Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 10.2s

Alternatives: 14

Speedup: 1.0×

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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{x + \frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}{x + -1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ x (/ (- 1.0 (* x 16.0)) (- 1.0 (* 4.0 (sqrt x))))) (+ x -1.0))))

C

double code(double x) {
	return 6.0 / ((x + ((1.0 - (x * 16.0)) / (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 - (x * 16.0d0)) / (1.0d0 - (4.0d0 * sqrt(x))))) / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return 6.0 / ((x + ((1.0 - (x * 16.0)) / (1.0 - (4.0 * Math.sqrt(x))))) / (x + -1.0));
}

Python

def code(x):
	return 6.0 / ((x + ((1.0 - (x * 16.0)) / (1.0 - (4.0 * math.sqrt(x))))) / (x + -1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(x + Float64(Float64(1.0 - Float64(x * 16.0)) / Float64(1.0 - Float64(4.0 * sqrt(x))))) / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / ((x + ((1.0 - (x * 16.0)) / (1.0 - (4.0 * sqrt(x))))) / (x + -1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(x + N[(N[(1.0 - N[(x * 16.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   15. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]
7. Add Preprocessing

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{6}{\frac{1 + t\_0}{x + -1}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0)
     (/ 6.0 (/ (+ 1.0 t_0) (+ x -1.0)))
     (* 6.0 (/ (+ x -1.0) (+ x t_0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 / ((1.0 + t_0) / (x + -1.0));
	} else {
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 6.0d0 / ((1.0d0 + t_0) / (x + (-1.0d0)))
    else
        tmp = 6.0d0 * ((x + (-1.0d0)) / (x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 / ((1.0 + t_0) / (x + -1.0));
	} else {
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 / ((1.0 + t_0) / (x + -1.0))
	else:
		tmp = 6.0 * ((x + -1.0) / (x + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(6.0 / Float64(Float64(1.0 + t_0) / Float64(x + -1.0)));
	else
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(x + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 / ((1.0 + t_0) / (x + -1.0));
	else
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(6.0 / N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, -1\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   7. Simplified 99.3%

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

   if 1 < 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right), 6\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      2. sqrt-lowering-sqrt.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   7. Simplified 98.0%

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

4. Final simplification 98.5%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.29:\\ \;\;\;\;\frac{-6}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (* 4.0 (sqrt x)))))
   (if (<= x 0.29) (/ -6.0 (+ 1.0 t_0)) (* 6.0 (/ (+ x -1.0) t_0)))))

C

double code(double x) {
	double t_0 = x + (4.0 * sqrt(x));
	double tmp;
	if (x <= 0.29) {
		tmp = -6.0 / (1.0 + t_0);
	} else {
		tmp = 6.0 * ((x + -1.0) / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (4.0d0 * sqrt(x))
    if (x <= 0.29d0) then
        tmp = (-6.0d0) / (1.0d0 + t_0)
    else
        tmp = 6.0d0 * ((x + (-1.0d0)) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + (4.0 * Math.sqrt(x));
	double tmp;
	if (x <= 0.29) {
		tmp = -6.0 / (1.0 + t_0);
	} else {
		tmp = 6.0 * ((x + -1.0) / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + (4.0 * math.sqrt(x))
	tmp = 0
	if x <= 0.29:
		tmp = -6.0 / (1.0 + t_0)
	else:
		tmp = 6.0 * ((x + -1.0) / t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x + Float64(4.0 * sqrt(x)))
	tmp = 0.0
	if (x <= 0.29)
		tmp = Float64(-6.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + (4.0 * sqrt(x));
	tmp = 0.0;
	if (x <= 0.29)
		tmp = -6.0 / (1.0 + t_0);
	else
		tmp = 6.0 * ((x + -1.0) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.29], N[(-6.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.28999999999999998

   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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(4 \cdot \sqrt{x} + \color{blue}{\left(x + 1\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(\left(4 \cdot \sqrt{x} + x\right) + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \sqrt{x} + x\right), \color{blue}{1}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \sqrt{x}\right), x\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right), x\right), 1\right)\right) \]

      6. sqrt-lowering-sqrt.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), x\right), 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), x\right), 1\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.3%

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

   if 0.28999999999999998 < 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right), 6\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      2. sqrt-lowering-sqrt.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   7. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.29:\\ \;\;\;\;\frac{-6}{1 + \left(x + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{x + 4 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (* 4.0 (sqrt x)))))
   (if (<= x 1.0) (/ -6.0 (+ 1.0 t_0)) (* 6.0 (/ x t_0)))))

C

double code(double x) {
	double t_0 = x + (4.0 * sqrt(x));
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + t_0);
	} else {
		tmp = 6.0 * (x / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (4.0d0 * sqrt(x))
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + t_0)
    else
        tmp = 6.0d0 * (x / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + (4.0 * Math.sqrt(x));
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + t_0);
	} else {
		tmp = 6.0 * (x / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + (4.0 * math.sqrt(x))
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + t_0)
	else:
		tmp = 6.0 * (x / t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x + Float64(4.0 * sqrt(x)))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(6.0 * Float64(x / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + (4.0 * sqrt(x));
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + t_0);
	else
		tmp = 6.0 * (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(4 \cdot \sqrt{x} + \color{blue}{\left(x + 1\right)}\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(\left(4 \cdot \sqrt{x} + x\right) + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\left(4 \cdot \sqrt{x} + x\right), \color{blue}{1}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \sqrt{x}\right), x\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right), x\right), 1\right)\right) \]

      6. sqrt-lowering-sqrt.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), x\right), 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), x\right), 1\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.3%

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

   if 1 < 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right), 6\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      2. sqrt-lowering-sqrt.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   7. Simplified 98.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   9. Step-by-step derivation

   10. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{t\_0 + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0) (/ -6.0 (+ t_0 (+ x 1.0))) (* 6.0 (/ x (+ x t_0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (t_0 + (x + 1.0));
	} else {
		tmp = 6.0 * (x / (x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (t_0 + (x + 1.0d0))
    else
        tmp = 6.0d0 * (x / (x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (t_0 + (x + 1.0));
	} else {
		tmp = 6.0 * (x / (x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (t_0 + (x + 1.0))
	else:
		tmp = 6.0 * (x / (x + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(t_0 + Float64(x + 1.0)));
	else
		tmp = Float64(6.0 * Float64(x / Float64(x + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (t_0 + (x + 1.0));
	else
		tmp = 6.0 * (x / (x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(-6.0 / N[(t$95$0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   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

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

   if 1 < 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right), 6\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      2. sqrt-lowering-sqrt.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   7. Simplified 98.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   9. Step-by-step derivation

   10. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \frac{1}{-1 + \sqrt{x} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{x + 4 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (1.0 / (-1.0 + (math.sqrt(x) * -4.0)))
	else:
		tmp = 6.0 * (x / (x + (4.0 * math.sqrt(x))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * (1.0 / (-1.0 + (sqrt(x) * -4.0)));
	else
		tmp = 6.0 * (x / (x + (4.0 * sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{1 + 4 \cdot \sqrt{x}}\right)}, 6\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{1 + 4 \cdot \sqrt{x}}\right), 6\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{1 + 4 \cdot \sqrt{x}}\right)\right), 6\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}\right), 6\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(-1 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(4 \cdot -1\right) \cdot \sqrt{x}\right)\right), 6\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + -4 \cdot \sqrt{x}\right)\right), 6\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(-4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot -4\right)\right)\right), 6\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), -4\right)\right)\right), 6\right) \]

      14. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right), 6\right) \]

   7. Simplified 99.3%

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

   if 1 < 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right), 6\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      2. sqrt-lowering-sqrt.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   7. Simplified 98.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   9. Step-by-step derivation

   10. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 7: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \frac{1}{-1 + \sqrt{x} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \frac{1}{-1 + \frac{-4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (1.0 / (-1.0 + (math.sqrt(x) * -4.0)))
	else:
		tmp = -6.0 * (1.0 / (-1.0 + (-4.0 / math.sqrt(x))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * (1.0 / (-1.0 + (sqrt(x) * -4.0)));
	else
		tmp = -6.0 * (1.0 / (-1.0 + (-4.0 / sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{1 + 4 \cdot \sqrt{x}}\right)}, 6\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{1 + 4 \cdot \sqrt{x}}\right), 6\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{1 + 4 \cdot \sqrt{x}}\right)\right), 6\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}\right), 6\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(-1 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(4 \cdot -1\right) \cdot \sqrt{x}\right)\right), 6\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + -4 \cdot \sqrt{x}\right)\right), 6\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(-4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot -4\right)\right)\right), 6\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), -4\right)\right)\right), 6\right) \]

      14. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right), 6\right) \]

   7. Simplified 99.3%

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

   if 1 < 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

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(4 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 1\right)}\right) \]

   5. Simplified 97.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{-1}^{3} + {\left(\sqrt{\frac{1}{x}} \cdot -4\right)}^{3}}{-1 \cdot -1 + \left(\left(\sqrt{\frac{1}{x}} \cdot -4\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right) - -1 \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}} \cdot -6 \]

      4. clear-num N/A

         \[\leadsto \frac{-1 \cdot -1 + \left(\left(\sqrt{\frac{1}{x}} \cdot -4\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right) - -1 \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}{{-1}^{3} + {\left(\sqrt{\frac{1}{x}} \cdot -4\right)}^{3}} \cdot -6 \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot -1 + \left(\left(\sqrt{\frac{1}{x}} \cdot -4\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right) - -1 \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}{{-1}^{3} + {\left(\sqrt{\frac{1}{x}} \cdot -4\right)}^{3}}\right), \color{blue}{-6}\right) \]

   7. Applied egg-rr 97.8%

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

4. Final simplification 98.4%

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

Alternative 8: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \frac{1}{-1 + \sqrt{x} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * (1.0 / (-1.0 + (math.sqrt(x) * -4.0)))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * (1.0 / (-1.0 + (sqrt(x) * -4.0)));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(6.0 * N[(1.0 / N[(-1.0 + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $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 x < 1

   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{1 + 4 \cdot \sqrt{x}}\right)}, 6\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{1 + 4 \cdot \sqrt{x}}\right), 6\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{1 + 4 \cdot \sqrt{x}}\right)\right), 6\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}\right), 6\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(\mathsf{neg}\left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + 4 \cdot \left(-1 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + \left(4 \cdot -1\right) \cdot \sqrt{x}\right)\right), 6\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 + -4 \cdot \sqrt{x}\right)\right), 6\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(-4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot -4\right)\right)\right), 6\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), -4\right)\right)\right), 6\right) \]

      14. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right), 6\right) \]

   7. Simplified 99.3%

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

   if 1 < 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

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(4 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 1\right)}\right) \]

   5. Simplified 97.8%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(-1 + \sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(-1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot -4}\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + -1 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(-1 \cdot \left(-4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot -4\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right)\right)\right) \]

      11. sqrt-div N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{1}{\sqrt{\color{blue}{x}}}\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\frac{4}{\color{blue}{\sqrt{x}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(4, \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 97.8%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   7. Applied egg-rr 97.8%

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

4. Final simplification 98.4%

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

Alternative 9: 99.9% accurate, 1.0× speedup

Math

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

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(6.0 * Float64(Float64(x + -1.0) / Float64(x + Float64(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[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), 6\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), 6\right) \]

   12. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), 6\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 10: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{-0.16666666666666666 + \sqrt{x} \cdot -0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 / ((-0.16666666666666666d0) + (sqrt(x) * (-0.6666666666666666d0)))
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (-0.16666666666666666 + (Math.sqrt(x) * -0.6666666666666666));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 / (-0.16666666666666666 + (math.sqrt(x) * -0.6666666666666666))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 / (-0.16666666666666666 + (sqrt(x) * -0.6666666666666666));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(1.0 / N[(-0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $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 x < 1

   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

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot -4\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \left(4 \cdot \color{blue}{-1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(4 \cdot -1\right)}\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{4} \cdot -1\right)\right)\right)\right) \]

      14. metadata-eval 99.2%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 99.2%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1 + \sqrt{x} \cdot -4}{6}\right)}\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(-1 + \sqrt{x} \cdot -4\right)\right)}{\color{blue}{\mathsf{neg}\left(6\right)}}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)}{\mathsf{neg}\left(\color{blue}{6}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + -1 \cdot \left(\sqrt{x} \cdot -4\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + -1 \cdot \left(-4 \cdot \sqrt{x}\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + \left(-1 \cdot -4\right) \cdot \sqrt{x}}{\mathsf{neg}\left(6\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + 4 \cdot \sqrt{x}}{\mathsf{neg}\left(6\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + 4 \cdot \sqrt{x}}{-6}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + 4 \cdot \sqrt{x}\right), \color{blue}{-6}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right), -6\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), -6\right)\right) \]

      14. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), -6\right)\right) \]

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + 4 \cdot \sqrt{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \frac{-1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right) \cdot \frac{-1}{6}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right)} \cdot \frac{-1}{6}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\left(4 \cdot \sqrt{x}\right) \cdot \frac{-1}{6}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\sqrt{x} \cdot 4\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\sqrt{x} \cdot \color{blue}{\left(4 \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\sqrt{x} \cdot \frac{-2}{3}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{-2}{3}}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-2}{3}\right)\right)\right) \]

   10. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{-0.16666666666666666 + \sqrt{x} \cdot -0.6666666666666666}} \]

   if 1 < 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

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-6, \color{blue}{\left(4 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 1\right)}\right) \]

   5. Simplified 97.8%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(-1 + \sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(-1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot -4}\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(1 + -1 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot -4\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(-1 \cdot \left(-4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot -4\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right)\right)\right) \]

      11. sqrt-div N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{1}{\sqrt{\color{blue}{x}}}\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\frac{4}{\color{blue}{\sqrt{x}}}\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(4, \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 97.8%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   7. Applied egg-rr 97.8%

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

Alternative 11: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.83:\\ \;\;\;\;\frac{1}{-0.16666666666666666 + \sqrt{x} \cdot -0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.83)
   (/ 1.0 (+ -0.16666666666666666 (* (sqrt x) -0.6666666666666666)))
   (/ 6.0 (/ x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.83) {
		tmp = 1.0 / (-0.16666666666666666 + (sqrt(x) * -0.6666666666666666));
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.83d0) then
        tmp = 1.0d0 / ((-0.16666666666666666d0) + (sqrt(x) * (-0.6666666666666666d0)))
    else
        tmp = 6.0d0 / (x / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.83) {
		tmp = 1.0 / (-0.16666666666666666 + (Math.sqrt(x) * -0.6666666666666666));
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.83:
		tmp = 1.0 / (-0.16666666666666666 + (math.sqrt(x) * -0.6666666666666666))
	else:
		tmp = 6.0 / (x / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.83)
		tmp = Float64(1.0 / Float64(-0.16666666666666666 + Float64(sqrt(x) * -0.6666666666666666)));
	else
		tmp = Float64(6.0 / Float64(x / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.83)
		tmp = 1.0 / (-0.16666666666666666 + (sqrt(x) * -0.6666666666666666));
	else
		tmp = 6.0 / (x / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.83], N[(1.0 / N[(-0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.82999999999999996

   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

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot -4\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \left(4 \cdot \color{blue}{-1}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(4 \cdot -1\right)}\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{4} \cdot -1\right)\right)\right)\right) \]

      14. metadata-eval 99.2%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 99.2%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1 + \sqrt{x} \cdot -4}{6}\right)}\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(-1 + \sqrt{x} \cdot -4\right)\right)}{\color{blue}{\mathsf{neg}\left(6\right)}}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)}{\mathsf{neg}\left(\color{blue}{6}\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + -1 \cdot \left(\sqrt{x} \cdot -4\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + -1 \cdot \left(-4 \cdot \sqrt{x}\right)}{\mathsf{neg}\left(6\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + \left(-1 \cdot -4\right) \cdot \sqrt{x}}{\mathsf{neg}\left(6\right)}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + 4 \cdot \sqrt{x}}{\mathsf{neg}\left(6\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1 + 4 \cdot \sqrt{x}}{-6}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + 4 \cdot \sqrt{x}\right), \color{blue}{-6}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right), -6\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), -6\right)\right) \]

      14. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), -6\right)\right) \]

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + 4 \cdot \sqrt{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot \frac{-1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right) \cdot \frac{-1}{6}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right)} \cdot \frac{-1}{6}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\left(4 \cdot \sqrt{x}\right) \cdot \frac{-1}{6}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\sqrt{x} \cdot 4\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\sqrt{x} \cdot \color{blue}{\left(4 \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\sqrt{x} \cdot \frac{-2}{3}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{-2}{3}}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-2}{3}\right)\right)\right) \]

   10. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{-0.16666666666666666 + \sqrt{x} \cdot -0.6666666666666666}} \]

   if 0.82999999999999996 < 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. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 94.6%

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

Alternative 12: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.061:\\ \;\;\;\;-6 + -6 \cdot \left(\sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.061)
   (+ -6.0 (* -6.0 (* (sqrt x) -4.0)))
   (/ 6.0 (/ x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.061) {
		tmp = -6.0 + (-6.0 * (sqrt(x) * -4.0));
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.061d0) then
        tmp = (-6.0d0) + ((-6.0d0) * (sqrt(x) * (-4.0d0)))
    else
        tmp = 6.0d0 / (x / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.061) {
		tmp = -6.0 + (-6.0 * (Math.sqrt(x) * -4.0));
	} else {
		tmp = 6.0 / (x / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.061:
		tmp = -6.0 + (-6.0 * (math.sqrt(x) * -4.0))
	else:
		tmp = 6.0 / (x / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.061)
		tmp = Float64(-6.0 + Float64(-6.0 * Float64(sqrt(x) * -4.0)));
	else
		tmp = Float64(6.0 / Float64(x / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.061)
		tmp = -6.0 + (-6.0 * (sqrt(x) * -4.0));
	else
		tmp = 6.0 / (x / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.061], N[(-6.0 + N[(-6.0 * N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.060999999999999999

   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. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-6, \color{blue}{\left(\left(-4 \cdot \sqrt{x}\right) \cdot -6\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(\left(-4 \cdot \sqrt{x}\right), \color{blue}{-6}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(\left(\sqrt{x} \cdot -4\right), -6\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), -4\right), -6\right)\right) \]

      9. sqrt-lowering-sqrt.f64 99.1%

         \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right), -6\right)\right) \]

   9. Simplified 99.1%

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

   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. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 94.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.061:\\ \;\;\;\;-6 + -6 \cdot \left(\sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{x}{x + -1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 6.0 / (x / (x + -1.0));
}

Python

def code(x):
	return 6.0 / (x / (x + -1.0))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(6.0 / N[(x / 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. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   15. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, -1\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 56.9%

   \[\leadsto \frac{6}{\frac{\color{blue}{x}}{x + -1}} \]
10. Add Preprocessing

Alternative 14: 48.1% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

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. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}{1 - 4 \cdot \sqrt{x}}\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(1 - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(4 \cdot \sqrt{x}\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   10. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \left(-4 \cdot -4\right)\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \left(1 - 4 \cdot \sqrt{x}\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   15. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, 16\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{6}{\frac{x + \color{blue}{\frac{1 - x \cdot 16}{1 - 4 \cdot \sqrt{x}}}}{x + -1}} \]

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{6} \]
8. Step-by-step derivation

9. Simplified 55.4%

   \[\leadsto \color{blue}{6} \]
10. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× 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 2024160 
(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)))))