2isqrt (example 3.6)

Percentage Accurate: 38.8% → 98.7%

Time: 7.8s

Alternatives: 6

Speedup: 1.4×

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: 38.8% 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: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x} - \frac{0.125}{x \cdot x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (/ 0.5 x) (/ 0.125 (* x x))) (sqrt (+ x 1.0))))

C

double code(double x) {
	return ((0.5 / x) - (0.125 / (x * x))) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 / x) - (0.125d0 / (x * x))) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((0.5 / x) - (0.125 / (x * x))) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((0.5 / x) - (0.125 / (x * x))) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 / x) - Float64(0.125 / Float64(x * x))) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 / x) - (0.125 / (x * x))) / sqrt((x + 1.0));
end

Wolfram

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

Derivation

1. Initial program 37.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 98.7

      \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 98.7%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
8. Step-by-step derivation

9. Applied rewrites 98.7%

   \[\leadsto \frac{\frac{0.5}{x} - \color{blue}{\frac{\frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
10. Step-by-step derivation

11. Applied rewrites 98.7%

    \[\leadsto \frac{\frac{0.5}{x} - \frac{0.125}{\color{blue}{x \cdot x}}}{\sqrt{x + 1}} \]
12. 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 37.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.125d0 / x)) / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((0.5 - (0.125 / x)) / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
end

Wolfram

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

Derivation

1. Initial program 37.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 98.7

      \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 98.7%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 4: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.6

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

7. Applied rewrites 97.6%

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

Alternative 5: 81.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.9%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 83.9

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

5. Applied rewrites 83.9%

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

7. Applied rewrites 95.9%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 95.9%

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

12. Applied rewrites 81.0%

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

Alternative 6: 37.3% 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 37.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 36.2%

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

Developer Target 1: 38.9% 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 2024306 
(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)))))