Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.8% → 99.9%

Time: 8.1s

Alternatives: 8

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \cdot 6 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (- x 1.0) (fma (sqrt x) 4.0 (+ 1.0 x))) 6.0))

C

double code(double x) {
	return ((x - 1.0) / fma(sqrt(x), 4.0, (1.0 + x))) * 6.0;
}

Julia

function code(x)
	return Float64(Float64(Float64(x - 1.0) / fma(sqrt(x), 4.0, Float64(1.0 + x))) * 6.0)
end

Wolfram

code[x_] := N[(N[(N[(x - 1.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{6 \cdot \frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6} \]

   6. lower-/.f64 100.0

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot 6 \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x - 1}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot 6 \]

   8. +-commutative N/A

      \[\leadsto \frac{x - 1}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \cdot 6 \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{x - 1}{\color{blue}{4 \cdot \sqrt{x}} + \left(x + 1\right)} \cdot 6 \]

   10. *-commutative N/A

       \[\leadsto \frac{x - 1}{\color{blue}{\sqrt{x} \cdot 4} + \left(x + 1\right)} \cdot 6 \]

   11. lower-fma.f64 100.0

       \[\leadsto \frac{x - 1}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot 6 \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \cdot 6 \]

   13. +-commutative N/A

       \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1 + x}\right)} \cdot 6 \]

   14. lower-+.f64 100.0

       \[\leadsto \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1 + x}\right)} \cdot 6 \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \cdot 6} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5)
   (/ (fma 6.0 x -6.0) (fma (sqrt x) 4.0 1.0))
   (/ (* x 6.0) (fma 4.0 (sqrt x) (+ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = fma(6.0, x, -6.0) / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (x * 6.0) / fma(4.0, sqrt(x), (1.0 + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(fma(6.0, x, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(x * 6.0) / fma(4.0, sqrt(x), Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(N[(6.0 * x + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(6 + \left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{6 \cdot x + \left(\left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right) \cdot x}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \left(\color{blue}{-6} \cdot \frac{1}{x}\right) \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6 \cdot \left(\frac{1}{x} \cdot x\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{6 \cdot x + -6 \cdot \color{blue}{1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, x, -6\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(1 + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(1 + x\right)} \]

      7. lower-fma.f64 100.0

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

      4. lower-sqrt.f64 98.5

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4, 1\right)} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

   if 3.5 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(6 + \left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{6 \cdot x + \left(\left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right) \cdot x}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \left(\color{blue}{-6} \cdot \frac{1}{x}\right) \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6 \cdot \left(\frac{1}{x} \cdot x\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{6 \cdot x + -6 \cdot \color{blue}{1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 99.7

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, x, -6\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(1 + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(1 + x\right)} \]

      7. lower-fma.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \]

      2. lower-*.f64 97.3

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \]

   10. Applied rewrites 97.3%

       \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 27:\\ \;\;\;\;\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-24}{\sqrt{x}} + 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 27.0)
   (/ (fma 6.0 x -6.0) (fma (sqrt x) 4.0 1.0))
   (+ (/ -24.0 (sqrt x)) 6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 27.0) {
		tmp = fma(6.0, x, -6.0) / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (-24.0 / sqrt(x)) + 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 27.0)
		tmp = Float64(fma(6.0, x, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(-24.0 / sqrt(x)) + 6.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 27.0], N[(N[(6.0 * x + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 27

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(6 + \left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{6 \cdot x + \left(\left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right) \cdot x}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \left(\color{blue}{-6} \cdot \frac{1}{x}\right) \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6 \cdot \left(\frac{1}{x} \cdot x\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{6 \cdot x + -6 \cdot \color{blue}{1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{6 \cdot x + \color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, x, -6\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(1 + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(1 + x\right)} \]

      7. lower-fma.f64 100.0

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

      4. lower-sqrt.f64 98.5

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4, 1\right)} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

   if 27 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      3. flip-+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   4. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

      3. metadata-eval N/A

         \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

      9. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.2%

      \[\leadsto \frac{-24}{\sqrt{x}} + \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 15:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-24}{\sqrt{x}} + 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 15.0)
   (/ -6.0 (fma (sqrt x) 4.0 (+ 1.0 x)))
   (+ (/ -24.0 (sqrt x)) 6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 15.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (1.0 + x));
	} else {
		tmp = (-24.0 / sqrt(x)) + 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 15.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(1.0 + x)));
	else
		tmp = Float64(Float64(-24.0 / sqrt(x)) + 6.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 15.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 15

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x}} + \left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{\sqrt{x} \cdot 4} + \left(x + 1\right)} \]

      5. lower-fma.f64 98.5

         \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1 + x}\right)} \]

      8. lift-+.f64 98.5

         \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1 + x}\right)} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}} \]

   if 15 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      3. flip-+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   4. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

      3. metadata-eval N/A

         \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

      9. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.2%

      \[\leadsto \frac{-24}{\sqrt{x}} + \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma 6.0 x -6.0) (fma 4.0 (sqrt x) (+ 1.0 x))))

C

double code(double x) {
	return fma(6.0, x, -6.0) / fma(4.0, sqrt(x), (1.0 + x));
}

Julia

function code(x)
	return Float64(fma(6.0, x, -6.0) / fma(4.0, sqrt(x), Float64(1.0 + x)))
end

Wolfram

code[x_] := N[(N[(6.0 * x + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left(6 + \left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{6 \cdot x + \left(\left(\mathsf{neg}\left(6\right)\right) \cdot \frac{1}{x}\right) \cdot x}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{6 \cdot x + \left(\color{blue}{-6} \cdot \frac{1}{x}\right) \cdot x}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{6 \cdot x + \color{blue}{-6 \cdot \left(\frac{1}{x} \cdot x\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   5. lft-mult-inverse N/A

      \[\leadsto \frac{6 \cdot x + -6 \cdot \color{blue}{1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{6 \cdot x + \color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   7. lower-fma.f64 99.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, x, -6\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

5. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\left(1 + x\right)} + 4 \cdot \sqrt{x}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(1 + x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(1 + x\right)} \]

   7. lower-fma.f64 99.8

      \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1 + x\right)}} \]
8. Add Preprocessing

Alternative 6: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-24}{\sqrt{x}} + 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 14.0) (/ -6.0 (fma (sqrt x) 4.0 1.0)) (+ (/ -24.0 (sqrt x)) 6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 14.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = (-24.0 / sqrt(x)) + 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 14.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(Float64(-24.0 / sqrt(x)) + 6.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 14.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 14

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{\sqrt{x} \cdot 4} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

      5. lower-sqrt.f64 98.5

         \[\leadsto \frac{-6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4, 1\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}} \]

   if 14 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      3. flip-+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   4. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

      3. metadata-eval N/A

         \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

      9. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.2%

      \[\leadsto \frac{-24}{\sqrt{x}} + \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.061:\\ \;\;\;\;\mathsf{fma}\left(24, \sqrt{x}, -6\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-24}{\sqrt{x}} + 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.061) (fma 24.0 (sqrt x) -6.0) (+ (/ -24.0 (sqrt x)) 6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.061) {
		tmp = fma(24.0, sqrt(x), -6.0);
	} else {
		tmp = (-24.0 / sqrt(x)) + 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.061)
		tmp = fma(24.0, sqrt(x), -6.0);
	else
		tmp = Float64(Float64(-24.0 / sqrt(x)) + 6.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.061], N[(24.0 * N[Sqrt[x], $MachinePrecision] + -6.0), $MachinePrecision], N[(N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.060999999999999999

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      3. flip-+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

      3. metadata-eval N/A

         \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

      9. lower-/.f64 6.7

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(-4 \cdot \sqrt{x} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{-6 \cdot \left(-4 \cdot \sqrt{x}\right) + -6 \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot -4\right) \cdot \sqrt{x}} + -6 \cdot 1 \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{24} \cdot \sqrt{x} + -6 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-24\right)\right)} \cdot \sqrt{x} + -6 \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(-24\right)\right) \cdot \sqrt{x} + \color{blue}{-6} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(-24\right), \sqrt{x}, -6\right)} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{24}, \sqrt{x}, -6\right) \]

      9. lower-sqrt.f64 98.4

         \[\leadsto \mathsf{fma}\left(24, \color{blue}{\sqrt{x}}, -6\right) \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(24, \sqrt{x}, -6\right)} \]

   if 0.060999999999999999 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      3. flip-+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   4. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

      3. metadata-eval N/A

         \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

      9. lower-/.f64 97.2

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.2%

      \[\leadsto \frac{-24}{\sqrt{x}} + \color{blue}{6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(24, \sqrt{x}, -6\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 24.0 (sqrt x) -6.0))

C

double code(double x) {
	return fma(24.0, sqrt(x), -6.0);
}

Julia

function code(x)
	return fma(24.0, sqrt(x), -6.0)
end

Wolfram

code[x_] := N[(24.0 * N[Sqrt[x], $MachinePrecision] + -6.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   3. flip-+ N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{\left(x + 1\right) - 4 \cdot \sqrt{x}}}} \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)} \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(6 \cdot \left(x - 1\right)\right) \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right)}{\left(x + 1\right) \cdot \left(x + 1\right) - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}} \]

4. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\frac{\left(\left(x - 1\right) \cdot 6\right) \cdot \mathsf{fma}\left(-4, \sqrt{x}, 1 + x\right)}{{\left(1 + x\right)}^{2} - 16 \cdot x}} \]

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

      \[\leadsto 6 \cdot \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + 1 \cdot 6} \]

   3. metadata-eval N/A

      \[\leadsto \left(-4 \cdot \sqrt{\frac{1}{x}}\right) \cdot 6 + \color{blue}{6} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)} \cdot 6 + 6 \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-4 \cdot 6\right)} + 6 \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-24} + 6 \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, -24, 6\right) \]

   9. lower-/.f64 56.5

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, -24, 6\right) \]

7. Applied rewrites 56.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, -24, 6\right)} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto -6 \cdot \color{blue}{\left(-4 \cdot \sqrt{x} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{-6 \cdot \left(-4 \cdot \sqrt{x}\right) + -6 \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-6 \cdot -4\right) \cdot \sqrt{x}} + -6 \cdot 1 \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{24} \cdot \sqrt{x} + -6 \cdot 1 \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-24\right)\right)} \cdot \sqrt{x} + -6 \cdot 1 \]

   6. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(-24\right)\right) \cdot \sqrt{x} + \color{blue}{-6} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(-24\right), \sqrt{x}, -6\right)} \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{24}, \sqrt{x}, -6\right) \]

   9. lower-sqrt.f64 48.0

      \[\leadsto \mathsf{fma}\left(24, \color{blue}{\sqrt{x}}, -6\right) \]

10. Applied rewrites 48.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(24, \sqrt{x}, -6\right)} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0))))

C

double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / (((x + 1.0d0) + (4.0d0 * sqrt(x))) / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * Math.sqrt(x))) / (x - 1.0));
}

Python

def code(x):
	return 6.0 / (((x + 1.0) + (4.0 * math.sqrt(x))) / (x - 1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))) / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1))))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))