2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.8%

Time: 7.0s

Alternatives: 6

Speedup: 0.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+15)
   (/ (- (+ 1.0 x) x) (+ (sqrt x) (sqrt (+ 1.0 x))))
   (* (sqrt (pow x -1.0)) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+15) {
		tmp = ((1.0 + x) - x) / (sqrt(x) + sqrt((1.0 + x)));
	} else {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+15) then
        tmp = ((1.0d0 + x) - x) / (sqrt(x) + sqrt((1.0d0 + x)))
    else
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+15) {
		tmp = ((1.0 + x) - x) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	} else {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+15:
		tmp = ((1.0 + x) - x) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	else:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+15)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	else
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+15)
		tmp = ((1.0 + x) - x) / (sqrt(x) + sqrt((1.0 + x)));
	else
		tmp = sqrt((x ^ -1.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+15], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e15

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 5e15 < x

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (sqrt (pow x -1.0)) 0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.5

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

   \[\leadsto \sqrt{{x}^{-1}} \cdot 0.5 \]
7. Add Preprocessing

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{-0.0625}{x}, \frac{-1}{\sqrt{x}}, \mathsf{fma}\left(0.5, \sqrt{x}, \frac{-0.125}{\sqrt{x}}\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (/ -0.0625 x) (/ -1.0 (sqrt x)) (fma 0.5 (sqrt x) (/ -0.125 (sqrt x))))
  x))

C

double code(double x) {
	return fma((-0.0625 / x), (-1.0 / sqrt(x)), fma(0.5, sqrt(x), (-0.125 / sqrt(x)))) / x;
}

Julia

function code(x)
	return Float64(fma(Float64(-0.0625 / x), Float64(-1.0 / sqrt(x)), fma(0.5, sqrt(x), Float64(-0.125 / sqrt(x)))) / x)
end

Wolfram

code[x_] := N[(N[(N[(-0.0625 / x), $MachinePrecision] * N[(-1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[x], $MachinePrecision] + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{-1}{x} \cdot \frac{\frac{-1}{x}}{x}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)}{x} \]
8. Step-by-step derivation

9. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{x}}{-\sqrt{x}}, 0.0625, \mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)\right)}{x} \]
10. Step-by-step derivation

11. Applied rewrites 99.1%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.0625}{x}, \frac{-1}{\sqrt{x}}, \mathsf{fma}\left(0.5, \sqrt{x}, \frac{-0.125}{\sqrt{x}}\right)\right)}{x} \]
12. Add Preprocessing

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma (sqrt x) 0.5 (/ -0.125 (sqrt x))) x))

C

double code(double x) {
	return fma(sqrt(x), 0.5, (-0.125 / sqrt(x))) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(x), 0.5, Float64(-0.125 / sqrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-1}{8}} + \frac{1}{2} \cdot \sqrt{x}}{x} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{-1}{8}, \color{blue}{\frac{1}{2} \cdot \sqrt{x}}\right)}{x} \]

   7. lower-sqrt.f64 98.7

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -0.125, 0.5 \cdot \sqrt{x}\right)}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \frac{-0.125}{\sqrt{x}}\right)}{x}} \]
8. Add Preprocessing

Alternative 5: 4.4% accurate, 1.4× speedup

Math

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

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 7.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 4.7

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

5. Applied rewrites 4.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.7%

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

Alternative 6: 1.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.6%

   \[\leadsto \color{blue}{1} - \sqrt{x} \]
6. Add Preprocessing

Developer Target 1: 98.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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