2isqrt (example 3.6)

Percentage Accurate: 38.1% → 99.6%

Time: 15.6s

Alternatives: 7

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. flip-- 38.9%

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

   2. frac-times 22.7%

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

   3. metadata-eval 22.7%

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

   4. add-sqr-sqrt 19.9%

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

   5. frac-times 26.4%

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

   6. metadata-eval 26.4%

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

   7. add-sqr-sqrt 39.1%

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

   8. +-commutative 39.1%

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

   9. inv-pow 39.1%

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

   10. sqrt-pow2 39.1%

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

   11. metadata-eval 39.1%

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

   12. pow1/2 39.1%

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

   13. pow-flip 39.1%

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

   14. +-commutative 39.1%

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

   15. metadata-eval 39.1%

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

4. Applied egg-rr 39.1%

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

   1. frac-sub 41.8%

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

   2. div-inv 41.8%

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

   3. *-un-lft-identity 41.8%

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

6. Applied egg-rr 41.8%

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

   1. associate-*r/ 41.8%

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

   2. *-rgt-identity 41.8%

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

   3. *-rgt-identity 41.8%

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

   4. associate--l+ 81.1%

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

   5. +-inverses 81.1%

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

   6. metadata-eval 81.1%

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

   7. associate-/r* 83.0%

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

8. Simplified 83.0%

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

   1. div-inv 82.9%

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

   2. associate-/l* 99.4%

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

10. Applied egg-rr 99.4%

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

    1. associate-/r* 99.5%

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

    2. associate-*r/ 99.6%

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

    3. *-rgt-identity 99.6%

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

    4. distribute-lft-in 99.5%

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

    5. *-commutative 99.5%

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

    6. fma-define 99.6%

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

    7. *-commutative 99.6%

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

    8. pow-plus 99.7%

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

    9. metadata-eval 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \frac{\frac{1}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)} \]
14. 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 38.9%

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

   1. flip-- 38.9%

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

   2. frac-times 22.7%

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

   3. metadata-eval 22.7%

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

   4. add-sqr-sqrt 19.9%

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

   5. frac-times 26.4%

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

   6. metadata-eval 26.4%

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

   7. add-sqr-sqrt 39.1%

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

   8. +-commutative 39.1%

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

   9. inv-pow 39.1%

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

   10. sqrt-pow2 39.1%

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

   11. metadata-eval 39.1%

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

   12. pow1/2 39.1%

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

   13. pow-flip 39.1%

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

   14. +-commutative 39.1%

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

   15. metadata-eval 39.1%

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

4. Applied egg-rr 39.1%

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

   1. frac-sub 41.8%

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

   2. div-inv 41.8%

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

   3. *-un-lft-identity 41.8%

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

6. Applied egg-rr 41.8%

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

   1. associate-*r/ 41.8%

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

   2. *-rgt-identity 41.8%

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

   3. *-rgt-identity 41.8%

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

   4. associate--l+ 81.1%

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

   5. +-inverses 81.1%

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

   6. metadata-eval 81.1%

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

   7. associate-/r* 83.0%

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

8. Simplified 83.0%

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

   1. div-inv 82.9%

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

   2. associate-/l* 99.4%

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

10. Applied egg-rr 99.4%

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

    1. associate-/r* 99.5%

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

    2. associate-*r/ 99.6%

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

    3. *-rgt-identity 99.6%

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

    4. distribute-lft-in 99.5%

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

    5. *-commutative 99.5%

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

    6. fma-define 99.6%

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

    7. *-commutative 99.6%

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

    8. pow-plus 99.7%

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

    9. metadata-eval 99.7%

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

12. Simplified 99.7%

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

    1. fma-undefine 99.6%

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

    2. unpow1/2 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 3: 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 38.9%

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

   1. sub-neg 38.9%

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

   2. inv-pow 38.9%

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

   3. sqrt-pow2 30.0%

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

   4. metadata-eval 30.0%

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

   5. distribute-neg-frac 30.0%

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

   6. metadata-eval 30.0%

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

   7. +-commutative 30.0%

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

4. Applied egg-rr 30.0%

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

   1. *-rgt-identity 30.0%

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

   2. metadata-eval 30.0%

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

   3. distribute-rgt-neg-in 30.0%

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

   4. unsub-neg 30.0%

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

   5. associate-*l/ 30.0%

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

   6. metadata-eval 30.0%

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

   7. unpow1/2 30.0%

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

   8. exp-to-pow 7.5%

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

   9. log1p-undefine 7.5%

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

   10. *-commutative 7.5%

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

   11. exp-neg 7.5%

       \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]

   12. *-commutative 7.5%

       \[\leadsto {x}^{-0.5} - e^{-\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.5}} \]

   13. distribute-rgt-neg-in 7.5%

       \[\leadsto {x}^{-0.5} - e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(-0.5\right)}} \]

   14. log1p-undefine 7.5%

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

   15. metadata-eval 7.5%

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

   16. exp-to-pow 39.0%

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

6. Simplified 39.0%

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

   1. flip-- 39.0%

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

   2. pow-prod-up 20.1%

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

   3. metadata-eval 20.1%

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

   4. inv-pow 20.4%

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

   5. pow-prod-up 38.8%

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

   6. metadata-eval 38.8%

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

   7. inv-pow 39.1%

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

   8. frac-sub 41.8%

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

   9. *-un-lft-identity 41.8%

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

   10. *-rgt-identity 41.8%

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

   11. associate-/r* 41.8%

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

   12. *-rgt-identity 41.8%

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

   13. *-rgt-identity 41.8%

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

   14. associate--l+ 81.1%

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

   15. +-inverses 81.1%

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

   16. metadata-eval 81.1%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in x around inf 98.9%

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

    1. +-commutative 98.9%

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

    2. associate-+l+ 98.9%

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

    3. distribute-rgt-out 98.9%

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

    4. metadata-eval 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 4: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{x}^{0.5}} \cdot \frac{1}{2 + x \cdot 2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 1.0 (pow x 0.5)) (/ 1.0 (+ 2.0 (* x 2.0)))))

C

double code(double x) {
	return (1.0 / pow(x, 0.5)) * (1.0 / (2.0 + (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x ** 0.5d0)) * (1.0d0 / (2.0d0 + (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.pow(x, 0.5)) * (1.0 / (2.0 + (x * 2.0)));
}

Python

def code(x):
	return (1.0 / math.pow(x, 0.5)) * (1.0 / (2.0 + (x * 2.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / (x ^ 0.5)) * Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x ^ 0.5)) * (1.0 / (2.0 + (x * 2.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

   1. flip-- 38.9%

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

   2. frac-times 22.7%

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

   3. metadata-eval 22.7%

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

   4. add-sqr-sqrt 19.9%

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

   5. frac-times 26.4%

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

   6. metadata-eval 26.4%

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

   7. add-sqr-sqrt 39.1%

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

   8. +-commutative 39.1%

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

   9. inv-pow 39.1%

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

   10. sqrt-pow2 39.1%

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

   11. metadata-eval 39.1%

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

   12. pow1/2 39.1%

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

   13. pow-flip 39.1%

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

   14. +-commutative 39.1%

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

   15. metadata-eval 39.1%

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

4. Applied egg-rr 39.1%

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

   1. frac-sub 41.8%

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

   2. div-inv 41.8%

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

   3. *-un-lft-identity 41.8%

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

6. Applied egg-rr 41.8%

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

   1. associate-*r/ 41.8%

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

   2. *-rgt-identity 41.8%

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

   3. *-rgt-identity 41.8%

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

   4. associate--l+ 81.1%

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

   5. +-inverses 81.1%

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

   6. metadata-eval 81.1%

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

   7. associate-/r* 83.0%

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

8. Simplified 83.0%

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

9. Taylor expanded in x around inf 81.0%

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

    1. div-inv 80.9%

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

    2. inv-pow 80.9%

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

    3. metadata-eval 80.9%

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

    4. pow-prod-up 81.0%

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

    5. *-commutative 81.0%

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

    6. inv-pow 81.0%

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

    7. sqrt-pow1 81.0%

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

    8. metadata-eval 81.0%

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

    9. times-frac 97.6%

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

    10. pow-prod-up 97.4%

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

    11. metadata-eval 97.4%

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

    12. inv-pow 97.4%

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

11. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{{x}^{-0.5}} \cdot \frac{\frac{1}{1 + x}}{2}} \]
12. Step-by-step derivation

    1. associate-/l/ 97.5%

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

    2. pow-plus 97.6%

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

    3. metadata-eval 97.6%

       \[\leadsto \frac{1}{{x}^{\color{blue}{0.5}}} \cdot \frac{\frac{1}{1 + x}}{2} \]

    4. associate-/l/ 97.6%

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

    5. distribute-lft-in 97.6%

       \[\leadsto \frac{1}{{x}^{0.5}} \cdot \frac{1}{\color{blue}{2 \cdot 1 + 2 \cdot x}} \]

    6. metadata-eval 97.6%

       \[\leadsto \frac{1}{{x}^{0.5}} \cdot \frac{1}{\color{blue}{2} + 2 \cdot x} \]

13. Simplified 97.6%

    \[\leadsto \color{blue}{\frac{1}{{x}^{0.5}} \cdot \frac{1}{2 + 2 \cdot x}} \]

14. Final simplification 97.6%

    \[\leadsto \frac{1}{{x}^{0.5}} \cdot \frac{1}{2 + x \cdot 2} \]
15. Add Preprocessing

Alternative 5: 37.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.8e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.8e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.8d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.8e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.8e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.8e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.8e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.8e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999985e153

   1. Initial program 10.8%

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

      1. flip-- 10.8%

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

      2. frac-times 11.0%

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

      3. metadata-eval 11.0%

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

      4. add-sqr-sqrt 11.2%

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

      5. frac-times 11.3%

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

      6. metadata-eval 11.3%

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

      7. add-sqr-sqrt 11.3%

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

      8. +-commutative 11.3%

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

      9. inv-pow 11.3%

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

      10. sqrt-pow2 11.3%

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

      11. metadata-eval 11.3%

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

      12. pow1/2 11.3%

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

      13. pow-flip 11.3%

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

      14. +-commutative 11.3%

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

      15. metadata-eval 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. frac-sub 16.8%

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

      2. div-inv 16.8%

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

      3. *-un-lft-identity 16.8%

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

   6. Applied egg-rr 16.8%

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

      1. associate-*r/ 16.8%

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

      2. *-rgt-identity 16.8%

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

      3. *-rgt-identity 16.8%

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

      4. associate--l+ 99.2%

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

      5. +-inverses 99.2%

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

      6. metadata-eval 99.2%

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

      7. associate-/r* 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 8.6%

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

       1. distribute-rgt-in 8.6%

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

       2. *-lft-identity 8.6%

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

       3. pow-plus 8.6%

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

       4. metadata-eval 8.6%

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

       5. unpow1/2 8.6%

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

   11. Simplified 8.6%

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

   if 4.79999999999999985e153 < x

   1. Initial program 64.5%

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

   3. Taylor expanded in x around inf 64.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. sub-neg 38.9%

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

   2. inv-pow 38.9%

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

   3. sqrt-pow2 30.0%

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

   4. metadata-eval 30.0%

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

   5. distribute-neg-frac 30.0%

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

   6. metadata-eval 30.0%

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

   7. +-commutative 30.0%

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

4. Applied egg-rr 30.0%

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

   1. *-rgt-identity 30.0%

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

   2. metadata-eval 30.0%

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

   3. distribute-rgt-neg-in 30.0%

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

   4. unsub-neg 30.0%

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

   5. associate-*l/ 30.0%

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

   6. metadata-eval 30.0%

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

   7. unpow1/2 30.0%

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

   8. exp-to-pow 7.5%

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

   9. log1p-undefine 7.5%

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

   10. *-commutative 7.5%

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

   11. exp-neg 7.5%

       \[\leadsto {x}^{-0.5} - \color{blue}{e^{-0.5 \cdot \mathsf{log1p}\left(x\right)}} \]

   12. *-commutative 7.5%

       \[\leadsto {x}^{-0.5} - e^{-\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.5}} \]

   13. distribute-rgt-neg-in 7.5%

       \[\leadsto {x}^{-0.5} - e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \left(-0.5\right)}} \]

   14. log1p-undefine 7.5%

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

   15. metadata-eval 7.5%

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

   16. exp-to-pow 39.0%

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

6. Simplified 39.0%

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

   1. flip-- 39.0%

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

   2. pow-prod-up 20.1%

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

   3. metadata-eval 20.1%

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

   4. inv-pow 20.4%

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

   5. pow-prod-up 38.8%

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

   6. metadata-eval 38.8%

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

   7. inv-pow 39.1%

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

   8. frac-sub 41.8%

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

   9. *-un-lft-identity 41.8%

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

   10. *-rgt-identity 41.8%

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

   11. associate-/r* 41.8%

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

   12. *-rgt-identity 41.8%

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

   13. *-rgt-identity 41.8%

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

   14. associate--l+ 81.1%

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

   15. +-inverses 81.1%

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

   16. metadata-eval 81.1%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in x around inf 97.6%

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

10. Final simplification 97.6%

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

Alternative 7: 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 38.9%

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

3. Taylor expanded in x around inf 35.8%

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

4. Final simplification 35.8%

   \[\leadsto 0 \]
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 2024048 
(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)))))