2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.7%

Time: 10.6s

Alternatives: 10

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: 38.7% 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: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\sqrt{1 + x}}}{\mathsf{fma}\left(x, 2, 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (sqrt (+ 1.0 x))) (fma x 2.0 0.5)))

C

double code(double x) {
	return (1.0 / sqrt((1.0 + x))) / fma(x, 2.0, 0.5);
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(Float64(1.0 + x))) / fma(x, 2.0, 0.5))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * 2.0 + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

4. Applied egg-rr 39.7%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. accelerator-lowering-fma.f64 82.4

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

6. Applied egg-rr 82.4%

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

7. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. rgt-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 99.0

      \[\leadsto \frac{\frac{1}{\sqrt{1 + x}}}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5\right)}} \]

9. Simplified 99.0%

   \[\leadsto \frac{\frac{1}{\sqrt{1 + x}}}{\color{blue}{\mathsf{fma}\left(x, 2, 0.5\right)}} \]
10. Add Preprocessing

Alternative 2: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

4. Applied egg-rr 39.7%

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

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. /-lowering-/.f64 98.1

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

7. Simplified 98.1%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
8. Add Preprocessing

Alternative 3: 97.6% 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 38.2%

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

4. Applied egg-rr 39.7%

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

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. /-lowering-/.f64 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 98.0

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

10. Simplified 98.0%

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

Alternative 4: 96.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

4. Applied egg-rr 39.7%

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

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. /-lowering-/.f64 98.1

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

7. Simplified 98.1%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. +-lowering-+.f64 97.1

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

9. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\frac{0.5}{x \cdot \sqrt{1 + x}}} \]
10. 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 38.2%

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

4. Applied egg-rr 39.7%

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

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. /-lowering-/.f64 98.1

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

7. Simplified 98.1%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. +-lowering-+.f64 97.1

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

9. Applied egg-rr 97.1%

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

10. Taylor expanded in x around inf

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

    1. sqrt-lowering-sqrt.f64 97.1

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

12. Simplified 97.1%

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

Alternative 6: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, 2.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.8e+153) (/ 1.0 (fma x 2.25 0.5)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.8e+153) {
		tmp = 1.0 / fma(x, 2.25, 0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.8e+153)
		tmp = Float64(1.0 / fma(x, 2.25, 0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.8e+153], N[(1.0 / N[(x * 2.25 + 0.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.7999999999999995e153

   1. Initial program 9.0%

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

   4. Applied egg-rr 12.1%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. +-lowering-+.f64 10.8

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

   7. Simplified 10.8%

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

      1. associate-/l/ N/A

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

      2. associate--l+ N/A

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

      3. +-inverses N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 98.2

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + \frac{9}{4} \cdot x}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{9}{4} \cdot x + \frac{1}{2}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{9}{4}} + \frac{1}{2}} \]

       3. accelerator-lowering-fma.f64 8.5

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, 2.25, 0.5\right)}} \]

   12. Simplified 8.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, 2.25, 0.5\right)}} \]

   if 6.7999999999999995e153 < x

   1. Initial program 66.1%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 53.0

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

   5. Simplified 53.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
   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. +-inverses 66.1

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

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.9375, 2.25\right), 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (fma x (fma x 0.9375 2.25) 0.5)))

C

double code(double x) {
	return 1.0 / fma(x, fma(x, 0.9375, 2.25), 0.5);
}

Julia

function code(x)
	return Float64(1.0 / fma(x, fma(x, 0.9375, 2.25), 0.5))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(x * 0.9375 + 2.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. +-lowering-+.f64 39.1

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

7. Simplified 39.1%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 97.7

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

9. Applied egg-rr 97.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + x \cdot \left(\frac{9}{4} + \frac{15}{16} \cdot x\right)}} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\frac{9}{4} + \frac{15}{16} \cdot x\right) + \frac{1}{2}}} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{9}{4} + \frac{15}{16} \cdot x, \frac{1}{2}\right)}} \]

    3. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{15}{16} \cdot x + \frac{9}{4}}, \frac{1}{2}\right)} \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{15}{16}} + \frac{9}{4}, \frac{1}{2}\right)} \]

    5. accelerator-lowering-fma.f64 38.0

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.9375, 2.25\right)}, 0.5\right)} \]

12. Simplified 38.0%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.9375, 2.25\right), 0.5\right)}} \]
13. Add Preprocessing

Alternative 8: 37.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

4. Applied egg-rr 39.7%

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

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. /-lowering-/.f64 98.1

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

7. Simplified 98.1%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. +-lowering-+.f64 97.1

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

9. Applied egg-rr 97.1%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. accelerator-lowering-fma.f64 38.0

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

12. Simplified 38.0%

    \[\leadsto \frac{0.5}{x \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}} \]
13. Add Preprocessing

Alternative 9: 37.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.5e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.5d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.5e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.5e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999972e153

   1. Initial program 9.0%

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

   4. Applied egg-rr 12.1%

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

   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. /-lowering-/.f64 96.4

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 8.5

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

   10. Simplified 8.5%

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

   if 6.49999999999999972e153 < x

   1. Initial program 66.1%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 53.0

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

   5. Simplified 53.0%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
   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. +-inverses 66.1

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

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 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 38.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. sqrt-lowering-sqrt.f64 N/A

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

   2. /-lowering-/.f64 29.3

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

5. Simplified 29.3%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
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. +-inverses 35.9

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

7. Applied egg-rr 35.9%

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

Developer Target 1: 98.5% 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: 38.7% 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 2024194 
(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)))))