2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.6%

Time: 10.1s

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: 38.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= x 2e+130)
     (/ 1.0 (* (+ x (sqrt (fma x x x))) t_0))
     (/ (/ 0.5 x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (x <= 2e+130) {
		tmp = 1.0 / ((x + sqrt(fma(x, x, x))) * t_0);
	} else {
		tmp = (0.5 / x) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (x <= 2e+130)
		tmp = Float64(1.0 / Float64(Float64(x + sqrt(fma(x, x, x))) * t_0));
	else
		tmp = Float64(Float64(0.5 / x) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2e+130], N[(1.0 / N[(N[(x + N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e130

   1. Initial program 11.5%

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

   4. Applied egg-rr 18.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   if 2.0000000000000001e130 < x

   1. Initial program 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.8

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

   7. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right) \cdot \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

4. Applied egg-rr 42.1%

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

5. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 41.0

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

7. Simplified 41.0%

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

   1. associate--l+ N/A

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

   2. +-inverses N/A

      \[\leadsto \frac{1 + \color{blue}{0}}{\left(x + \frac{1}{2}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]

   3. metadata-eval 97.9

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

9. Applied egg-rr 97.9%

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

10. Final simplification 97.9%

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

Alternative 3: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

4. Applied egg-rr 42.1%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.5

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

   \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{x + 1}} \]
9. Add Preprocessing

Alternative 4: 97.6% 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 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. metadata-eval N/A

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

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

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

   3. unsub-neg N/A

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

   4. distribute-neg-in N/A

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

   5. +-commutative N/A

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

   6. lower-/.f64 N/A

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

5. Simplified 82.6%

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

6. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 82.4

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

8. Simplified 82.4%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\sqrt{x}}}{x \cdot x} \]
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. lift-*.f64 N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

10. Applied egg-rr 96.6%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. inv-pow N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. unpow-prod-down N/A

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

    10. metadata-eval N/A

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

    11. inv-pow N/A

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

    12. div-inv N/A

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

    13. lower-/.f64 97.3

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

12. Applied egg-rr 97.3%

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

Alternative 5: 96.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. metadata-eval N/A

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

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

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

   3. unsub-neg N/A

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

   4. distribute-neg-in N/A

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

   5. +-commutative N/A

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

   6. lower-/.f64 N/A

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

5. Simplified 82.6%

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

6. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 82.4

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

8. Simplified 82.4%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\sqrt{x}}}{x \cdot x} \]
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. lift-*.f64 N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

10. Applied egg-rr 96.6%

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

Alternative 6: 81.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. metadata-eval N/A

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

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

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

   3. unsub-neg N/A

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

   4. distribute-neg-in N/A

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

   5. +-commutative N/A

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

   6. lower-/.f64 N/A

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

5. Simplified 82.6%

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

6. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 82.4

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

8. Simplified 82.4%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\sqrt{x}}}{x \cdot x} \]
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. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. times-frac N/A

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

   7. lift-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. metadata-eval N/A

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

   10. sqrt-div N/A

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

   11. /-rgt-identity N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lower-*.f64 83.2

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-/l/ N/A

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

   17. lift-*.f64 N/A

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

   18. lower-/.f64 82.3

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

10. Applied egg-rr 82.3%

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

11. Final simplification 82.3%

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

Alternative 7: 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 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Simplified 5.7%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. div-inv N/A

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

   5. rgt-mult-inverse N/A

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

   6. un-div-inv N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. /-rgt-identity N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. sqrt-unprod N/A

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

   15. lift-*.f64 N/A

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

   16. sqrt-undiv N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-/.f64 37.0

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

7. Applied egg-rr 37.0%

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

Alternative 8: 35.8% accurate, 49.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 26.7

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

5. Simplified 26.7%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-inverses 35.7

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

7. Applied egg-rr 35.7%

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

Developer Target 1: 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]

Developer Target 2: 38.8% 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 2024215 
(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)))))