2sqrt (example 3.1)

Percentage Accurate: 6.9% → 99.0%

Time: 4.8s

Alternatives: 8

Speedup: 1.2×

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: 6.9% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (sqrt x) 0.5 (fma 0.0625 (pow x -1.5) (/ -0.125 (sqrt x)))) x))

C

double code(double x) {
	return fma(sqrt(x), 0.5, fma(0.0625, pow(x, -1.5), (-0.125 / sqrt(x)))) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(x), 0.5, fma(0.0625, (x ^ -1.5), Float64(-0.125 / sqrt(x)))) / x)
end

Wolfram

code[x_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(0.0625 * N[Power[x, -1.5], $MachinePrecision] + N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 98.6%

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

7. Applied rewrites 98.6%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, \mathsf{fma}\left(0.0625, {x}^{-1.5}, \frac{-0.125}{\sqrt{x}}\right)\right)}{\color{blue}{x}} \]
8. Add Preprocessing

Alternative 2: 97.8% 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 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 97.1

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

5. Applied rewrites 97.1%

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

6. Final simplification 97.1%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{x}} - \frac{\frac{0.125}{\sqrt{x}}}{x} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 / sqrt(x)) - ((0.125d0 / sqrt(x)) / x)
end function

Java

public static double code(double x) {
	return (0.5 / Math.sqrt(x)) - ((0.125 / Math.sqrt(x)) / x);
}

Python

def code(x):
	return (0.5 / math.sqrt(x)) - ((0.125 / math.sqrt(x)) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (0.5 / sqrt(x)) - ((0.125 / sqrt(x)) / x);
end

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.3%

   \[\leadsto \frac{0.5}{\sqrt{x}} - \color{blue}{\frac{\frac{0.125}{\sqrt{x}}}{x}} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -(((-0.25d0) / x) + 1.0d0) * ((-0.5d0) / sqrt(x))
end function

Java

public static double code(double x) {
	return -((-0.25 / x) + 1.0) * (-0.5 / Math.sqrt(x));
}

Python

def code(x):
	return -((-0.25 / x) + 1.0) * (-0.5 / math.sqrt(x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = -((-0.25 / x) + 1.0) * (-0.5 / sqrt(x));
end

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.3%

   \[\leadsto \frac{0.5}{\sqrt{x}} - \color{blue}{\frac{\frac{0.125}{\sqrt{x}}}{x}} \]

9. Applied rewrites 98.3%

   \[\leadsto -\left(\frac{-0.25}{x} + 1\right) \cdot \frac{-0.5}{\sqrt{x}} \]

10. Final simplification 98.3%

    \[\leadsto \left(-\left(\frac{-0.25}{x} + 1\right)\right) \cdot \frac{-0.5}{\sqrt{x}} \]
11. Add Preprocessing

Alternative 5: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 97.1

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 96.9%

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

Alternative 6: 4.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 4.6

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

5. Applied rewrites 4.6%

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

Alternative 7: 4.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(x, 0.5, sqrt(x))
end

Wolfram

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

Derivation

1. Initial program 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 4.6

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

5. Applied rewrites 4.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.5%

   \[\leadsto \mathsf{fma}\left(0.5, x, -\sqrt{x}\right) \]
9. Step-by-step derivation

10. Applied rewrites 4.6%

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

Alternative 8: 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 8.0%

   \[\sqrt{x + 1} - \sqrt{x} \]
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: 98.0% 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 2024332 
(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)))