2isqrt (example 3.6)

Percentage Accurate: 38.2% → 98.6%

Time: 10.5s

Alternatives: 8

Speedup: 1.8×

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: 38.2% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5, \sqrt{x} + \frac{\mathsf{fma}\left(x, 0.25, 1\right)}{x \cdot \sqrt{x}}, \frac{-0.5}{\sqrt{x}}\right)}{x} \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (fma 0.5 (+ (sqrt x) (/ (fma x 0.25 1.0) (* x (sqrt x)))) (/ -0.5 (sqrt x)))
   x)
  (/ 1.0 x)))

C

double code(double x) {
	return (fma(0.5, (sqrt(x) + (fma(x, 0.25, 1.0) / (x * sqrt(x)))), (-0.5 / sqrt(x))) / x) * (1.0 / x);
}

Julia

function code(x)
	return Float64(Float64(fma(0.5, Float64(sqrt(x) + Float64(fma(x, 0.25, 1.0) / Float64(x * sqrt(x)))), Float64(-0.5 / sqrt(x))) / x) * Float64(1.0 / x))
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[(N[Sqrt[x], $MachinePrecision] + N[(N[(x * 0.25 + 1.0), $MachinePrecision] / N[(x * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 84.4%

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

7. Applied egg-rr 99.0%

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

Alternative 2: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 83.9

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

5. Simplified 83.9%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

7. Applied egg-rr 98.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.7

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

9. Applied egg-rr 98.7%

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

Alternative 3: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 84.4%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 83.9

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

8. Simplified 83.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. div-inv N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. inv-pow N/A

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

   11. pow-prod-up N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow-flip N/A

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

   15. pow1/2 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. un-div-inv N/A

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

   18. lower-/.f64 98.6

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

10. Applied egg-rr 98.6%

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

Alternative 4: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Simplified 84.4%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 83.9

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

8. Simplified 83.9%

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

   1. lift-sqrt.f64 N/A

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

   2. times-frac N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. inv-pow N/A

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow-flip N/A

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

   14. pow1/2 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. un-div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. distribute-neg-frac N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 98.6

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

10. Applied egg-rr 98.6%

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

Alternative 5: 96.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 68.5

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

5. Simplified 68.5%

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

   1. lift-*.f64 N/A

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

   2. associate-/r* N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-/.f64 N/A

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

   6. sqrt-div N/A

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

   7. metadata-eval N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sqrt-unprod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-/r* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. un-div-inv N/A

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

   16. lower-/.f64 97.6

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

7. Applied egg-rr 97.6%

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

Alternative 6: 7.9% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (+ 0.5 (/ -0.5 x)) x))

C

double code(double x) {
	return (0.5 + (-0.5 / x)) / x;
}

Fortran

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

Java

public static double code(double x) {
	return (0.5 + (-0.5 / x)) / x;
}

Python

def code(x):
	return (0.5 + (-0.5 / x)) / x

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.5 / x)) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 83.9

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

5. Simplified 83.9%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

7. Applied egg-rr 98.5%

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

8. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)}{x} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{\frac{1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}}{x} \]

   6. metadata-eval N/A

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

   7. lower-/.f64 7.8

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

10. Simplified 7.8%

    \[\leadsto \frac{\color{blue}{0.5 + \frac{-0.5}{x}}}{x} \]
11. Add Preprocessing

Alternative 7: 7.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 0.5 / x;
}

Python

def code(x):
	return 0.5 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 68.5

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

5. Simplified 68.5%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 7.8

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

8. Simplified 7.8%

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

Alternative 8: 4.6% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 36.3%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 83.9

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

5. Simplified 83.9%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

7. Applied egg-rr 98.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2}} \]
9. Step-by-step derivation

10. Simplified 4.7%

    \[\leadsto \color{blue}{0.5} \]
11. Add Preprocessing

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.2% 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 2024214 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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

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