2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.3%

Time: 8.1s

Alternatives: 9

Speedup: 1.4×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.7% 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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return ((1.0 / t_0) * (1.0 / (t_0 + (x / sqrt(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 / t_0) * (1.0d0 / (t_0 + (x / sqrt(x))))) / sqrt(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 37.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 38.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 38.8

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

6. Applied rewrites 38.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.2%

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

    1. lift-sqrt.f64 N/A

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

    2. pow1/2 N/A

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

    3. metadata-eval N/A

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

    4. pow-plus N/A

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

    5. metadata-eval N/A

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

    6. pow-flip N/A

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

    7. pow1/2 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. associate-/r/ N/A

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

    10. clear-num N/A

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

    11. lift-/.f64 99.4

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x}} \cdot \frac{1}{\sqrt{1 + x} + \frac{x}{\sqrt{x}}}}{\sqrt{x}} \]
13. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (1.0 / (t_0 + sqrt(x))) / (t_0 * 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 / (t_0 + sqrt(x))) / (t_0 * sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 37.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 38.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 38.8

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

6. Applied rewrites 38.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-*r/ N/A

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

    5. associate-/l/ N/A

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

    6. lower-/.f64 N/A

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x} \cdot \sqrt{x}} \]
13. Add Preprocessing

Alternative 3: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (1.0 / t_0) / ((t_0 + sqrt(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 / t_0) / ((t_0 + sqrt(x)) * sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 37.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 38.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 38.8

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

6. Applied rewrites 38.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. lift-/.f64 N/A

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

    5. frac-times N/A

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

    6. lift-+.f64 N/A

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

    7. +-commutative N/A

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

    8. lift-sqrt.f64 N/A

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

    9. lift-+.f64 N/A

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

    10. lift-sqrt.f64 N/A

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

    11. lift-neg.f64 N/A

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

    12. lift-sqrt.f64 N/A

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\sqrt{1 + x}}}{\mathsf{fma}\left(-2, x, -0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ -1.0 (sqrt (+ 1.0 x))) (fma -2.0 x -0.5)))

C

double code(double x) {
	return (-1.0 / sqrt((1.0 + x))) / fma(-2.0, x, -0.5);
}

Julia

function code(x)
	return Float64(Float64(-1.0 / sqrt(Float64(1.0 + x))) / fma(-2.0, x, -0.5))
end

Wolfram

code[x_] := N[(N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * x + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. distribute-lft-in N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lft-mult-inverse N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

7. Applied rewrites 37.8%

   \[\leadsto \frac{\left(\left(x + 1\right) - x\right) \cdot \frac{-1}{\sqrt{x + 1}}}{\color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right)}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 37.8

       \[\leadsto \frac{-\left(\left(1 + x\right) - x\right)}{\color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right) \cdot \sqrt{x + 1}}} \]

9. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\frac{-\left(\left(1 + x\right) - x\right)}{\mathsf{fma}\left(-2, x, -0.5\right) \cdot \sqrt{1 + x}}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. lift-+.f64 N/A

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

    7. associate-/r* N/A

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

11. Applied rewrites 98.7%

    \[\leadsto \color{blue}{\frac{\frac{-1}{\sqrt{1 + x}}}{\mathsf{fma}\left(-2, x, -0.5\right)}} \]
12. Add Preprocessing

Alternative 5: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\mathsf{fma}\left(-2, x, -0.5\right)}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ -1.0 (fma -2.0 x -0.5)) (sqrt (+ 1.0 x))))

C

double code(double x) {
	return (-1.0 / fma(-2.0, x, -0.5)) / sqrt((1.0 + x));
}

Julia

function code(x)
	return Float64(Float64(-1.0 / fma(-2.0, x, -0.5)) / sqrt(Float64(1.0 + x)))
end

Wolfram

code[x_] := N[(N[(-1.0 / N[(-2.0 * x + -0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. distribute-lft-in N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lft-mult-inverse N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

7. Applied rewrites 37.8%

   \[\leadsto \frac{\left(\left(x + 1\right) - x\right) \cdot \frac{-1}{\sqrt{x + 1}}}{\color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right)}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 37.8

       \[\leadsto \frac{-\left(\left(1 + x\right) - x\right)}{\color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right) \cdot \sqrt{x + 1}}} \]

9. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\frac{-\left(\left(1 + x\right) - x\right)}{\mathsf{fma}\left(-2, x, -0.5\right) \cdot \sqrt{1 + x}}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lift-+.f64 N/A

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

    5. +-commutative N/A

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

    6. lift-+.f64 N/A

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

    7. lower-/.f64 N/A

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

11. Applied rewrites 98.7%

    \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(-2, x, -0.5\right)}}{\sqrt{1 + x}}} \]
12. Add Preprocessing

Alternative 6: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 37.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 38.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 38.8

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

6. Applied rewrites 38.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{3}{8} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{-\sqrt{x}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 98.1%

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

Alternative 7: 38.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (fma (/ (* x x) x) -2.0 -0.5)))

C

double code(double x) {
	return -1.0 / fma(((x * x) / x), -2.0, -0.5);
}

Julia

function code(x)
	return Float64(-1.0 / fma(Float64(Float64(x * x) / x), -2.0, -0.5))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(x * x), $MachinePrecision] / x), $MachinePrecision] * -2.0 + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. distribute-lft-in N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lft-mult-inverse N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

7. Applied rewrites 37.8%

   \[\leadsto \frac{\left(\left(x + 1\right) - x\right) \cdot \frac{-1}{\sqrt{x + 1}}}{\color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right)}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 37.0%

    \[\leadsto \frac{\color{blue}{-1}}{\mathsf{fma}\left(\frac{x \cdot x}{x}, -2, -0.5\right)} \]
11. Add Preprocessing

Alternative 8: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.5

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

5. Applied rewrites 5.5%

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

7. Applied rewrites 5.5%

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

9. Applied rewrites 5.5%

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

11. Applied rewrites 36.2%

    \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
12. 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 37.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.5

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

5. Applied rewrites 5.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
6. Add Preprocessing

Developer Target 1: 38.7% 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 2024331 
(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)))))