Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 11.1s

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: 99.7% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+l+ N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.9

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lower-+.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (+ x -1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -5.0)
   (/ -6.0 (+ 1.0 (fma 4.0 (sqrt x) x)))
   (/ (fma x -6.0 6.0) (- (* (sqrt x) -4.0) x))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x + -1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -5.0) {
		tmp = -6.0 / (1.0 + fma(4.0, sqrt(x), x));
	} else {
		tmp = fma(x, -6.0, 6.0) / ((sqrt(x) * -4.0) - x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x + -1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -5.0)
		tmp = Float64(-6.0 / Float64(1.0 + fma(4.0, sqrt(x), x)));
	else
		tmp = Float64(fma(x, -6.0, 6.0) / Float64(Float64(sqrt(x) * -4.0) - x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], -5.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -6.0 + 6.0), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision] - x), $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)))) < -5

   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 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.8%

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

   if -5 < (/.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 99.8%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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. 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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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 96.7

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

   7. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lift-+.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 96.6

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

   9. Applied rewrites 96.6%

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

4. Final simplification 97.7%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (+ x -1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -5.0)
   (/ -6.0 (+ 1.0 (fma 4.0 (sqrt x) x)))
   (/ (* 6.0 x) (fma (sqrt x) 4.0 (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x + -1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -5.0) {
		tmp = -6.0 / (1.0 + fma(4.0, sqrt(x), x));
	} else {
		tmp = (6.0 * x) / fma(sqrt(x), 4.0, (x + 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x + -1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -5.0)
		tmp = Float64(-6.0 / Float64(1.0 + fma(4.0, sqrt(x), x)));
	else
		tmp = Float64(Float64(6.0 * x) / fma(sqrt(x), 4.0, Float64(x + 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], -5.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x + 1.0), $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)))) < -5

   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 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.8%

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

   if -5 < (/.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 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.6

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

   7. Applied rewrites 96.6%

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

4. Final simplification 97.7%

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

Alternative 4: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+l+ N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.9

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lower-+.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-/.f64 99.9

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. lower-fma.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x + -1.0), $MachinePrecision] * N[(-6.0 / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0 + -1.0), $MachinePrecision] - 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. 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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. associate-+l+ N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.9

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

   15. lift--.f64 N/A

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

   16. sub-neg N/A

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

   17. lower-+.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-+.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate-*l/ N/A

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

   12. lift-+.f64 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. lift--.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 7: 52.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $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 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 51.6

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

5. Applied rewrites 51.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 51.6

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

7. Applied rewrites 51.6%

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

Alternative 8: 52.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 24.0 + -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. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. metadata-eval 48.8

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

5. Applied rewrites 48.8%

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

7. Applied rewrites 48.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 51.4%

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, \color{blue}{24}, -6\right) \]
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 2024233 
(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)))))