Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 14.4s

Alternatives: 11

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: 99.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $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 99.9

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f64 N/A

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

   10. metadata-eval 99.9

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

   11. lift-+.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. associate-+l+ N/A

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

   14. lower-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq 2:\\ \;\;\;\;\frac{x + -1}{\mathsf{fma}\left(0.6666666666666666, \sqrt{x}, 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + \frac{-4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) (* 4.0 (sqrt x)))) 2.0)
   (/ (+ x -1.0) (fma 0.6666666666666666 (sqrt x) 0.16666666666666666))
   (/ -6.0 (+ -1.0 (/ -4.0 (sqrt x))))))

C

double code(double x) {
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + (4.0 * sqrt(x)))) <= 2.0) {
		tmp = (x + -1.0) / fma(0.6666666666666666, sqrt(x), 0.16666666666666666);
	} else {
		tmp = -6.0 / (-1.0 + (-4.0 / sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= 2.0)
		tmp = Float64(Float64(x + -1.0) / fma(0.6666666666666666, sqrt(x), 0.16666666666666666));
	else
		tmp = Float64(-6.0 / Float64(-1.0 + Float64(-4.0 / sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(x + -1.0), $MachinePrecision] / N[(0.6666666666666666 * N[Sqrt[x], $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(-6.0 / N[(-1.0 + N[(-4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < 2

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.9

          \[\leadsto \frac{1}{\frac{\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right) \cdot \color{blue}{0.16666666666666666}}{x + -1}} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right) \cdot 0.16666666666666666}{x + -1}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 99.9

         \[\leadsto \color{blue}{\frac{x + -1}{\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right) \cdot 0.16666666666666666}} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-+r+ N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lower-fma.f64 99.9

          \[\leadsto \frac{x + -1}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)}, 0.16666666666666666\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(0.16666666666666666, \mathsf{fma}\left(4, \sqrt{x}, x\right), 0.16666666666666666\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x + -1}{\color{blue}{\frac{1}{6} + \frac{2}{3} \cdot \sqrt{x}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x + -1}{\color{blue}{\frac{2}{3} \cdot \sqrt{x} + \frac{1}{6}}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{x + -1}{\color{blue}{\mathsf{fma}\left(\frac{2}{3}, \sqrt{x}, \frac{1}{6}\right)}} \]

       3. lower-sqrt.f64 97.8

          \[\leadsto \frac{x + -1}{\mathsf{fma}\left(0.6666666666666666, \color{blue}{\sqrt{x}}, 0.16666666666666666\right)} \]

   11. Applied rewrites 97.8%

       \[\leadsto \frac{x + -1}{\color{blue}{\mathsf{fma}\left(0.6666666666666666, \sqrt{x}, 0.16666666666666666\right)}} \]

   if 2 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. rem-square-sqrt N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. neg-mul-1 N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

      \[\leadsto \frac{-6}{\frac{-4}{\sqrt{x}} + \color{blue}{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq 2:\\ \;\;\;\;\frac{x + -1}{\mathsf{fma}\left(0.6666666666666666, \sqrt{x}, 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-6}{-1 + \frac{-4}{\sqrt{x}}}\\ \end{array} \]
5. 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 2024229 
(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)))))