2isqrt (example 3.6)

Percentage Accurate: 39.8% → 98.4%

Time: 7.2s

Alternatives: 7

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: 39.8% 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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \mathsf{hypot}\left(\sqrt{x}, x\right)\right)}^{-1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(x + 1.0)) + sqrt(x)) * hypot(sqrt(x), x)) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((x + 1.0)) + sqrt(x)) * hypot(sqrt(x), x)) ^ -1.0;
end

Wolfram

code[x_] := N[Power[N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. flip-- N/A

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

   6. associate-/l/ N/A

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

4. Applied rewrites 38.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Final simplification 99.2%

   \[\leadsto {\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \mathsf{hypot}\left(\sqrt{x}, x\right)\right)}^{-1} \]
8. Add Preprocessing

Alternative 2: 5.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Applied rewrites 5.7%

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

6. Final simplification 5.7%

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

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{0.25}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (+ (/ 0.25 (* x x)) 0.5) (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((((0.25 / (x * x)) + 0.5) - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((0.25d0 / (x * x)) + 0.5d0) - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((((0.25 / (x * x)) + 0.5) - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((((0.25 / (x * x)) + 0.5) - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.25 / Float64(x * x)) + 0.5) - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((((0.25 / (x * x)) + 0.5) - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 36.4

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}}{\sqrt{x}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}}{x}}{\sqrt{x}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)}{x}}{\sqrt{x}} \]

   10. div-add-rev N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{x \cdot x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{1}{8} + \frac{1}{4}}{x}}}{x}}{\sqrt{x}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{1}{4}}{x \cdot x} + \frac{1}{2}\right) - \frac{\color{blue}{\frac{3}{8}}}{x}}{x}}{\sqrt{x}} \]

   12. lower-/.f64 98.4

       \[\leadsto \frac{\frac{\left(\frac{0.25}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

7. Applied rewrites 98.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.25}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 36.4

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{1} \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}{\sqrt{x}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{x \cdot \left(\frac{1}{8} + \frac{1}{4}\right)}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{x \cdot \color{blue}{\frac{3}{8}}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{3}{8} \cdot x}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{3}{8} \cdot x}{\color{blue}{x \cdot x}}}{x}}{\sqrt{x}} \]

   10. times-frac N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{3}{8}}{x} \cdot \frac{x}{x}}}{x}}{\sqrt{x}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{1}{8} + \frac{1}{4}}}{x} \cdot \frac{x}{x}}{x}}{\sqrt{x}} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)} \cdot \frac{x}{x}}{x}}{\sqrt{x}} \]

   13. *-rgt-identity N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \frac{\color{blue}{x \cdot 1}}{x}}{x}}{\sqrt{x}} \]

   14. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}}{x}}{\sqrt{x}} \]

   15. rgt-mult-inverse N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \color{blue}{1}}{x}}{\sqrt{x}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot 1}}{x}}{\sqrt{x}} \]

   17. div-add-rev N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} + \frac{1}{4}}{x}} \cdot 1}{x}}{\sqrt{x}} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{3}{8}}}{x} \cdot 1}{x}}{\sqrt{x}} \]

   19. lower-/.f64 98.3

       \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.375}{x}} \cdot 1}{x}}{\sqrt{x}} \]

7. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x} \cdot 1}{x}}}{\sqrt{x}} \]

8. Final simplification 98.3%

   \[\leadsto \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \]
9. Add Preprocessing

Alternative 5: 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 36.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 36.4

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.6

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

7. Applied rewrites 97.6%

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

Alternative 6: 38.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Applied rewrites 5.7%

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

7. Applied rewrites 34.7%

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

Alternative 7: 38.2% 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 36.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. Applied rewrites 5.7%

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

7. Applied rewrites 5.7%

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

9. Applied rewrites 34.7%

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

Developer Target 1: 39.8% 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 2024337 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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