sqrt(x + 9) - sqrt(x)

Percentage Accurate: 53.2% → 99.8%
Time: 4.1s
Alternatives: 7
Speedup: 1.1×

Specification

?
\[0 \leq x \land x \leq 1.79 \cdot 10^{+308}\]
\[\begin{array}{l} \\ \sqrt{x + 9} - \sqrt{x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 9.0)) (sqrt x)))
double code(double x) {
	return sqrt((x + 9.0)) - sqrt(x);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 9.0d0)) - sqrt(x)
end function

public static double code(double x) {
	return Math.sqrt((x + 9.0)) - Math.sqrt(x);
}

def code(x):
	return math.sqrt((x + 9.0)) - math.sqrt(x)

function code(x)
	return Float64(sqrt(Float64(x + 9.0)) - sqrt(x))
end

function tmp = code(x)
	tmp = sqrt((x + 9.0)) - sqrt(x);
end

code[x_] := N[(N[Sqrt[N[(x + 9.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{x + 9} - \sqrt{x}
\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 7 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: 53.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{x + 9} - \sqrt{x} \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 9.0)) (sqrt x)))
double code(double x) {
	return sqrt((x + 9.0)) - sqrt(x);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 9.0d0)) - sqrt(x)
end function

public static double code(double x) {
	return Math.sqrt((x + 9.0)) - Math.sqrt(x);
}

def code(x):
	return math.sqrt((x + 9.0)) - math.sqrt(x)

function code(x)
	return Float64(sqrt(Float64(x + 9.0)) - sqrt(x))
end

function tmp = code(x)
	tmp = sqrt((x + 9.0)) - sqrt(x);
end

code[x_] := N[(N[Sqrt[N[(x + 9.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{x + 9} - \sqrt{x}
\end{array}

Alternative 1: 99.8% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \frac{9}{\sqrt{x} + \sqrt{9 + x}} \end{array} \]
(FPCore (x) :precision binary64 (/ 9.0 (+ (sqrt x) (sqrt (+ 9.0 x)))))
double code(double x) {
	return 9.0 / (sqrt(x) + sqrt((9.0 + x)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 9.0d0 / (sqrt(x) + sqrt((9.0d0 + x)))
end function

public static double code(double x) {
	return 9.0 / (Math.sqrt(x) + Math.sqrt((9.0 + x)));
}

def code(x):
	return 9.0 / (math.sqrt(x) + math.sqrt((9.0 + x)))

function code(x)
	return Float64(9.0 / Float64(sqrt(x) + sqrt(Float64(9.0 + x))))
end

function tmp = code(x)
	tmp = 9.0 / (sqrt(x) + sqrt((9.0 + x)));
end

code[x_] := N[(9.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(9.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{9}{\sqrt{x} + \sqrt{9 + x}}
\end{array}
Derivation

  1. Initial program 52.9%

    \[\sqrt{x + 9} - \sqrt{x} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{\sqrt{x + 9} - \sqrt{x}} \]

    2. flip--N/A

      \[\leadsto \color{blue}{\frac{\sqrt{x + 9} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}}} \]

    3. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\sqrt{x + 9} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}}} \]

    4. lift-sqrt.f64N/A

      \[\leadsto \frac{\color{blue}{\sqrt{x + 9}} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

    5. lift-sqrt.f64N/A

      \[\leadsto \frac{\sqrt{x + 9} \cdot \color{blue}{\sqrt{x + 9}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

    6. rem-square-sqrtN/A

      \[\leadsto \frac{\color{blue}{\left(x + 9\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

    7. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 9\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

    8. lift-sqrt.f64N/A

      \[\leadsto \frac{\left(x + 9\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 9} + \sqrt{x}} \]

    9. rem-square-sqrtN/A

      \[\leadsto \frac{\left(x + 9\right) - \color{blue}{x}}{\sqrt{x + 9} + \sqrt{x}} \]

    10. lower--.f64N/A

      \[\leadsto \frac{\color{blue}{\left(x + 9\right) - x}}{\sqrt{x + 9} + \sqrt{x}} \]

    11. lift-+.f64N/A

      \[\leadsto \frac{\color{blue}{\left(x + 9\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

    12. +-commutativeN/A

      \[\leadsto \frac{\color{blue}{\left(9 + x\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

    13. lower-+.f64N/A

      \[\leadsto \frac{\color{blue}{\left(9 + x\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

    14. +-commutativeN/A

      \[\leadsto \frac{\left(9 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 9}}} \]

    15. lower-+.f6454.5

      \[\leadsto \frac{\left(9 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 9}}} \]

    16. lift-+.f64N/A

      \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{x + 9}}} \]

    17. +-commutativeN/A

      \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{9 + x}}} \]

    18. lower-+.f6454.5

      \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{9 + x}}} \]

  4. Applied rewrites54.5%

    \[\leadsto \color{blue}{\frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{9 + x}}} \]

  5. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{9}}{\sqrt{x} + \sqrt{9 + x}} \]
  6. Step-by-step derivation

    1. Applied rewrites99.8%

      \[\leadsto \frac{\color{blue}{9}}{\sqrt{x} + \sqrt{9 + x}} \]
    2. Add Preprocessing

    Alternative 2: 98.3% accurate, 0.2× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 9} - \sqrt{x} \leq 0.001:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right) - \sqrt{x}\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 9.0)) (sqrt x)) 0.001)
   (* (sqrt (pow x -1.0)) 4.5)
   (- (fma (fma -0.004629629629629629 x 0.16666666666666666) x 3.0) (sqrt x))))
    double code(double x) {
	double tmp;
	if ((sqrt((x + 9.0)) - sqrt(x)) <= 0.001) {
		tmp = sqrt(pow(x, -1.0)) * 4.5;
	} else {
		tmp = fma(fma(-0.004629629629629629, x, 0.16666666666666666), x, 3.0) - sqrt(x);
	}
	return tmp;
}

    function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 9.0)) - sqrt(x)) <= 0.001)
		tmp = Float64(sqrt((x ^ -1.0)) * 4.5);
	else
		tmp = Float64(fma(fma(-0.004629629629629629, x, 0.16666666666666666), x, 3.0) - sqrt(x));
	end
	return tmp
end

    code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 9.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 4.5), $MachinePrecision], N[(N[(N[(-0.004629629629629629 * x + 0.16666666666666666), $MachinePrecision] * x + 3.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\sqrt{x + 9} - \sqrt{x} \leq 0.001:\\
\;\;\;\;\sqrt{{x}^{-1}} \cdot 4.5\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right) - \sqrt{x}\\


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

    2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x)) < 1e-3

      1. Initial program 5.8%

        \[\sqrt{x + 9} - \sqrt{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. *-commutativeN/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{9}{2}} \]

        2. lower-*.f64N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{9}{2}} \]

        3. lower-sqrt.f64N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{9}{2} \]

        4. lower-/.f6498.7

          \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot 4.5 \]

      5. Applied rewrites98.7%

        \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 4.5} \]

      if 1e-3 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x))

      1. Initial program 100.0%

        \[\sqrt{x + 9} - \sqrt{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{\left(3 + x \cdot \left(\frac{1}{6} + \frac{-1}{216} \cdot x\right)\right)} - \sqrt{x} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{216} \cdot x\right) + 3\right)} - \sqrt{x} \]

        2. *-commutativeN/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + \frac{-1}{216} \cdot x\right) \cdot x} + 3\right) - \sqrt{x} \]

        3. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{-1}{216} \cdot x, x, 3\right)} - \sqrt{x} \]

        4. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{216} \cdot x + \frac{1}{6}}, x, 3\right) - \sqrt{x} \]

        5. lower-fma.f6498.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right)}, x, 3\right) - \sqrt{x} \]

      5. Applied rewrites98.2%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right)} - \sqrt{x} \]
    3. Recombined 2 regimes into one program.

    4. Final simplification98.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 9} - \sqrt{x} \leq 0.001:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right) - \sqrt{x}\\ \end{array} \]
    5. Add Preprocessing

    Alternative 3: 99.3% accurate, 0.5× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 9} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{4.5 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 9.0)) (sqrt x))))
   (if (<= t_0 1e-5) (/ (* 4.5 (sqrt x)) x) t_0)))
    double code(double x) {
	double t_0 = sqrt((x + 9.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (4.5 * sqrt(x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 9.0d0)) - sqrt(x)
    if (t_0 <= 1d-5) then
        tmp = (4.5d0 * sqrt(x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

    public static double code(double x) {
	double t_0 = Math.sqrt((x + 9.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (4.5 * Math.sqrt(x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

    def code(x):
	t_0 = math.sqrt((x + 9.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 1e-5:
		tmp = (4.5 * math.sqrt(x)) / x
	else:
		tmp = t_0
	return tmp

    function code(x)
	t_0 = Float64(sqrt(Float64(x + 9.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(Float64(4.5 * sqrt(x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

    function tmp_2 = code(x)
	t_0 = sqrt((x + 9.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = (4.5 * sqrt(x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

    code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 9.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[(4.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

    \begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{x + 9} - \sqrt{x}\\
\mathbf{if}\;t\_0 \leq 10^{-5}:\\
\;\;\;\;\frac{4.5 \cdot \sqrt{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;t\_0\\


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

    2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x)) < 1.00000000000000008e-5

      1. Initial program 5.3%

        \[\sqrt{x + 9} - \sqrt{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

        \[\leadsto \color{blue}{\frac{\frac{-81}{8} \cdot \sqrt{\frac{1}{x}} + \frac{9}{2} \cdot \sqrt{x}}{x}} \]
      4. Step-by-step derivation

        1. lower-/.f64N/A

          \[\leadsto \color{blue}{\frac{\frac{-81}{8} \cdot \sqrt{\frac{1}{x}} + \frac{9}{2} \cdot \sqrt{x}}{x}} \]

        2. lower-fma.f64N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\frac{1}{x}}, \frac{9}{2} \cdot \sqrt{x}\right)}}{x} \]

        3. lower-sqrt.f64N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{9}{2} \cdot \sqrt{x}\right)}{x} \]

        4. lower-/.f64N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{9}{2} \cdot \sqrt{x}\right)}{x} \]

        5. lower-*.f64N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{9}{2} \cdot \sqrt{x}}\right)}{x} \]

        6. lower-sqrt.f6499.5

          \[\leadsto \frac{\mathsf{fma}\left(-10.125, \sqrt{\frac{1}{x}}, 4.5 \cdot \color{blue}{\sqrt{x}}\right)}{x} \]

      5. Applied rewrites99.5%

        \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-10.125, \sqrt{\frac{1}{x}}, 4.5 \cdot \sqrt{x}\right)}{x}} \]

      6. Taylor expanded in x around -inf

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

        1. Applied rewrites99.0%

          \[\leadsto \frac{4.5 \cdot \sqrt{x}}{x} \]

        if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x))

        1. Initial program 99.7%

          \[\sqrt{x + 9} - \sqrt{x} \]
        2. Add Preprocessing
      8. Recombined 2 regimes into one program.
      9. Add Preprocessing

      Alternative 4: 98.2% accurate, 0.5× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 9} - \sqrt{x} \leq 0.001:\\ \;\;\;\;\frac{4.5 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right) - \sqrt{x}\\ \end{array} \end{array} \]
      (FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 9.0)) (sqrt x)) 0.001)
   (/ (* 4.5 (sqrt x)) x)
   (- (fma (fma -0.004629629629629629 x 0.16666666666666666) x 3.0) (sqrt x))))
      double code(double x) {
	double tmp;
	if ((sqrt((x + 9.0)) - sqrt(x)) <= 0.001) {
		tmp = (4.5 * sqrt(x)) / x;
	} else {
		tmp = fma(fma(-0.004629629629629629, x, 0.16666666666666666), x, 3.0) - sqrt(x);
	}
	return tmp;
}

      function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 9.0)) - sqrt(x)) <= 0.001)
		tmp = Float64(Float64(4.5 * sqrt(x)) / x);
	else
		tmp = Float64(fma(fma(-0.004629629629629629, x, 0.16666666666666666), x, 3.0) - sqrt(x));
	end
	return tmp
end

      code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 9.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(4.5 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(-0.004629629629629629 * x + 0.16666666666666666), $MachinePrecision] * x + 3.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\sqrt{x + 9} - \sqrt{x} \leq 0.001:\\
\;\;\;\;\frac{4.5 \cdot \sqrt{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right) - \sqrt{x}\\


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

      2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x)) < 1e-3

        1. Initial program 5.8%

          \[\sqrt{x + 9} - \sqrt{x} \]
        2. Add Preprocessing

        3. Taylor expanded in x around inf

          \[\leadsto \color{blue}{\frac{\frac{-81}{8} \cdot \sqrt{\frac{1}{x}} + \frac{9}{2} \cdot \sqrt{x}}{x}} \]
        4. Step-by-step derivation

          1. lower-/.f64N/A

            \[\leadsto \color{blue}{\frac{\frac{-81}{8} \cdot \sqrt{\frac{1}{x}} + \frac{9}{2} \cdot \sqrt{x}}{x}} \]

          2. lower-fma.f64N/A

            \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\frac{1}{x}}, \frac{9}{2} \cdot \sqrt{x}\right)}}{x} \]

          3. lower-sqrt.f64N/A

            \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{9}{2} \cdot \sqrt{x}\right)}{x} \]

          4. lower-/.f64N/A

            \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{9}{2} \cdot \sqrt{x}\right)}{x} \]

          5. lower-*.f64N/A

            \[\leadsto \frac{\mathsf{fma}\left(\frac{-81}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{9}{2} \cdot \sqrt{x}}\right)}{x} \]

          6. lower-sqrt.f6499.5

            \[\leadsto \frac{\mathsf{fma}\left(-10.125, \sqrt{\frac{1}{x}}, 4.5 \cdot \color{blue}{\sqrt{x}}\right)}{x} \]

        5. Applied rewrites99.5%

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-10.125, \sqrt{\frac{1}{x}}, 4.5 \cdot \sqrt{x}\right)}{x}} \]

        6. Taylor expanded in x around -inf

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

          1. Applied rewrites98.7%

            \[\leadsto \frac{4.5 \cdot \sqrt{x}}{x} \]

          if 1e-3 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 9 binary64))) (sqrt.f64 x))

          1. Initial program 100.0%

            \[\sqrt{x + 9} - \sqrt{x} \]
          2. Add Preprocessing

          3. Taylor expanded in x around 0

            \[\leadsto \color{blue}{\left(3 + x \cdot \left(\frac{1}{6} + \frac{-1}{216} \cdot x\right)\right)} - \sqrt{x} \]
          4. Step-by-step derivation

            1. +-commutativeN/A

              \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{216} \cdot x\right) + 3\right)} - \sqrt{x} \]

            2. *-commutativeN/A

              \[\leadsto \left(\color{blue}{\left(\frac{1}{6} + \frac{-1}{216} \cdot x\right) \cdot x} + 3\right) - \sqrt{x} \]

            3. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{-1}{216} \cdot x, x, 3\right)} - \sqrt{x} \]

            4. +-commutativeN/A

              \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{216} \cdot x + \frac{1}{6}}, x, 3\right) - \sqrt{x} \]

            5. lower-fma.f6498.2

              \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right)}, x, 3\right) - \sqrt{x} \]

          5. Applied rewrites98.2%

            \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.004629629629629629, x, 0.16666666666666666\right), x, 3\right)} - \sqrt{x} \]
        8. Recombined 2 regimes into one program.
        9. Add Preprocessing

        Alternative 5: 58.4% accurate, 1.1× speedup?

        \[\begin{array}{l} \\ \frac{9}{3 + \sqrt{x}} \end{array} \]
        (FPCore (x) :precision binary64 (/ 9.0 (+ 3.0 (sqrt x))))
        double code(double x) {
	return 9.0 / (3.0 + sqrt(x));
}

        real(8) function code(x)
    real(8), intent (in) :: x
    code = 9.0d0 / (3.0d0 + sqrt(x))
end function

        public static double code(double x) {
	return 9.0 / (3.0 + Math.sqrt(x));
}

        def code(x):
	return 9.0 / (3.0 + math.sqrt(x))

        function code(x)
	return Float64(9.0 / Float64(3.0 + sqrt(x)))
end

        function tmp = code(x)
	tmp = 9.0 / (3.0 + sqrt(x));
end

        code[x_] := N[(9.0 / N[(3.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

        \begin{array}{l}

\\
\frac{9}{3 + \sqrt{x}}
\end{array}
        Derivation

        1. Initial program 52.9%

          \[\sqrt{x + 9} - \sqrt{x} \]
        2. Add Preprocessing
        3. Step-by-step derivation

          1. lift--.f64N/A

            \[\leadsto \color{blue}{\sqrt{x + 9} - \sqrt{x}} \]

          2. flip--N/A

            \[\leadsto \color{blue}{\frac{\sqrt{x + 9} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}}} \]

          3. lower-/.f64N/A

            \[\leadsto \color{blue}{\frac{\sqrt{x + 9} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}}} \]

          4. lift-sqrt.f64N/A

            \[\leadsto \frac{\color{blue}{\sqrt{x + 9}} \cdot \sqrt{x + 9} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

          5. lift-sqrt.f64N/A

            \[\leadsto \frac{\sqrt{x + 9} \cdot \color{blue}{\sqrt{x + 9}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

          6. rem-square-sqrtN/A

            \[\leadsto \frac{\color{blue}{\left(x + 9\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

          7. lift-sqrt.f64N/A

            \[\leadsto \frac{\left(x + 9\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 9} + \sqrt{x}} \]

          8. lift-sqrt.f64N/A

            \[\leadsto \frac{\left(x + 9\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 9} + \sqrt{x}} \]

          9. rem-square-sqrtN/A

            \[\leadsto \frac{\left(x + 9\right) - \color{blue}{x}}{\sqrt{x + 9} + \sqrt{x}} \]

          10. lower--.f64N/A

            \[\leadsto \frac{\color{blue}{\left(x + 9\right) - x}}{\sqrt{x + 9} + \sqrt{x}} \]

          11. lift-+.f64N/A

            \[\leadsto \frac{\color{blue}{\left(x + 9\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

          12. +-commutativeN/A

            \[\leadsto \frac{\color{blue}{\left(9 + x\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

          13. lower-+.f64N/A

            \[\leadsto \frac{\color{blue}{\left(9 + x\right)} - x}{\sqrt{x + 9} + \sqrt{x}} \]

          14. +-commutativeN/A

            \[\leadsto \frac{\left(9 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 9}}} \]

          15. lower-+.f6454.5

            \[\leadsto \frac{\left(9 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{x + 9}}} \]

          16. lift-+.f64N/A

            \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{x + 9}}} \]

          17. +-commutativeN/A

            \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{9 + x}}} \]

          18. lower-+.f6454.5

            \[\leadsto \frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{\color{blue}{9 + x}}} \]

        4. Applied rewrites54.5%

          \[\leadsto \color{blue}{\frac{\left(9 + x\right) - x}{\sqrt{x} + \sqrt{9 + x}}} \]

        5. Taylor expanded in x around 0

          \[\leadsto \color{blue}{\frac{9}{3 + \sqrt{x}}} \]
        6. Step-by-step derivation

          1. lower-/.f64N/A

            \[\leadsto \color{blue}{\frac{9}{3 + \sqrt{x}}} \]

          2. lower-+.f64N/A

            \[\leadsto \frac{9}{\color{blue}{3 + \sqrt{x}}} \]

          3. lower-sqrt.f6458.0

            \[\leadsto \frac{9}{3 + \color{blue}{\sqrt{x}}} \]

        7. Applied rewrites58.0%

          \[\leadsto \color{blue}{\frac{9}{3 + \sqrt{x}}} \]
        8. Add Preprocessing

        Alternative 6: 51.8% accurate, 1.4× speedup?

        \[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666, x, 3 - \sqrt{x}\right) \end{array} \]
        (FPCore (x) :precision binary64 (fma 0.16666666666666666 x (- 3.0 (sqrt x))))
        double code(double x) {
	return fma(0.16666666666666666, x, (3.0 - sqrt(x)));
}

        function code(x)
	return fma(0.16666666666666666, x, Float64(3.0 - sqrt(x)))
end

        code[x_] := N[(0.16666666666666666 * x + N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

        \begin{array}{l}

\\
\mathsf{fma}\left(0.16666666666666666, x, 3 - \sqrt{x}\right)
\end{array}
        Derivation

        1. Initial program 52.9%

          \[\sqrt{x + 9} - \sqrt{x} \]
        2. Add Preprocessing

        3. Taylor expanded in x around 0

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

          1. +-commutativeN/A

            \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x + 3\right)} - \sqrt{x} \]

          2. associate--l+N/A

            \[\leadsto \color{blue}{\frac{1}{6} \cdot x + \left(3 - \sqrt{x}\right)} \]

          3. lower-fma.f64N/A

            \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x, 3 - \sqrt{x}\right)} \]

          4. lower--.f64N/A

            \[\leadsto \mathsf{fma}\left(\frac{1}{6}, x, \color{blue}{3 - \sqrt{x}}\right) \]

          5. lower-sqrt.f6451.1

            \[\leadsto \mathsf{fma}\left(0.16666666666666666, x, 3 - \color{blue}{\sqrt{x}}\right) \]

        5. Applied rewrites51.1%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 3 - \sqrt{x}\right)} \]
        6. Add Preprocessing

        Alternative 7: 49.8% accurate, 1.9× speedup?

        \[\begin{array}{l} \\ 3 - \sqrt{x} \end{array} \]
        (FPCore (x) :precision binary64 (- 3.0 (sqrt x)))
        double code(double x) {
	return 3.0 - sqrt(x);
}

        real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 - sqrt(x)
end function

        public static double code(double x) {
	return 3.0 - Math.sqrt(x);
}

        def code(x):
	return 3.0 - math.sqrt(x)

        function code(x)
	return Float64(3.0 - sqrt(x))
end

        function tmp = code(x)
	tmp = 3.0 - sqrt(x);
end

        code[x_] := N[(3.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

        \begin{array}{l}

\\
3 - \sqrt{x}
\end{array}
        Derivation

        1. Initial program 52.9%

          \[\sqrt{x + 9} - \sqrt{x} \]
        2. Add Preprocessing

        3. Taylor expanded in x around 0

          \[\leadsto \color{blue}{3} - \sqrt{x} \]
        4. Step-by-step derivation

          1. Applied rewrites49.4%

            \[\leadsto \color{blue}{3} - \sqrt{x} \]
          2. Add Preprocessing

          Reproduce

          ?
          herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x + 9) - sqrt(x)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1.79e+308))
  (- (sqrt (+ x 9.0)) (sqrt x)))