Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 11.3s

Alternatives: 15

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. flip-- N/A

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

   2. lift-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-+.f64 N/A

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

   10. flip-- N/A

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

   11. lift--.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l/ N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -4.0) {
		tmp = fma(x, 6.0, -6.0) / fma(4.0, sqrt(x), 1.0);
	} else {
		tmp = 6.0 * ((x + -1.0) / fma(sqrt(x), 4.0, 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)))) <= -4.0)
		tmp = Float64(fma(x, 6.0, -6.0) / fma(4.0, sqrt(x), 1.0));
	else
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / fma(sqrt(x), 4.0, 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], -4.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + x), $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)))) < -4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 98.3

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

   5. Simplified 98.3%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 98.3

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 98.3

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

   7. Applied egg-rr 98.3%

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

   if -4 < (/.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-/r/ N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 97.2

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

   7. Simplified 97.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 97.2

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \frac{6}{\color{blue}{0 - \left(\sqrt{x} \cdot -4 - x\right)}} \cdot \left(x + -1\right) \]

      12. lift--.f64 N/A

          \[\leadsto \frac{6}{0 - \color{blue}{\left(\sqrt{x} \cdot -4 - x\right)}} \cdot \left(x + -1\right) \]

      13. associate-+l- N/A

          \[\leadsto \frac{6}{\color{blue}{\left(0 - \sqrt{x} \cdot -4\right) + x}} \cdot \left(x + -1\right) \]

      14. neg-sub0 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 97.2

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

   9. Applied egg-rr 97.2%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. associate-*r/ N/A

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

       9. div-inv N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. times-frac N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 97.4

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

       17. lift-fma.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-fma.f64 97.4

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

   11. Applied egg-rr 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -4:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{\mathsf{fma}\left(\sqrt{x}, 4, x\right)}\\ \end{array} \]
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 2024218 
(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)))))