2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.5%

Time: 8.6s

Alternatives: 10

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: 38.3% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{\sqrt{x}}{-x}}{\left(\left(-\sqrt{x}\right) - t\_0\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (/ (/ (sqrt x) (- x)) (* (- (- (sqrt x)) t_0) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (sqrt(x) / -x) / ((-sqrt(x) - t_0) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = (sqrt(x) / -x) / ((-sqrt(x) - t_0) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return (Math.sqrt(x) / -x) / ((-Math.sqrt(x) - t_0) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return (math.sqrt(x) / -x) / ((-math.sqrt(x) - t_0) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(sqrt(x) / Float64(-x)) / Float64(Float64(Float64(-sqrt(x)) - t_0) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (sqrt(x) / -x) / ((-sqrt(x) - t_0) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sqrt[x], $MachinePrecision] / (-x)), $MachinePrecision] / N[(N[((-N[Sqrt[x], $MachinePrecision]) - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.8%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{-\sqrt{\frac{1}{x}}}{\left(\left(-\sqrt{x}\right) - t\_0\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (/ (- (sqrt (/ 1.0 x))) (* (- (- (sqrt x)) t_0) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return -sqrt((1.0 / x)) / ((-sqrt(x) - t_0) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = -sqrt((1.0d0 / x)) / ((-sqrt(x) - t_0) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return -Math.sqrt((1.0 / x)) / ((-Math.sqrt(x) - t_0) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return -math.sqrt((1.0 / x)) / ((-math.sqrt(x) - t_0) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(-sqrt(Float64(1.0 / x))) / Float64(Float64(Float64(-sqrt(x)) - t_0) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = -sqrt((1.0 / x)) / ((-sqrt(x) - t_0) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(N[((-N[Sqrt[x], $MachinePrecision]) - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.8%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{-1}{\sqrt{x}}}{\left(\left(-\sqrt{x}\right) - t\_0\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (/ (/ -1.0 (sqrt x)) (* (- (- (sqrt x)) t_0) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (-1.0 / sqrt(x)) / ((-sqrt(x) - t_0) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 + x))
    code = ((-1.0d0) / sqrt(x)) / ((-sqrt(x) - t_0) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return (-1.0 / Math.sqrt(x)) / ((-Math.sqrt(x) - t_0) * t_0);
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return (-1.0 / math.sqrt(x)) / ((-math.sqrt(x) - t_0) * t_0)

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(-1.0 / sqrt(x)) / Float64(Float64(Float64(-sqrt(x)) - t_0) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (-1.0 / sqrt(x)) / ((-sqrt(x) - t_0) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(-1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Sqrt[x], $MachinePrecision]) - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.8%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-/.f64 N/A

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

    2. frac-2neg N/A

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

    3. lower-/.f64 N/A

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

11. Applied rewrites 99.1%

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

12. Final simplification 99.1%

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

Alternative 4: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625}{x}}{x}}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - ((0.125d0 - (0.0625d0 / x)) / x)) / x) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return ((0.5 - ((0.125 - (0.0625 / x)) / x)) / x) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return ((0.5 - ((0.125 - (0.0625 / x)) / x)) / x) / math.sqrt((1.0 + x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((0.5 - ((0.125 - (0.0625 / x)) / x)) / x) / sqrt((1.0 + x));
end

Wolfram

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

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-sqrt.f64 N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lift-/.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lift-neg.f64 N/A

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

   13. frac-2neg N/A

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

   14. un-div-inv N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-+.f64 N/A

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

6. Applied rewrites 34.4%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125 - \frac{0.0625}{x}}{x}}{x}}}{\sqrt{x + 1}} \]

10. Final simplification 98.9%

    \[\leadsto \frac{\frac{0.5 - \frac{0.125 - \frac{0.0625}{x}}{x}}{x}}{\sqrt{1 + x}} \]
11. Add Preprocessing

Alternative 5: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (sqrt (/ 1.0 x))) (fma -2.0 x -1.5)))

C

double code(double x) {
	return -sqrt((1.0 / x)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-sqrt(Float64(1.0 / x))) / fma(-2.0, x, -1.5))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 35.8%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. distribute-lft-in N/A

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

    3. distribute-neg-in N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. distribute-lft-neg-in N/A

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

    7. metadata-eval N/A

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

    8. unpow2 N/A

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

    9. associate-*r* N/A

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

    10. lft-mult-inverse N/A

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

    11. *-lft-identity N/A

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

    12. *-commutative N/A

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

    13. *-commutative N/A

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

    14. lower-fma.f64 N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left({x}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}} \]

    15. *-commutative N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}}\right)\right)} \]

    16. associate-*l* N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right)\right)} \]

    17. lft-mult-inverse N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right)} \]

12. Applied rewrites 98.5%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]
13. Add Preprocessing

Alternative 6: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around -inf

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

8. Applied rewrites 97.1%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1.25}{x}, \sqrt{\frac{1}{x}}, \sqrt{\frac{1}{x}}\right) \cdot 0.5}{\color{blue}{x}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 97.2%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} \]
12. Add Preprocessing

Alternative 7: 97.9% 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 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-sqrt.f64 N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lift-/.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lift-neg.f64 N/A

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

   13. frac-2neg N/A

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

   14. un-div-inv N/A

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

   15. lift-sqrt.f64 N/A

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

   16. lift-+.f64 N/A

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

6. Applied rewrites 34.4%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 97.1

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

9. Applied rewrites 97.1%

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

10. Final simplification 97.1%

    \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{1 + x}} \]
11. Add Preprocessing

Alternative 8: 81.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 77.5%

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

Alternative 9: 36.8% 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 34.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 32.4%

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

Alternative 10: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

Developer Target 1: 38.3% 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 2024276 
(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)))))