2isqrt (example 3.6)

Percentage Accurate: 39.0% → 99.4%

Time: 7.5s

Alternatives: 8

Speedup: 1.5×

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: 39.0% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{-\sqrt{{x}^{-1}}}{\left(\sqrt{x} + t\_0\right) \cdot \left(-t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (/ (- (sqrt (pow x -1.0))) (* (+ (sqrt x) t_0) (- t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return -sqrt(pow(x, -1.0)) / ((sqrt(x) + t_0) * -t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = -sqrt((x ** (-1.0d0))) / ((sqrt(x) + t_0) * -t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	return -Math.sqrt(Math.pow(x, -1.0)) / ((Math.sqrt(x) + t_0) * -t_0);
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	return -math.sqrt(math.pow(x, -1.0)) / ((math.sqrt(x) + t_0) * -t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64(Float64(-sqrt((x ^ -1.0))) / Float64(Float64(sqrt(x) + t_0) * Float64(-t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = -sqrt((x ^ -1.0)) / ((sqrt(x) + t_0) * -t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, N[((-N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]) / N[(N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 38.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (sqrt (pow x -1.0))) (fma -2.0 x -1.5)))

C

double code(double x) {
	return -sqrt(pow(x, -1.0)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-sqrt((x ^ -1.0))) / fma(-2.0, x, -1.5))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 38.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(2 \cdot \frac{1}{x} + \frac{3}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lft-mult-inverse N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. associate-*l* N/A

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

   13. lft-mult-inverse N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right)} \]

   15. metadata-eval 99.3

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \color{blue}{-1.5}\right)} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]

11. Final simplification 99.3%

    \[\leadsto \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \]
12. Add Preprocessing

Alternative 3: 5.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (pow x -1.0)))

C

double code(double x) {
	return sqrt(pow(x, -1.0));
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt(math.pow(x, -1.0))

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 36.7%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

6. Final simplification 5.6%

   \[\leadsto \sqrt{{x}^{-1}} \]
7. Add Preprocessing

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\sqrt{x}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ -1.0 (sqrt x)) (fma -2.0 x -1.5)))

C

double code(double x) {
	return (-1.0 / sqrt(x)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-1.0 / sqrt(x)) / fma(-2.0, x, -1.5))
end

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 38.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(2 \cdot \frac{1}{x} + \frac{3}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lft-mult-inverse N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. associate-*l* N/A

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

   13. lft-mult-inverse N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right)} \]

   15. metadata-eval 99.3

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \color{blue}{-1.5}\right)} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 99.2%

    \[\leadsto \color{blue}{\frac{\frac{-1}{\sqrt{x}}}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]
13. Add Preprocessing

Alternative 5: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(-2, x, -1.5\right) \cdot \sqrt{x}} \end{array} \]

FPCore

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

C

double code(double x) {
	return -1.0 / (fma(-2.0, x, -1.5) * sqrt(x));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(-2.0, x, -1.5) * sqrt(x)))
end

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. sqrt-div N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower--.f64 6.7

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

4. Applied rewrites 6.7%

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

6. Applied rewrites 36.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. +-inverses N/A

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

   7. associate-+r- N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift--.f64 N/A

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

   17. flip-+ N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. distribute-lft-in N/A

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

    3. distribute-neg-in N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. distribute-lft-neg-in N/A

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

    7. metadata-eval N/A

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

    8. unpow2 N/A

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

    9. associate-*r* N/A

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

    10. lft-mult-inverse N/A

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

    11. *-lft-identity N/A

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

    12. lower-fma.f64 N/A

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

    13. *-commutative N/A

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

    14. associate-*l* N/A

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

    15. lft-mult-inverse N/A

        \[\leadsto \frac{-1}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \cdot \sqrt{x}} \]

    16. metadata-eval N/A

        \[\leadsto \frac{-1}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \cdot \sqrt{x}} \]

11. Applied rewrites 98.0%

    \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)} \cdot \sqrt{x}} \]

12. Final simplification 98.0%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(-2, x, -1.5\right) \cdot \sqrt{x}} \]
13. Add Preprocessing

Alternative 6: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\left(-2 \cdot x\right) \cdot \sqrt{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. sqrt-div N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower--.f64 6.7

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

4. Applied rewrites 6.7%

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

6. Applied rewrites 36.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. +-inverses N/A

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

   7. associate-+r- N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift--.f64 N/A

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

   17. flip-+ N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 96.9

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

11. Applied rewrites 96.9%

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

12. Final simplification 96.9%

    \[\leadsto \frac{-1}{\left(-2 \cdot x\right) \cdot \sqrt{x}} \]
13. Add Preprocessing

Alternative 7: 81.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 84.2%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 83.2%

   \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
9. Add Preprocessing

Alternative 8: 37.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

7. Applied rewrites 35.7%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 39.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))