2isqrt (example 3.6)

Percentage Accurate: 38.1% → 99.6%

Time: 13.5s

Alternatives: 3

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

   1. flip-- 39.9%

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

   2. div-inv 39.9%

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

   3. frac-times 23.8%

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

   4. metadata-eval 23.8%

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

   5. add-sqr-sqrt 24.5%

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

   6. frac-times 24.2%

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

   7. metadata-eval 24.2%

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

   8. add-sqr-sqrt 40.0%

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

   9. +-commutative 40.0%

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

   10. inv-pow 40.0%

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

   11. sqrt-pow2 40.0%

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

   12. metadata-eval 40.0%

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

   13. pow1/2 40.0%

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

   14. pow-flip 40.0%

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

   15. +-commutative 40.0%

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

   16. metadata-eval 40.0%

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

4. Applied egg-rr 40.0%

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

   1. associate-*r/ 40.0%

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

   2. *-rgt-identity 40.0%

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

6. Simplified 40.0%

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

   1. frac-sub 41.9%

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

   2. *-un-lft-identity 41.9%

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

8. Applied egg-rr 41.9%

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

   1. *-un-lft-identity 41.9%

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

   2. associate-/r* 41.9%

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

   3. *-rgt-identity 41.9%

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

   4. associate--l+ 85.7%

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

   5. +-inverses 85.7%

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

   6. metadata-eval 85.7%

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

   7. +-commutative 85.7%

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

   8. +-commutative 85.7%

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

10. Applied egg-rr 85.7%

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

    1. *-lft-identity 85.7%

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

    2. associate-/l/ 99.6%

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

    3. +-commutative 99.6%

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

    4. +-commutative 99.6%

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

12. Simplified 99.6%

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

    1. *-un-lft-identity 99.6%

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

    2. associate-/l/ 98.6%

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

    3. *-commutative 98.6%

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

    4. associate-*l* 98.7%

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

    5. +-commutative 98.7%

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

    6. *-commutative 98.7%

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

    7. distribute-lft-in 98.7%

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

14. Applied egg-rr 98.7%

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

    1. *-lft-identity 98.7%

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

    2. associate-/r* 99.6%

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

    3. +-commutative 99.6%

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

    4. +-commutative 99.6%

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

16. Simplified 99.6%

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

Alternative 2: 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 39.9%

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

3. Taylor expanded in x around inf 81.9%

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

   1. distribute-lft-out-- 81.9%

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

5. Simplified 81.9%

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

   1. *-commutative 81.9%

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

   2. unpow2 81.9%

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

   3. times-frac 97.7%

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

   4. inv-pow 97.7%

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

   5. sqrt-pow1 97.7%

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

   6. metadata-eval 97.7%

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

7. Applied egg-rr 97.7%

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

8. Taylor expanded in x around inf 97.5%

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

   1. neg-mul-1 97.5%

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

10. Simplified 97.5%

    \[\leadsto \frac{\color{blue}{-\sqrt{x}}}{x} \cdot \frac{-0.5}{x} \]
11. Step-by-step derivation

    1. distribute-frac-neg 97.5%

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

    2. pow1/2 97.5%

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

    3. pow1 97.5%

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

    4. pow-div 97.6%

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

    5. metadata-eval 97.6%

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

    6. distribute-lft-neg-out 97.6%

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

    7. metadata-eval 97.6%

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

    8. pow-div 97.5%

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

    9. pow1/2 97.5%

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

    10. add-sqr-sqrt 97.5%

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

    11. sqr-neg 97.5%

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

    12. sqrt-unprod 0.0%

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

    13. add-sqr-sqrt 35.9%

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

    14. pow1 35.9%

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

    15. *-commutative 35.9%

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

    16. div-inv 35.9%

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

    17. associate-*l* 35.9%

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

    18. inv-pow 35.9%

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

    19. metadata-eval 35.9%

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

    20. pow-prod-up 35.9%

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

12. Applied egg-rr 98.0%

    \[\leadsto \color{blue}{--0.5 \cdot {x}^{-1.5}} \]
13. Step-by-step derivation

    1. distribute-lft-neg-in 98.0%

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

    2. metadata-eval 98.0%

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

14. Simplified 98.0%

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

Alternative 3: 35.2% 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 39.9%

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

   1. add-cube-cbrt 10.2%

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

   2. associate-*l* 10.2%

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

   3. frac-2neg 10.2%

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

   4. metadata-eval 10.2%

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

   5. div-inv 10.2%

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

   6. distribute-neg-frac2 10.2%

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

   7. prod-diff 6.5%

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

4. Applied egg-rr 6.8%

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

6. Simplified 6.8%

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

7. Taylor expanded in x around inf 37.0%

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

   1. distribute-rgt1-in 37.0%

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

   2. metadata-eval 37.0%

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

   3. mul0-lft 37.0%

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

9. Simplified 37.0%

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

Developer target: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024101 
(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)))))