Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 10.3s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-+l+ N/A

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

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

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

   10. remove-double-neg N/A

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

   11. sub-neg N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.4%

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

3. Simplified 99.4%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 4\\ \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{6}{1 + t\_0} \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + 1\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 4.0)))
   (if (<= x 3.4)
     (* (/ 6.0 (+ 1.0 t_0)) (+ x -1.0))
     (* 6.0 (/ x (+ (+ x 1.0) t_0))))))

C

double code(double x) {
	double t_0 = sqrt(x) * 4.0;
	double tmp;
	if (x <= 3.4) {
		tmp = (6.0 / (1.0 + t_0)) * (x + -1.0);
	} else {
		tmp = 6.0 * (x / ((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 = sqrt(x) * 4.0d0
    if (x <= 3.4d0) then
        tmp = (6.0d0 / (1.0d0 + t_0)) * (x + (-1.0d0))
    else
        tmp = 6.0d0 * (x / ((x + 1.0d0) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(x) * 4.0;
	double tmp;
	if (x <= 3.4) {
		tmp = (6.0 / (1.0 + t_0)) * (x + -1.0);
	} else {
		tmp = 6.0 * (x / ((x + 1.0) + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(x) * 4.0
	tmp = 0
	if x <= 3.4:
		tmp = (6.0 / (1.0 + t_0)) * (x + -1.0)
	else:
		tmp = 6.0 * (x / ((x + 1.0) + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(x) * 4.0)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(Float64(6.0 / Float64(1.0 + t_0)) * Float64(x + -1.0));
	else
		tmp = Float64(6.0 * Float64(x / Float64(Float64(x + 1.0) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(x) * 4.0;
	tmp = 0.0;
	if (x <= 3.4)
		tmp = (6.0 / (1.0 + t_0)) * (x + -1.0);
	else
		tmp = 6.0 * (x / ((x + 1.0) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[x, 3.4], N[(N[(6.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(N[(x + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 96.9%

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

   7. Simplified 96.9%

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

   8. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. +-lowering-+.f64 96.9%

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

   10. Simplified 96.9%

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

   if 3.39999999999999991 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 96.6%

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

   7. Simplified 96.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. +-lowering-+.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\frac{6}{1 + \sqrt{x} \cdot 4} \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + 1\right) + \sqrt{x} \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 96.6%

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

   7. Simplified 96.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-+r- N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. +-lowering-+.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

4. Final simplification 97.2%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (x + (1.0 - (sqrt(x) * -4.0)));
	} else {
		tmp = 6.0 / ((x + (sqrt(x) * 4.0)) / 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) / (x + (1.0d0 - (sqrt(x) * (-4.0d0))))
    else
        tmp = 6.0d0 / ((x + (sqrt(x) * 4.0d0)) / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 97.4%

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

   10. Simplified 97.4%

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

4. Final simplification 97.2%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 97.4%

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

   10. Simplified 97.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. sqrt-lowering-sqrt.f64 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 97.2%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt(x) * 4.0;
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + t_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 = sqrt(x) * 4.0d0
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + t_0)
    else
        tmp = 6.0d0 * (x / (x + t_0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 96.8%

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

   7. Simplified 96.8%

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

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 97.4%

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

   10. Simplified 97.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. sqrt-lowering-sqrt.f64 97.4%

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

   12. Applied egg-rr 97.4%

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

4. Final simplification 97.2%

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

Alternative 7: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 8: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ 1.0 (* (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 + (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 + (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 + (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 + (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(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 + (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[(N[Sqrt[x], $MachinePrecision] * 4.0), $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. Step-by-step derivation

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 96.8%

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

   7. Simplified 96.8%

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

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 97.4%

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

   7. Simplified 97.4%

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

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

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

      2. *-commutative N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. unsub-neg N/A

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

      9. distribute-neg-frac N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. sqrt-div N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. sqrt-div N/A

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

      17. metadata-eval N/A

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

      18. div-inv N/A

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

      19. distribute-neg-frac N/A

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

      20. /-lowering-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. sqrt-lowering-sqrt.f64 97.4%

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

   9. Applied egg-rr 97.4%

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

Alternative 9: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -1.5 (sqrt x)) (* (sqrt x) 1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = sqrt(x) * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-1.5d0) / sqrt(x)
    else
        tmp = sqrt(x) * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / Math.sqrt(x);
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 / math.sqrt(x)
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = Float64(sqrt(x) * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -1.5 / sqrt(x);
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $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. Step-by-step derivation

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. /-lowering-/.f64 7.2%

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

   10. Simplified 7.2%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot -1.5} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{-3}{2}}{\color{blue}{\sqrt{x}}} \]

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

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

       6. sqrt-lowering-sqrt.f64 7.2%

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

   12. Applied egg-rr 7.2%

       \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]

   if 1 < x

   1. Initial program 99.0%

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

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

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-+l+ N/A

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

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

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

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

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

      3. sqrt-lowering-sqrt.f64 7.0%

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

   10. Simplified 7.0%

       \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -6 + \sqrt{x} \cdot 24 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -6.0 (* (sqrt x) 24.0)))

C

double code(double x) {
	return -6.0 + (sqrt(x) * 24.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-6.0d0) + (sqrt(x) * 24.0d0)
end function

Java

public static double code(double x) {
	return -6.0 + (Math.sqrt(x) * 24.0);
}

Python

def code(x):
	return -6.0 + (math.sqrt(x) * 24.0)

Julia

function code(x)
	return Float64(-6.0 + Float64(sqrt(x) * 24.0))
end

MATLAB

function tmp = code(x)
	tmp = -6.0 + (sqrt(x) * 24.0);
end

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-+l+ N/A

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

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

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

   10. remove-double-neg N/A

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

   11. sub-neg N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot -4\right)\right)\right)\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \left(\sqrt{x} \cdot 4\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{4}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 44.5%

      \[\leadsto \mathsf{/.f64}\left(-6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), 4\right)\right)\right) \]

7. Simplified 44.5%

   \[\leadsto \color{blue}{\frac{-6}{1 + \sqrt{x} \cdot 4}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-6}{1 + 4 \cdot \color{blue}{\sqrt{x}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{-6}{1 + \left(\mathsf{neg}\left(-4\right)\right) \cdot \sqrt{\color{blue}{x}}} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{-6}{1 - \color{blue}{-4 \cdot \sqrt{x}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{-6}{1 - \sqrt{x} \cdot \color{blue}{-4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{-6}{1 - -4 \cdot \color{blue}{\sqrt{x}}} \]

   6. cancel-sign-sub-inv N/A

      \[\leadsto \frac{-6}{1 + \color{blue}{\left(\mathsf{neg}\left(-4\right)\right) \cdot \sqrt{x}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{\color{blue}{x}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{-6}{1 + \sqrt{x} \cdot \color{blue}{4}} \]

   9. flip-+ N/A

      \[\leadsto \frac{-6}{\frac{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}{\color{blue}{1 - \sqrt{x} \cdot 4}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{-6}{\frac{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}{1 - 4 \cdot \color{blue}{\sqrt{x}}}} \]

   11. cancel-sign-sub-inv N/A

       \[\leadsto \frac{-6}{\frac{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}{1 + \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{-6}{\frac{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}{1 + -4 \cdot \sqrt{\color{blue}{x}}}} \]

   13. *-commutative N/A

       \[\leadsto \frac{-6}{\frac{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}{1 + \sqrt{x} \cdot \color{blue}{-4}}} \]

   14. associate-/r/ N/A

       \[\leadsto \frac{-6}{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)} \cdot \color{blue}{\left(1 + \sqrt{x} \cdot -4\right)} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-6}{1 \cdot 1 - \left(\sqrt{x} \cdot 4\right) \cdot \left(\sqrt{x} \cdot 4\right)}\right), \color{blue}{\left(1 + \sqrt{x} \cdot -4\right)}\right) \]

9. Applied egg-rr 44.5%

   \[\leadsto \color{blue}{\frac{-6}{1 - x \cdot 16} \cdot \left(1 - 4 \cdot \sqrt{x}\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-6 \cdot \left(1 - 4 \cdot \sqrt{x}\right)} \]
11. 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-lft-in N/A

       \[\leadsto -6 \cdot 1 + \color{blue}{-6 \cdot \left(-4 \cdot \sqrt{x}\right)} \]

    4. metadata-eval N/A

       \[\leadsto -6 + \color{blue}{-6} \cdot \left(-4 \cdot \sqrt{x}\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-6, \color{blue}{\left(-6 \cdot \left(-4 \cdot \sqrt{x}\right)\right)}\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(-6, \left(\left(-6 \cdot -4\right) \cdot \color{blue}{\sqrt{x}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(\left(-6 \cdot -4\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(24, \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

    9. sqrt-lowering-sqrt.f64 47.1%

       \[\leadsto \mathsf{+.f64}\left(-6, \mathsf{*.f64}\left(24, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

12. Simplified 47.1%

    \[\leadsto \color{blue}{-6 + 24 \cdot \sqrt{x}} \]

13. Final simplification 47.1%

    \[\leadsto -6 + \sqrt{x} \cdot 24 \]
14. Add Preprocessing

Alternative 11: 4.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt x) 1.5))

C

double code(double x) {
	return sqrt(x) * 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(x) * 1.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt(x) * 1.5;
}

Python

def code(x):
	return math.sqrt(x) * 1.5

Julia

function code(x)
	return Float64(sqrt(x) * 1.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(x) * 1.5;
end

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x - 1\right)\right), \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(6 \cdot x + 6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(6 \cdot x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\left(\color{blue}{x} + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), \left(6 \cdot -1\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(\left(x + \color{blue}{1}\right) + 4 \cdot \sqrt{x}\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \left(x + \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \left(1 - \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{x} \cdot 4\right)\right)\right)\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval 99.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(6, x\right), -6\right), \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{6 \cdot x + -6}{x + \left(1 - \sqrt{x} \cdot -4\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 - -4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(6, \left(1 + \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(-4 \cdot \sqrt{\frac{1}{x}}\right)\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\sqrt{\frac{1}{x}} \cdot -4\right)\right)\right)\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{x}} \cdot 4\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{4}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), 4\right)\right)\right) \]

   9. /-lowering-/.f64 54.5%

      \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), 4\right)\right)\right) \]

7. Simplified 54.5%

   \[\leadsto \color{blue}{\frac{6}{1 + \sqrt{\frac{1}{x}} \cdot 4}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

   3. sqrt-lowering-sqrt.f64 4.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

10. Simplified 4.7%

    \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
11. 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 2024145 
(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)))))