2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.5%

Time: 8.4s

Alternatives: 9

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, 0.5× speedup

Math

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

FPCore

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

C

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

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)) / (-t_0 * (t_0 + (x / sqrt(x))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = -sqrt((1.0 / x)) / (-t_0 * (t_0 + (x / sqrt(x))));
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[((-t$95$0) * N[(t$95$0 + N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.4%

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

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. inv-pow N/A

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

   13. div-inv N/A

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

   14. lower-/.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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)) / (-t_0 * (t_0 + (x / sqrt(x))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (-1.0 / sqrt(x)) / (-t_0 * (t_0 + (x / sqrt(x))));
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[((-t$95$0) * N[(t$95$0 + N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.4%

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

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. inv-pow N/A

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

   13. div-inv N/A

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

   14. lower-/.f64 99.4

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

9. Applied rewrites 99.4%

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = -sqrt((1.0 / x)) / ((t_0 + sqrt(x)) * -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[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.4%

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

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (-1.0 / sqrt(x)) / ((t_0 + sqrt(x)) * -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[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.4%

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

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 5: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 - \frac{0.5 - \frac{0.375}{x}}{x}}{x}}{\sqrt{1 + x} + \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- 1.0 (/ (- 0.5 (/ 0.375 x)) x)) x) (+ (sqrt (+ 1.0 x)) (sqrt x))))

C

double code(double x) {
	return ((1.0 - ((0.5 - (0.375 / x)) / x)) / x) / (sqrt((1.0 + x)) + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 - ((0.5d0 - (0.375d0 / x)) / x)) / x) / (sqrt((1.0d0 + x)) + sqrt(x))
end function

Java

public static double code(double x) {
	return ((1.0 - ((0.5 - (0.375 / x)) / x)) / x) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

Python

def code(x):
	return ((1.0 - ((0.5 - (0.375 / x)) / x)) / x) / (math.sqrt((1.0 + x)) + math.sqrt(x))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.375 / x)) / x)) / x) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 - ((0.5 - (0.375 / x)) / x)) / x) / (sqrt((1.0 + x)) + sqrt(x));
end

Wolfram

code[x_] := N[(N[(N[(1.0 - N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 98.6

       \[\leadsto \frac{\frac{\left(\frac{0.375}{x \cdot x} + 1\right) - \color{blue}{\frac{0.5}{x}}}{x}}{\sqrt{x} + \sqrt{x + 1}} \]

7. Applied rewrites 98.6%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.375}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}}{\sqrt{x} + \sqrt{x + 1}} \]

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

    \[\leadsto \frac{\frac{1 - \frac{0.5 - \frac{0.375}{x}}{x}}{x}}{\sqrt{1 + x} + \sqrt{x}} \]
12. Add Preprocessing

Alternative 6: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.7%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(2 \cdot \frac{1}{x} + \frac{3}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}} \]
9. 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-rgt-in N/A

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

   3. distribute-neg-in N/A

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

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

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

   5. distribute-rgt-neg-in N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. rem-square-sqrt N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*r/ N/A

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

   12. unpow2 N/A

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

   13. rem-square-sqrt N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

10. Applied rewrites 81.0%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(\frac{-2}{x}, x \cdot x, -1.5\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 98.4%

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

Alternative 7: 80.4% 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 38.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.9%

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

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 79.4%

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

Alternative 8: 36.6% 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 38.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 35.8%

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

Alternative 9: 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 38.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 2024332 
(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)))))