2isqrt (example 3.6)

Percentage Accurate: 38.0% → 98.9%

Time: 14.1s

Alternatives: 5

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.0% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} - \frac{0.375}{{x}^{2.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 0.5 (pow x -1.5)) (/ 0.375 (pow x 2.5))))

C

double code(double x) {
	return (0.5 * pow(x, -1.5)) - (0.375 / pow(x, 2.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-1.5d0))) - (0.375d0 / (x ** 2.5d0))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -1.5)) - (0.375 / Math.pow(x, 2.5));
}

Python

def code(x):
	return (0.5 * math.pow(x, -1.5)) - (0.375 / math.pow(x, 2.5))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -1.5)) - Float64(0.375 / (x ^ 2.5)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -1.5)) - (0.375 / (x ^ 2.5));
end

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[x, 2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.3%

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

3. Taylor expanded in x around inf 78.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(-\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)}}{x} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. *-commutative 98.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

   5. distribute-rgt-out 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.125 + 0.5\right)}}{x}}{x} \]

   6. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. div-sub 98.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} - 0.375 \cdot \frac{{x}^{-1.5}}{x}} \]
9. Step-by-step derivation

   1. clear-num 98.7%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - 0.375 \cdot \color{blue}{\frac{1}{\frac{x}{{x}^{-1.5}}}} \]

   2. un-div-inv 98.7%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \color{blue}{\frac{0.375}{\frac{x}{{x}^{-1.5}}}} \]

   3. pow1 98.7%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{\frac{\color{blue}{{x}^{1}}}{{x}^{-1.5}}} \]

   4. pow-div 98.7%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{\color{blue}{{x}^{\left(1 - -1.5\right)}}} \]

   5. metadata-eval 98.7%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{{x}^{\color{blue}{2.5}}} \]

10. Applied egg-rr 98.7%

    \[\leadsto 0.5 \cdot {x}^{-1.5} - \color{blue}{\frac{0.375}{{x}^{2.5}}} \]

11. Final simplification 98.7%

    \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{{x}^{2.5}} \]
12. Add Preprocessing

Alternative 2: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

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

C

double code(double x) {
	return (0.5 * sqrt((1.0 / x))) / x;
}

Fortran

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

Java

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

Python

def code(x):
	return (0.5 * math.sqrt((1.0 / x))) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

3. Taylor expanded in x around inf 78.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} + -1 \cdot \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(-\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}\right)}}{x} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{\frac{1}{x}} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}}{x} \]

   4. *-commutative 98.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} - \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x} \]

   5. distribute-rgt-out 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.125 + 0.5\right)}}{x}}{x} \]

   6. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. inv-pow 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\color{blue}{{x}^{-1}}}}{x} \cdot 0.375}{x} \]

   2. sqrt-pow1 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x} \cdot 0.375}{x} \]

   3. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{{x}^{\color{blue}{-0.5}}}{x} \cdot 0.375}{x} \]

   4. *-un-lft-identity 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\left(1 \cdot \frac{{x}^{-0.5}}{x}\right)} \cdot 0.375}{x} \]

   5. associate-*r/ 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{1 \cdot {x}^{-0.5}}{x}} \cdot 0.375}{x} \]

   6. add-sqr-sqrt 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{1 \cdot {x}^{-0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot 0.375}{x} \]

   7. times-frac 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right)} \cdot 0.375}{x} \]

   8. metadata-eval 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right) \cdot 0.375}{x} \]

   9. sqrt-div 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right) \cdot 0.375}{x} \]

   10. inv-pow 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(\sqrt{\color{blue}{{x}^{-1}}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right) \cdot 0.375}{x} \]

   11. sqrt-pow1 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right) \cdot 0.375}{x} \]

   12. metadata-eval 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{\color{blue}{-0.5}} \cdot \frac{{x}^{-0.5}}{\sqrt{x}}\right) \cdot 0.375}{x} \]

   13. un-div-inv 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \color{blue}{\left({x}^{-0.5} \cdot \frac{1}{\sqrt{x}}\right)}\right) \cdot 0.375}{x} \]

   14. metadata-eval 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}\right)\right) \cdot 0.375}{x} \]

   15. sqrt-div 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)\right) \cdot 0.375}{x} \]

   16. inv-pow 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot \sqrt{\color{blue}{{x}^{-1}}}\right)\right) \cdot 0.375}{x} \]

   17. sqrt-pow1 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)\right) \cdot 0.375}{x} \]

   18. metadata-eval 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left({x}^{-0.5} \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right)\right) \cdot 0.375}{x} \]

   19. cube-unmult 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{{\left({x}^{-0.5}\right)}^{3}} \cdot 0.375}{x} \]

   20. *-un-lft-identity 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\left(1 \cdot {\left({x}^{-0.5}\right)}^{3}\right)} \cdot 0.375}{x} \]

   21. pow-pow 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(1 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right) \cdot 0.375}{x} \]

   22. metadata-eval 98.4%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.375}{x} \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.375}{x} \]
9. Step-by-step derivation

   1. *-lft-identity 98.4%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{{x}^{-1.5}} \cdot 0.375}{x} \]

10. Simplified 98.4%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{{x}^{-1.5}} \cdot 0.375}{x} \]

11. Taylor expanded in x around inf 97.1%

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

12. Final simplification 97.1%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
13. Add Preprocessing

Alternative 3: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

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

   1. inv-pow 40.3%

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

   2. add-sqr-sqrt 25.9%

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

   3. unpow-prod-down 22.8%

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

   4. frac-2neg 22.8%

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

   5. metadata-eval 22.8%

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

   6. div-inv 22.8%

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

   7. distribute-neg-frac2 22.8%

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

   8. prod-diff 8.0%

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

4. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left({\left(1 + x\right)}^{-0.5}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]

6. Simplified 24.4%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2} - {\left(1 + x\right)}^{-0.5}} \]
7. Step-by-step derivation

   1. flip-- 24.4%

      \[\leadsto \color{blue}{\frac{{\left({x}^{0.25}\right)}^{-2} \cdot {\left({x}^{0.25}\right)}^{-2} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{\left({x}^{0.25}\right)}^{-2} + {\left(1 + x\right)}^{-0.5}}} \]

8. Applied egg-rr 40.5%

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

9. Taylor expanded in x around inf 65.7%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
10. Step-by-step derivation

    1. associate-/r* 65.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

    2. metadata-eval 65.7%

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

    3. div-inv 65.7%

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

    4. metadata-eval 65.7%

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

    5. unpow3 65.7%

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

    6. sqrt-prod 77.1%

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

    7. sqrt-unprod 95.9%

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

    8. add-sqr-sqrt 96.2%

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

    9. /-rgt-identity 96.2%

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

    10. frac-times 97.0%

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

    11. clear-num 97.0%

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

    12. metadata-eval 97.0%

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

    13. sqrt-div 97.0%

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

    14. associate-*l/ 97.1%

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

    15. *-un-lft-identity 97.1%

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

    16. associate-/l* 97.1%

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

    17. *-commutative 97.1%

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

    18. associate-/l* 97.0%

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

    19. inv-pow 97.0%

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

    20. sqrt-pow1 97.1%

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

    21. metadata-eval 97.1%

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

11. Applied egg-rr 97.1%

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

12. Final simplification 97.1%

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

Alternative 4: 96.3% accurate, 2.0× speedup

Math

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

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 40.3%

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

   1. inv-pow 40.3%

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

   2. add-sqr-sqrt 25.9%

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

   3. unpow-prod-down 22.8%

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

   4. frac-2neg 22.8%

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

   5. metadata-eval 22.8%

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

   6. div-inv 22.8%

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

   7. distribute-neg-frac2 22.8%

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

   8. prod-diff 8.0%

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

4. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left({\left(1 + x\right)}^{-0.5}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]

6. Simplified 24.4%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2} - {\left(1 + x\right)}^{-0.5}} \]
7. Step-by-step derivation

   1. flip-- 24.4%

      \[\leadsto \color{blue}{\frac{{\left({x}^{0.25}\right)}^{-2} \cdot {\left({x}^{0.25}\right)}^{-2} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{\left({x}^{0.25}\right)}^{-2} + {\left(1 + x\right)}^{-0.5}}} \]

8. Applied egg-rr 40.5%

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

9. Taylor expanded in x around inf 65.7%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
10. Step-by-step derivation

    1. inv-pow 65.7%

       \[\leadsto \color{blue}{{\left(2 \cdot \sqrt{{x}^{3}}\right)}^{-1}} \]

    2. unpow-prod-down 65.7%

       \[\leadsto \color{blue}{{2}^{-1} \cdot {\left(\sqrt{{x}^{3}}\right)}^{-1}} \]

    3. metadata-eval 65.7%

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

    4. sqrt-pow1 96.3%

       \[\leadsto 0.5 \cdot {\color{blue}{\left({x}^{\left(\frac{3}{2}\right)}\right)}}^{-1} \]

    5. metadata-eval 96.3%

       \[\leadsto 0.5 \cdot {\left({x}^{\color{blue}{1.5}}\right)}^{-1} \]

11. Applied egg-rr 96.3%

    \[\leadsto \color{blue}{0.5 \cdot {\left({x}^{1.5}\right)}^{-1}} \]
12. Step-by-step derivation

    1. unpow-1 96.3%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{{x}^{1.5}}} \]

    2. associate-*r/ 96.3%

       \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{{x}^{1.5}}} \]

    3. metadata-eval 96.3%

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

13. Simplified 96.3%

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

14. Final simplification 96.3%

    \[\leadsto \frac{0.5}{{x}^{1.5}} \]
15. Add Preprocessing

Alternative 5: 5.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 40.3%

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

3. Taylor expanded in x around 0 5.7%

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

   1. pow1/2 5.7%

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

   2. inv-pow 5.7%

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

   3. pow-pow 5.7%

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

   4. metadata-eval 5.7%

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

   5. *-un-lft-identity 5.7%

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

5. Applied egg-rr 5.7%

   \[\leadsto \color{blue}{1 \cdot {x}^{-0.5}} \]
6. Step-by-step derivation

   1. *-lft-identity 5.7%

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

7. Simplified 5.7%

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

8. Final simplification 5.7%

   \[\leadsto {x}^{-0.5} \]
9. 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 2024068 
(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)))))