sqrt(x*(x-1))/(x-2)

Percentage Accurate: 99.9% → 99.9%
Time: 3.7s
Alternatives: 12
Speedup: 1.0×

Specification

?
\[-10000000000 \leq x \land x \leq 10000000000\]
\[\begin{array}{l} \\ \frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \end{array} \]
(FPCore (x) :precision binary64 (/ (sqrt (* x (- x 1.0))) (- x 2.0)))
double code(double x) {
	return sqrt((x * (x - 1.0))) / (x - 2.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * (x - 1.0d0))) / (x - 2.0d0)
end function

public static double code(double x) {
	return Math.sqrt((x * (x - 1.0))) / (x - 2.0);
}

def code(x):
	return math.sqrt((x * (x - 1.0))) / (x - 2.0)

function code(x)
	return Float64(sqrt(Float64(x * Float64(x - 1.0))) / Float64(x - 2.0))
end

function tmp = code(x)
	tmp = sqrt((x * (x - 1.0))) / (x - 2.0);
end

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

\begin{array}{l}

\\
\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 12 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \end{array} \]
(FPCore (x) :precision binary64 (/ (sqrt (* x (- x 1.0))) (- x 2.0)))
double code(double x) {
	return sqrt((x * (x - 1.0))) / (x - 2.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * (x - 1.0d0))) / (x - 2.0d0)
end function

public static double code(double x) {
	return Math.sqrt((x * (x - 1.0))) / (x - 2.0);
}

def code(x):
	return math.sqrt((x * (x - 1.0))) / (x - 2.0)

function code(x)
	return Float64(sqrt(Float64(x * Float64(x - 1.0))) / Float64(x - 2.0))
end

function tmp = code(x)
	tmp = sqrt((x * (x - 1.0))) / (x - 2.0);
end

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

\begin{array}{l}

\\
\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2}
\end{array}

Alternative 1: 99.9% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \frac{\left(x - -2\right) \cdot \sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)} \end{array} \]
(FPCore (x)
 :precision binary64
 (/ (* (- x -2.0) (sqrt (* (- x 1.0) x))) (fma x x -4.0)))
double code(double x) {
	return ((x - -2.0) * sqrt(((x - 1.0) * x))) / fma(x, x, -4.0);
}

function code(x)
	return Float64(Float64(Float64(x - -2.0) * sqrt(Float64(Float64(x - 1.0) * x))) / fma(x, x, -4.0))
end

code[x_] := N[(N[(N[(x - -2.0), $MachinePrecision] * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\left(x - -2\right) \cdot \sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)}
\end{array}
Derivation

  1. Initial program 99.9%

    \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-/.f64N/A

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

    2. lift--.f64N/A

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

    3. flip--N/A

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

    4. associate-/r/N/A

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

    5. associate-*l/N/A

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

    6. lower-/.f64N/A

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

    7. *-commutativeN/A

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

    8. lower-*.f64N/A

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

    9. metadata-evalN/A

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

    10. metadata-evalN/A

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

    11. sub-negN/A

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

    12. lower--.f64N/A

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

    13. metadata-evalN/A

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

    14. lift-*.f64N/A

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

    15. *-commutativeN/A

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

    16. lower-*.f64N/A

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

    17. sub-negN/A

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

    18. lower-fma.f64N/A

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

    19. metadata-evalN/A

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

    20. metadata-eval99.9

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

  4. Applied rewrites99.9%

    \[\leadsto \color{blue}{\frac{\left(x - -2\right) \cdot \sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)}} \]
  5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \leq -5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{x \cdot x - x}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1}, \frac{2.875}{x} + 1.5, 1\right)\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= (/ (sqrt (* x (- x 1.0))) (- x 2.0)) -5e-155)
   (/ (sqrt (- (* x x) x)) -2.0)
   (fma (pow x -1.0) (+ (/ 2.875 x) 1.5) 1.0)))
double code(double x) {
	double tmp;
	if ((sqrt((x * (x - 1.0))) / (x - 2.0)) <= -5e-155) {
		tmp = sqrt(((x * x) - x)) / -2.0;
	} else {
		tmp = fma(pow(x, -1.0), ((2.875 / x) + 1.5), 1.0);
	}
	return tmp;
}

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x * Float64(x - 1.0))) / Float64(x - 2.0)) <= -5e-155)
		tmp = Float64(sqrt(Float64(Float64(x * x) - x)) / -2.0);
	else
		tmp = fma((x ^ -1.0), Float64(Float64(2.875 / x) + 1.5), 1.0);
	end
	return tmp
end

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision], -5e-155], N[(N[Sqrt[N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] / -2.0), $MachinePrecision], N[(N[Power[x, -1.0], $MachinePrecision] * N[(N[(2.875 / x), $MachinePrecision] + 1.5), $MachinePrecision] + 1.0), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \leq -5 \cdot 10^{-155}:\\
\;\;\;\;\frac{\sqrt{x \cdot x - x}}{-2}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({x}^{-1}, \frac{2.875}{x} + 1.5, 1\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (/.f64 (sqrt.f64 (*.f64 x (-.f64 x #s(literal 1 binary64)))) (-.f64 x #s(literal 2 binary64))) < -4.9999999999999999e-155

    1. Initial program 99.9%

      \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-*.f64N/A

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

      2. lift--.f64N/A

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

      3. distribute-lft-out--N/A

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

      4. *-rgt-identityN/A

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

      5. lower--.f64N/A

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

      6. lower-*.f6499.9

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

    4. Applied rewrites99.9%

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

    5. Taylor expanded in x around 0

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

      1. Applied rewrites94.0%

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

      if -4.9999999999999999e-155 < (/.f64 (sqrt.f64 (*.f64 x (-.f64 x #s(literal 1 binary64)))) (-.f64 x #s(literal 2 binary64)))

      1. Initial program 99.3%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

        \[\leadsto \color{blue}{1 + \left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\frac{23}{8}}{{x}^{2}}\right)} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\frac{23}{8}}{{x}^{2}}\right) + 1} \]

        2. unpow2N/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\frac{23}{8}}{\color{blue}{x \cdot x}}\right) + 1 \]

        3. associate-/r*N/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \color{blue}{\frac{\frac{\frac{23}{8}}{x}}{x}}\right) + 1 \]

        4. metadata-evalN/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\frac{\color{blue}{\frac{23}{8} \cdot 1}}{x}}{x}\right) + 1 \]

        5. associate-*r/N/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x}}}{x}\right) + 1 \]

        6. associate-*l/N/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \color{blue}{\frac{\frac{23}{8}}{x} \cdot \frac{1}{x}}\right) + 1 \]

        7. metadata-evalN/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \frac{\color{blue}{\frac{23}{8} \cdot 1}}{x} \cdot \frac{1}{x}\right) + 1 \]

        8. associate-*r/N/A

          \[\leadsto \left(\frac{3}{2} \cdot \frac{1}{x} + \color{blue}{\left(\frac{23}{8} \cdot \frac{1}{x}\right)} \cdot \frac{1}{x}\right) + 1 \]

        9. distribute-rgt-outN/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}\right)} + 1 \]

        10. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, \frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}, 1\right)} \]

        11. lower-/.f64N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, \frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}, 1\right) \]

        12. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}, 1\right) \]

        13. lower-+.f64N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}, 1\right) \]

        14. associate-*r/N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \color{blue}{\frac{\frac{23}{8} \cdot 1}{x}} + \frac{3}{2}, 1\right) \]

        15. metadata-evalN/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{\color{blue}{\frac{23}{8}}}{x} + \frac{3}{2}, 1\right) \]

        16. lower-/.f6492.1

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \color{blue}{\frac{2.875}{x}} + 1.5, 1\right) \]

      5. Applied rewrites92.1%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, \frac{2.875}{x} + 1.5, 1\right)} \]
    7. Recombined 2 regimes into one program.

    8. Final simplification94.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \leq -5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{x \cdot x - x}}{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1}, \frac{2.875}{x} + 1.5, 1\right)\\ \end{array} \]
    9. Add Preprocessing

    Alternative 3: 99.9% accurate, 0.8× speedup?

    \[\begin{array}{l} \\ \frac{\sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)} \cdot \left(x - -2\right) \end{array} \]
    (FPCore (x)
 :precision binary64
 (* (/ (sqrt (* (- x 1.0) x)) (fma x x -4.0)) (- x -2.0)))
    double code(double x) {
	return (sqrt(((x - 1.0) * x)) / fma(x, x, -4.0)) * (x - -2.0);
}

    function code(x)
	return Float64(Float64(sqrt(Float64(Float64(x - 1.0) * x)) / fma(x, x, -4.0)) * Float64(x - -2.0))
end

    code[x_] := N[(N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision] * N[(x - -2.0), $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
\frac{\sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)} \cdot \left(x - -2\right)
\end{array}
    Derivation

    1. Initial program 99.9%

      \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-/.f64N/A

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

      2. lift--.f64N/A

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

      3. flip--N/A

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

      4. associate-/r/N/A

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

      5. lower-*.f64N/A

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

      6. lower-/.f64N/A

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

      7. lift-*.f64N/A

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

      8. *-commutativeN/A

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

      9. lower-*.f64N/A

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

      10. sub-negN/A

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

      11. lower-fma.f64N/A

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

      12. metadata-evalN/A

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

      13. metadata-evalN/A

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

      14. metadata-evalN/A

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

      15. metadata-evalN/A

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

      16. sub-negN/A

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

      17. lower--.f64N/A

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

      18. metadata-eval99.9

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

    4. Applied rewrites99.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(x - 1\right) \cdot x}}{\mathsf{fma}\left(x, x, -4\right)} \cdot \left(x - -2\right)} \]
    5. Add Preprocessing

    Alternative 4: 95.1% accurate, 0.9× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-x}}{x - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-0.5 - \frac{0.125}{x}\right) + x}{x - 2}\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= x -1e-308)
   (/ (sqrt (- x)) (- x 2.0))
   (/ (+ (- -0.5 (/ 0.125 x)) x) (- x 2.0))))
    double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = sqrt(-x) / (x - 2.0);
	} else {
		tmp = ((-0.5 - (0.125 / x)) + x) / (x - 2.0);
	}
	return tmp;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-308)) then
        tmp = sqrt(-x) / (x - 2.0d0)
    else
        tmp = (((-0.5d0) - (0.125d0 / x)) + x) / (x - 2.0d0)
    end if
    code = tmp
end function

    public static double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = Math.sqrt(-x) / (x - 2.0);
	} else {
		tmp = ((-0.5 - (0.125 / x)) + x) / (x - 2.0);
	}
	return tmp;
}

    def code(x):
	tmp = 0
	if x <= -1e-308:
		tmp = math.sqrt(-x) / (x - 2.0)
	else:
		tmp = ((-0.5 - (0.125 / x)) + x) / (x - 2.0)
	return tmp

    function code(x)
	tmp = 0.0
	if (x <= -1e-308)
		tmp = Float64(sqrt(Float64(-x)) / Float64(x - 2.0));
	else
		tmp = Float64(Float64(Float64(-0.5 - Float64(0.125 / x)) + x) / Float64(x - 2.0));
	end
	return tmp
end

    function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-308)
		tmp = sqrt(-x) / (x - 2.0);
	else
		tmp = ((-0.5 - (0.125 / x)) + x) / (x - 2.0);
	end
	tmp_2 = tmp;
end

    code[x_] := If[LessEqual[x, -1e-308], N[(N[Sqrt[(-x)], $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\
\;\;\;\;\frac{\sqrt{-x}}{x - 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-0.5 - \frac{0.125}{x}\right) + x}{x - 2}\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if x < -9.9999999999999991e-309

      1. Initial program 99.9%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

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

        1. mul-1-negN/A

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

        2. lower-neg.f6494.4

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

      5. Applied rewrites94.4%

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

      if -9.9999999999999991e-309 < x

      1. Initial program 99.2%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. +-commutativeN/A

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

        2. distribute-rgt-inN/A

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

        3. *-lft-identityN/A

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

        4. lower-+.f64N/A

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

        5. associate-*r/N/A

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

        6. associate-*l/N/A

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

        7. associate-/l*N/A

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

        8. *-rgt-identityN/A

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

        9. associate-*r/N/A

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

        10. rgt-mult-inverseN/A

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

        11. *-rgt-identityN/A

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

        12. distribute-lft-inN/A

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

        13. metadata-evalN/A

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

        14. mul-1-negN/A

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

        15. unsub-negN/A

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

        16. lower--.f64N/A

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

        17. associate-*r/N/A

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

        18. metadata-evalN/A

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

        19. lower-/.f6486.2

          \[\leadsto \frac{\left(-0.5 - \color{blue}{\frac{0.125}{x}}\right) + x}{x - 2} \]

      5. Applied rewrites86.2%

        \[\leadsto \frac{\color{blue}{\left(-0.5 - \frac{0.125}{x}\right) + x}}{x - 2} \]
    3. Recombined 2 regimes into one program.
    4. Add Preprocessing

    Alternative 5: 94.8% accurate, 1.0× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{-x}}{x - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + x}{x - 2}\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= x -1e-308) (/ (sqrt (- x)) (- x 2.0)) (/ (+ -0.5 x) (- x 2.0))))
    double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = sqrt(-x) / (x - 2.0);
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-308)) then
        tmp = sqrt(-x) / (x - 2.0d0)
    else
        tmp = ((-0.5d0) + x) / (x - 2.0d0)
    end if
    code = tmp
end function

    public static double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = Math.sqrt(-x) / (x - 2.0);
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    def code(x):
	tmp = 0
	if x <= -1e-308:
		tmp = math.sqrt(-x) / (x - 2.0)
	else:
		tmp = (-0.5 + x) / (x - 2.0)
	return tmp

    function code(x)
	tmp = 0.0
	if (x <= -1e-308)
		tmp = Float64(sqrt(Float64(-x)) / Float64(x - 2.0));
	else
		tmp = Float64(Float64(-0.5 + x) / Float64(x - 2.0));
	end
	return tmp
end

    function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-308)
		tmp = sqrt(-x) / (x - 2.0);
	else
		tmp = (-0.5 + x) / (x - 2.0);
	end
	tmp_2 = tmp;
end

    code[x_] := If[LessEqual[x, -1e-308], N[(N[Sqrt[(-x)], $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + x), $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\
\;\;\;\;\frac{\sqrt{-x}}{x - 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.5 + x}{x - 2}\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if x < -9.9999999999999991e-309

      1. Initial program 99.9%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

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

        1. mul-1-negN/A

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

        2. lower-neg.f6494.4

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

      5. Applied rewrites94.4%

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

      if -9.9999999999999991e-309 < x

      1. Initial program 99.2%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. sub-negN/A

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

        2. +-commutativeN/A

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

        3. distribute-rgt-inN/A

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

        4. distribute-lft-neg-inN/A

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

        5. metadata-evalN/A

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

        6. associate-*l*N/A

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

        7. lft-mult-inverseN/A

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

        8. metadata-evalN/A

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

        9. *-lft-identityN/A

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

        10. lower-+.f6476.5

          \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]

      5. Applied rewrites76.5%

        \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]
    3. Recombined 2 regimes into one program.
    4. Add Preprocessing

    Alternative 6: 99.9% accurate, 1.0× speedup?

    \[\begin{array}{l} \\ \frac{\sqrt{x \cdot x - x}}{x - 2} \end{array} \]
    (FPCore (x) :precision binary64 (/ (sqrt (- (* x x) x)) (- x 2.0)))
    double code(double x) {
	return sqrt(((x * x) - x)) / (x - 2.0);
}

    real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((x * x) - x)) / (x - 2.0d0)
end function

    public static double code(double x) {
	return Math.sqrt(((x * x) - x)) / (x - 2.0);
}

    def code(x):
	return math.sqrt(((x * x) - x)) / (x - 2.0)

    function code(x)
	return Float64(sqrt(Float64(Float64(x * x) - x)) / Float64(x - 2.0))
end

    function tmp = code(x)
	tmp = sqrt(((x * x) - x)) / (x - 2.0);
end

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

    \begin{array}{l}

\\
\frac{\sqrt{x \cdot x - x}}{x - 2}
\end{array}
    Derivation

    1. Initial program 99.9%

      \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-*.f64N/A

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

      2. lift--.f64N/A

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

      3. distribute-lft-out--N/A

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

      4. *-rgt-identityN/A

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

      5. lower--.f64N/A

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

      6. lower-*.f6499.9

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

    4. Applied rewrites99.9%

      \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x - x}}}{x - 2} \]
    5. Add Preprocessing

    Alternative 7: 99.9% accurate, 1.0× speedup?

    \[\begin{array}{l} \\ \frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \end{array} \]
    (FPCore (x) :precision binary64 (/ (sqrt (* x (- x 1.0))) (- x 2.0)))
    double code(double x) {
	return sqrt((x * (x - 1.0))) / (x - 2.0);
}

    real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * (x - 1.0d0))) / (x - 2.0d0)
end function

    public static double code(double x) {
	return Math.sqrt((x * (x - 1.0))) / (x - 2.0);
}

    def code(x):
	return math.sqrt((x * (x - 1.0))) / (x - 2.0)

    function code(x)
	return Float64(sqrt(Float64(x * Float64(x - 1.0))) / Float64(x - 2.0))
end

    function tmp = code(x)
	tmp = sqrt((x * (x - 1.0))) / (x - 2.0);
end

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

    \begin{array}{l}

\\
\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2}
\end{array}
    Derivation

    1. Initial program 99.9%

      \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
    2. Add Preprocessing
    3. Add Preprocessing

    Alternative 8: 10.9% accurate, 1.4× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{0.5 - x}{x - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + x}{x - 2}\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= x -1e-308) (/ (- 0.5 x) (- x 2.0)) (/ (+ -0.5 x) (- x 2.0))))
    double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = (0.5 - x) / (x - 2.0);
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-308)) then
        tmp = (0.5d0 - x) / (x - 2.0d0)
    else
        tmp = ((-0.5d0) + x) / (x - 2.0d0)
    end if
    code = tmp
end function

    public static double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = (0.5 - x) / (x - 2.0);
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    def code(x):
	tmp = 0
	if x <= -1e-308:
		tmp = (0.5 - x) / (x - 2.0)
	else:
		tmp = (-0.5 + x) / (x - 2.0)
	return tmp

    function code(x)
	tmp = 0.0
	if (x <= -1e-308)
		tmp = Float64(Float64(0.5 - x) / Float64(x - 2.0));
	else
		tmp = Float64(Float64(-0.5 + x) / Float64(x - 2.0));
	end
	return tmp
end

    function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-308)
		tmp = (0.5 - x) / (x - 2.0);
	else
		tmp = (-0.5 + x) / (x - 2.0);
	end
	tmp_2 = tmp;
end

    code[x_] := If[LessEqual[x, -1e-308], N[(N[(0.5 - x), $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + x), $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\
\;\;\;\;\frac{0.5 - x}{x - 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.5 + x}{x - 2}\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if x < -9.9999999999999991e-309

      1. Initial program 99.9%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around -inf

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

        1. mul-1-negN/A

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

        2. neg-sub0N/A

          \[\leadsto \frac{\color{blue}{0 - x \cdot \left(1 - \frac{1}{2} \cdot \frac{1}{x}\right)}}{x - 2} \]

        3. sub-negN/A

          \[\leadsto \frac{0 - x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{x - 2} \]

        4. +-commutativeN/A

          \[\leadsto \frac{0 - x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right) + 1\right)}}{x - 2} \]

        5. distribute-rgt-inN/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right) \cdot x + 1 \cdot x\right)}}{x - 2} \]

        6. distribute-lft-neg-inN/A

          \[\leadsto \frac{0 - \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{x}\right)} \cdot x + 1 \cdot x\right)}{x - 2} \]

        7. metadata-evalN/A

          \[\leadsto \frac{0 - \left(\left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{x}\right) \cdot x + 1 \cdot x\right)}{x - 2} \]

        8. associate-*l*N/A

          \[\leadsto \frac{0 - \left(\color{blue}{\frac{-1}{2} \cdot \left(\frac{1}{x} \cdot x\right)} + 1 \cdot x\right)}{x - 2} \]

        9. lft-mult-inverseN/A

          \[\leadsto \frac{0 - \left(\frac{-1}{2} \cdot \color{blue}{1} + 1 \cdot x\right)}{x - 2} \]

        10. metadata-evalN/A

          \[\leadsto \frac{0 - \left(\color{blue}{\frac{-1}{2}} + 1 \cdot x\right)}{x - 2} \]

        11. *-lft-identityN/A

          \[\leadsto \frac{0 - \left(\frac{-1}{2} + \color{blue}{x}\right)}{x - 2} \]

        12. associate--r+N/A

          \[\leadsto \frac{\color{blue}{\left(0 - \frac{-1}{2}\right) - x}}{x - 2} \]

        13. metadata-evalN/A

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

        14. lower--.f648.8

          \[\leadsto \frac{\color{blue}{0.5 - x}}{x - 2} \]

      5. Applied rewrites8.8%

        \[\leadsto \frac{\color{blue}{0.5 - x}}{x - 2} \]

      if -9.9999999999999991e-309 < x

      1. Initial program 99.2%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. sub-negN/A

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

        2. +-commutativeN/A

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

        3. distribute-rgt-inN/A

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

        4. distribute-lft-neg-inN/A

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

        5. metadata-evalN/A

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

        6. associate-*l*N/A

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

        7. lft-mult-inverseN/A

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

        8. metadata-evalN/A

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

        9. *-lft-identityN/A

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

        10. lower-+.f6476.5

          \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]

      5. Applied rewrites76.5%

        \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]
    3. Recombined 2 regimes into one program.
    4. Add Preprocessing

    Alternative 9: 10.0% accurate, 1.4× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + x}{x - 2}\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= x -1e-308) -1.0 (/ (+ -0.5 x) (- x 2.0))))
    double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = -1.0;
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-308)) then
        tmp = -1.0d0
    else
        tmp = ((-0.5d0) + x) / (x - 2.0d0)
    end if
    code = tmp
end function

    public static double code(double x) {
	double tmp;
	if (x <= -1e-308) {
		tmp = -1.0;
	} else {
		tmp = (-0.5 + x) / (x - 2.0);
	}
	return tmp;
}

    def code(x):
	tmp = 0
	if x <= -1e-308:
		tmp = -1.0
	else:
		tmp = (-0.5 + x) / (x - 2.0)
	return tmp

    function code(x)
	tmp = 0.0
	if (x <= -1e-308)
		tmp = -1.0;
	else
		tmp = Float64(Float64(-0.5 + x) / Float64(x - 2.0));
	end
	return tmp
end

    function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-308)
		tmp = -1.0;
	else
		tmp = (-0.5 + x) / (x - 2.0);
	end
	tmp_2 = tmp;
end

    code[x_] := If[LessEqual[x, -1e-308], -1.0, N[(N[(-0.5 + x), $MachinePrecision] / N[(x - 2.0), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.5 + x}{x - 2}\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if x < -9.9999999999999991e-309

      1. Initial program 99.9%

        \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
      2. Add Preprocessing

      3. Taylor expanded in x around -inf

        \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} - 1} \]
      4. Step-by-step derivation

        1. sub-negN/A

          \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)} \]

        2. metadata-evalN/A

          \[\leadsto -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \color{blue}{-1} \]

        3. +-commutativeN/A

          \[\leadsto \color{blue}{-1 + -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

        4. mul-1-negN/A

          \[\leadsto -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}\right)\right)} \]

        5. unsub-negN/A

          \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

        6. lower--.f64N/A

          \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

        7. lower-/.f64N/A

          \[\leadsto -1 - \color{blue}{\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

        8. +-commutativeN/A

          \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

        9. lower-+.f64N/A

          \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

        10. associate-*r/N/A

          \[\leadsto -1 - \frac{\color{blue}{\frac{\frac{23}{8} \cdot 1}{x}} + \frac{3}{2}}{x} \]

        11. metadata-evalN/A

          \[\leadsto -1 - \frac{\frac{\color{blue}{\frac{23}{8}}}{x} + \frac{3}{2}}{x} \]

        12. lower-/.f645.7

          \[\leadsto -1 - \frac{\color{blue}{\frac{2.875}{x}} + 1.5}{x} \]

      5. Applied rewrites5.7%

        \[\leadsto \color{blue}{-1 - \frac{\frac{2.875}{x} + 1.5}{x}} \]

      6. Taylor expanded in x around inf

        \[\leadsto -1 \]
      7. Step-by-step derivation

        1. Applied rewrites8.0%

          \[\leadsto -1 \]

        if -9.9999999999999991e-309 < x

        1. Initial program 99.2%

          \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
        2. Add Preprocessing

        3. Taylor expanded in x around inf

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

          1. sub-negN/A

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

          2. +-commutativeN/A

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

          3. distribute-rgt-inN/A

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

          4. distribute-lft-neg-inN/A

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

          5. metadata-evalN/A

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

          6. associate-*l*N/A

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

          7. lft-mult-inverseN/A

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

          8. metadata-evalN/A

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

          9. *-lft-identityN/A

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

          10. lower-+.f6476.5

            \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]

        5. Applied rewrites76.5%

          \[\leadsto \frac{\color{blue}{-0.5 + x}}{x - 2} \]
      8. Recombined 2 regimes into one program.
      9. Add Preprocessing

      Alternative 10: 9.8% accurate, 1.6× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1.5}{x} + 1\\ \end{array} \end{array} \]
      (FPCore (x) :precision binary64 (if (<= x 2.0) -1.0 (+ (/ 1.5 x) 1.0)))
      double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = -1.0;
	} else {
		tmp = (1.5 / x) + 1.0;
	}
	return tmp;
}

      real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = -1.0d0
    else
        tmp = (1.5d0 / x) + 1.0d0
    end if
    code = tmp
end function

      public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = -1.0;
	} else {
		tmp = (1.5 / x) + 1.0;
	}
	return tmp;
}

      def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = -1.0
	else:
		tmp = (1.5 / x) + 1.0
	return tmp

      function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(1.5 / x) + 1.0);
	end
	return tmp
end

      function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = -1.0;
	else
		tmp = (1.5 / x) + 1.0;
	end
	tmp_2 = tmp;
end

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

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 2:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\frac{1.5}{x} + 1\\


\end{array}
\end{array}
      Derivation
      1. Split input into 2 regimes

      2. if x < 2

        1. Initial program 99.9%

          \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
        2. Add Preprocessing

        3. Taylor expanded in x around -inf

          \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} - 1} \]
        4. Step-by-step derivation

          1. sub-negN/A

            \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)} \]

          2. metadata-evalN/A

            \[\leadsto -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \color{blue}{-1} \]

          3. +-commutativeN/A

            \[\leadsto \color{blue}{-1 + -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          4. mul-1-negN/A

            \[\leadsto -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}\right)\right)} \]

          5. unsub-negN/A

            \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          6. lower--.f64N/A

            \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          7. lower-/.f64N/A

            \[\leadsto -1 - \color{blue}{\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          8. +-commutativeN/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

          9. lower-+.f64N/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

          10. associate-*r/N/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{\frac{23}{8} \cdot 1}{x}} + \frac{3}{2}}{x} \]

          11. metadata-evalN/A

            \[\leadsto -1 - \frac{\frac{\color{blue}{\frac{23}{8}}}{x} + \frac{3}{2}}{x} \]

          12. lower-/.f645.7

            \[\leadsto -1 - \frac{\color{blue}{\frac{2.875}{x}} + 1.5}{x} \]

        5. Applied rewrites5.7%

          \[\leadsto \color{blue}{-1 - \frac{\frac{2.875}{x} + 1.5}{x}} \]

        6. Taylor expanded in x around inf

          \[\leadsto -1 \]
        7. Step-by-step derivation

          1. Applied rewrites8.1%

            \[\leadsto -1 \]

          if 2 < x

          1. Initial program 99.3%

            \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
          2. Add Preprocessing

          3. Taylor expanded in x around inf

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

            1. +-commutativeN/A

              \[\leadsto \color{blue}{\frac{3}{2} \cdot \frac{1}{x} + 1} \]

            2. lower-+.f64N/A

              \[\leadsto \color{blue}{\frac{3}{2} \cdot \frac{1}{x} + 1} \]

            3. associate-*r/N/A

              \[\leadsto \color{blue}{\frac{\frac{3}{2} \cdot 1}{x}} + 1 \]

            4. metadata-evalN/A

              \[\leadsto \frac{\color{blue}{\frac{3}{2}}}{x} + 1 \]

            5. lower-/.f6478.3

              \[\leadsto \color{blue}{\frac{1.5}{x}} + 1 \]

          5. Applied rewrites78.3%

            \[\leadsto \color{blue}{\frac{1.5}{x} + 1} \]
        8. Recombined 2 regimes into one program.
        9. Add Preprocessing

        Alternative 11: 7.8% accurate, 33.0× speedup?

        \[\begin{array}{l} \\ -1 \end{array} \]
        (FPCore (x) :precision binary64 -1.0)
        double code(double x) {
	return -1.0;
}

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

        public static double code(double x) {
	return -1.0;
}

        def code(x):
	return -1.0

        function code(x)
	return -1.0
end

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

        code[x_] := -1.0

        \begin{array}{l}

\\
-1
\end{array}
        Derivation

        1. Initial program 99.9%

          \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
        2. Add Preprocessing

        3. Taylor expanded in x around -inf

          \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} - 1} \]
        4. Step-by-step derivation

          1. sub-negN/A

            \[\leadsto \color{blue}{-1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)} \]

          2. metadata-evalN/A

            \[\leadsto -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x} + \color{blue}{-1} \]

          3. +-commutativeN/A

            \[\leadsto \color{blue}{-1 + -1 \cdot \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          4. mul-1-negN/A

            \[\leadsto -1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}\right)\right)} \]

          5. unsub-negN/A

            \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          6. lower--.f64N/A

            \[\leadsto \color{blue}{-1 - \frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          7. lower-/.f64N/A

            \[\leadsto -1 - \color{blue}{\frac{\frac{3}{2} + \frac{23}{8} \cdot \frac{1}{x}}{x}} \]

          8. +-commutativeN/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

          9. lower-+.f64N/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{23}{8} \cdot \frac{1}{x} + \frac{3}{2}}}{x} \]

          10. associate-*r/N/A

            \[\leadsto -1 - \frac{\color{blue}{\frac{\frac{23}{8} \cdot 1}{x}} + \frac{3}{2}}{x} \]

          11. metadata-evalN/A

            \[\leadsto -1 - \frac{\frac{\color{blue}{\frac{23}{8}}}{x} + \frac{3}{2}}{x} \]

          12. lower-/.f645.6

            \[\leadsto -1 - \frac{\color{blue}{\frac{2.875}{x}} + 1.5}{x} \]

        5. Applied rewrites5.6%

          \[\leadsto \color{blue}{-1 - \frac{\frac{2.875}{x} + 1.5}{x}} \]

        6. Taylor expanded in x around inf

          \[\leadsto -1 \]
        7. Step-by-step derivation

          1. Applied rewrites7.9%

            \[\leadsto -1 \]
          2. Add Preprocessing

          Alternative 12: 3.2% accurate, 33.0× speedup?

          \[\begin{array}{l} \\ 1 \end{array} \]
          (FPCore (x) :precision binary64 1.0)
          double code(double x) {
	return 1.0;
}

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

          public static double code(double x) {
	return 1.0;
}

          def code(x):
	return 1.0

          function code(x)
	return 1.0
end

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

          code[x_] := 1.0

          \begin{array}{l}

\\
1
\end{array}
          Derivation

          1. Initial program 99.9%

            \[\frac{\sqrt{x \cdot \left(x - 1\right)}}{x - 2} \]
          2. Add Preprocessing

          3. Taylor expanded in x around inf

            \[\leadsto \color{blue}{1} \]
          4. Step-by-step derivation

            1. Applied rewrites3.1%

              \[\leadsto \color{blue}{1} \]
            2. Add Preprocessing

            Reproduce

            ?
            herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x*(x-1))/(x-2)"
  :precision binary64
  :pre (and (<= -10000000000.0 x) (<= x 10000000000.0))
  (/ (sqrt (* x (- x 1.0))) (- x 2.0)))