%

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

   5. +-commutative 99.5%

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

   6. pow-plus 99.6%

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

   7. metadata-eval 99.6%

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

   8. unpow1/2 99.6%

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

   9. +-commutative 99.6%

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

10. Simplified 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

   1. flip-- 39.1%

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

   2. div-inv 39.1%

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

   3. frac-times 22.9%

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

   4. metadata-eval 22.9%

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

   5. add-sqr-sqrt 20.6%

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

   6. frac-times 27.0%

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

   7. metadata-eval 27.0%

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

   8. add-sqr-sqrt 39.3%

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

   9. +-commutative 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. metadata-eval 39.3%

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

   13. pow1/2 39.3%

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

   14. pow-flip 39.3%

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

   15. +-commutative 39.3%

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

   16. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. frac-sub 41.2%

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

   2. associate-/r* 41.2%

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

   3. *-un-lft-identity 41.2%

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

   4. *-rgt-identity 41.2%

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

   5. associate--l+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. clear-num 82.9%

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

   2. frac-times 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. +-inverses 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. /-rgt-identity 99.5%

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

   8. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. +-commutative 99.5%

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

   3. sqr-pow 99.4%

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

   4. +-commutative 99.4%

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

   5. sqr-pow 99.5%

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

   6. pow-plus 99.6%

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

   7. metadata-eval 99.6%

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

   8. pow1/2 99.6%

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

   9. +-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

   1. flip-- 39.1%

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

   2. div-inv 39.1%

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

   3. frac-times 22.9%

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

   4. metadata-eval 22.9%

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

   5. add-sqr-sqrt 20.6%

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

   6. frac-times 27.0%

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

   7. metadata-eval 27.0%

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

   8. add-sqr-sqrt 39.3%

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

   9. +-commutative 39.3%

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

   10. inv-pow 39.3%

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

   11. sqrt-pow2 39.3%

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

   12. metadata-eval 39.3%

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

   13. pow1/2 39.3%

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

   14. pow-flip 39.3%

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

   15. +-commutative 39.3%

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

   16. metadata-eval 39.3%

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

4. Applied egg-rr 39.3%

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

   1. frac-sub 41.2%

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

   2. associate-/r* 41.2%

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

   3. *-un-lft-identity 41.2%

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

   4. *-rgt-identity 41.2%

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

   5. associate--l+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. clear-num 82.9%

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

   2. frac-times 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. +-inverses 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. /-rgt-identity 99.5%

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

   8. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around inf 97.6%

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

    1. *-commutative 97.6%

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

11. Simplified 97.6%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\sqrt{x} \cdot 2}} \]
12. Add Preprocessing

Alternative 4: 35.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 39.1%

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

   1. add-sqr-sqrt 27.9%

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

   2. fma-neg 7.4%

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

   3. inv-pow 7.4%

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

   4. sqrt-pow1 7.4%

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

   5. metadata-eval 7.4%

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

   6. inv-pow 7.4%

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

   7. sqrt-pow1 7.4%

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

   8. metadata-eval 7.4%

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

   9. distribute-neg-frac 7.4%

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

   10. metadata-eval 7.4%

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

   11. +-commutative 7.4%

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

4. Applied egg-rr 7.4%

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

5. Taylor expanded in x around inf 36.3%

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

   1. distribute-rgt1-in 36.3%

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

   2. metadata-eval 36.3%

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

   3. mul0-lft 36.3%

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

7. Simplified 36.3%

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

Developer target: 98.5% 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 2024111 
(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)))))