Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 11.1s

Alternatives: 12

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

   1. metadata-eval 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.9%

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

   4. *-commutative 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. sub-neg 99.9%

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

   7. metadata-eval 99.9%

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

   8. *-un-lft-identity 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-define 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.28999999999999998

   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. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 0.28999999999999998 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 97.8%

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

4. Final simplification 97.8%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 97.8%

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

4. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   6. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 97.8%

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

4. Final simplification 97.8%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (x + (4.0 * sqrt(x))));
	} 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 + (x + (4.0d0 * sqrt(x))))
    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 + (x + (4.0 * Math.sqrt(x))));
	} 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 + (x + (4.0 * math.sqrt(x))))
	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(x + Float64(4.0 * sqrt(x)))));
	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 + (x + (4.0 * sqrt(x))));
	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[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   6. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. add-exp-log 97.6%

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

      2. log-div 97.7%

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

      3. log1p-define 97.7%

         \[\leadsto e^{\log 6 - \color{blue}{\mathsf{log1p}\left(4 \cdot \sqrt{\frac{1}{x}}\right)}} \]

      4. sqrt-div 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} \]

      5. metadata-eval 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} \]

      6. un-div-inv 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(\color{blue}{\frac{4}{\sqrt{x}}}\right)} \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{e^{\log 6 - \mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 97.7%

         \[\leadsto \color{blue}{\frac{e^{\log 6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}}} \]

      2. rem-exp-log 97.7%

         \[\leadsto \frac{\color{blue}{6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]

      3. log1p-undefine 97.7%

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

      4. rem-exp-log 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.8%

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

Alternative 6: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (x + (1.0 + (4.0 * sqrt(x))));
	} 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) / (x + (1.0d0 + (4.0d0 * sqrt(x))))
    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 / (x + (1.0 + (4.0 * Math.sqrt(x))));
	} 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 / (x + (1.0 + (4.0 * math.sqrt(x))))
	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(x + Float64(1.0 + Float64(4.0 * sqrt(x)))));
	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 / (x + (1.0 + (4.0 * sqrt(x))));
	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[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. add-exp-log 97.6%

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

      2. log-div 97.7%

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

      3. log1p-define 97.7%

         \[\leadsto e^{\log 6 - \color{blue}{\mathsf{log1p}\left(4 \cdot \sqrt{\frac{1}{x}}\right)}} \]

      4. sqrt-div 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} \]

      5. metadata-eval 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} \]

      6. un-div-inv 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(\color{blue}{\frac{4}{\sqrt{x}}}\right)} \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{e^{\log 6 - \mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 97.7%

         \[\leadsto \color{blue}{\frac{e^{\log 6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}}} \]

      2. rem-exp-log 97.7%

         \[\leadsto \frac{\color{blue}{6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]

      3. log1p-undefine 97.7%

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

      4. rem-exp-log 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.8%

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

Alternative 7: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.82) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (+ 6.0 (/ -6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 + (-6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.82d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 + ((-6.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 + (-6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.82:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 + (-6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 + Float64(-6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.82)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 + (-6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.82], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   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. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 0.819999999999999951 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. flip-+ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. swap-sqr 99.7%

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

      6. add-sqr-sqrt 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 95.1%

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

   8. Taylor expanded in x around 0 95.1%

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

      1. div-sub 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. sub-neg 95.1%

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

      5. associate-*l/ 95.2%

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

      6. metadata-eval 95.2%

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

      7. associate-*r/ 94.8%

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

      8. associate-*l* 95.0%

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

      9. lft-mult-inverse 95.3%

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

      10. metadata-eval 95.3%

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

      11. associate-*r/ 95.3%

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

      12. metadata-eval 95.3%

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

      13. distribute-neg-frac 95.3%

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

      14. metadata-eval 95.3%

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

   10. Simplified 95.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} 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 + (4.0d0 * sqrt(x)))
    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 + (4.0 * Math.sqrt(x)));
	} 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 + (4.0 * math.sqrt(x)))
	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(4.0 * sqrt(x))));
	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 + (4.0 * sqrt(x)));
	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[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. add-exp-log 97.6%

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

      2. log-div 97.7%

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

      3. log1p-define 97.7%

         \[\leadsto e^{\log 6 - \color{blue}{\mathsf{log1p}\left(4 \cdot \sqrt{\frac{1}{x}}\right)}} \]

      4. sqrt-div 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} \]

      5. metadata-eval 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(4 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} \]

      6. un-div-inv 97.7%

         \[\leadsto e^{\log 6 - \mathsf{log1p}\left(\color{blue}{\frac{4}{\sqrt{x}}}\right)} \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{e^{\log 6 - \mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 97.7%

         \[\leadsto \color{blue}{\frac{e^{\log 6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}}} \]

      2. rem-exp-log 97.7%

         \[\leadsto \frac{\color{blue}{6}}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)}} \]

      3. log1p-undefine 97.7%

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

      4. rem-exp-log 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 9: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.062) (+ -6.0 (* (sqrt x) 24.0)) (+ 6.0 (/ -6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.062) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.062d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0 + ((-6.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.062) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.062:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0 + (-6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.062)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(6.0 + Float64(-6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.062)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0 + (-6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.062], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.062

   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. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. metadata-eval 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. metadata-eval 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. +-commutative 97.5%

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

       2. associate-*r* 97.5%

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

       3. metadata-eval 97.5%

          \[\leadsto -6 + \color{blue}{24} \cdot \sqrt{x} \]

       4. *-commutative 97.5%

          \[\leadsto -6 + \color{blue}{\sqrt{x} \cdot 24} \]

   11. Simplified 97.5%

       \[\leadsto \color{blue}{-6 + \sqrt{x} \cdot 24} \]

   if 0.062 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. flip-+ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. swap-sqr 99.7%

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

      6. add-sqr-sqrt 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 95.1%

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

   8. Taylor expanded in x around 0 95.1%

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

      1. div-sub 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. sub-neg 95.1%

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

      5. associate-*l/ 95.2%

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

      6. metadata-eval 95.2%

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

      7. associate-*r/ 94.8%

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

      8. associate-*l* 95.0%

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

      9. lft-mult-inverse 95.3%

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

      10. metadata-eval 95.3%

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

      11. associate-*r/ 95.3%

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

      12. metadata-eval 95.3%

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

      13. distribute-neg-frac 95.3%

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

      14. metadata-eval 95.3%

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

   10. Simplified 95.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.4% accurate, 14.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) (/ -6.0 x) 6.0))

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / x;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / x
	else:
		tmp = 6.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

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. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 5.4%

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

   8. Taylor expanded in x around 0 5.4%

      \[\leadsto \color{blue}{\frac{-6}{x}} \]

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. flip-+ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. swap-sqr 99.7%

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

      6. add-sqr-sqrt 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 95.3%

      \[\leadsto \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

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

Alternative 11: 50.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 6 + \frac{-6}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 6.0 (/ -6.0 x)))

C

double code(double x) {
	return 6.0 + (-6.0 / x);
}

Fortran

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

Java

public static double code(double x) {
	return 6.0 + (-6.0 / x);
}

Python

def code(x):
	return 6.0 + (-6.0 / x)

Julia

function code(x)
	return Float64(6.0 + Float64(-6.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 + (-6.0 / x);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. flip-+ 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. swap-sqr 99.8%

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

   6. add-sqr-sqrt 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf 48.5%

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

8. Taylor expanded in x around 0 48.5%

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

   1. div-sub 48.5%

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

   2. metadata-eval 48.5%

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

   3. associate-*r/ 48.5%

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

   4. sub-neg 48.5%

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

   5. associate-*l/ 48.5%

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

   6. metadata-eval 48.5%

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

   7. associate-*r/ 48.4%

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

   8. associate-*l* 48.5%

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

   9. lft-mult-inverse 48.6%

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

   10. metadata-eval 48.6%

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

   11. associate-*r/ 48.6%

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

   12. metadata-eval 48.6%

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

   13. distribute-neg-frac 48.6%

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

   14. metadata-eval 48.6%

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

10. Simplified 48.6%

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

11. Final simplification 48.6%

    \[\leadsto 6 + \frac{-6}{x} \]
12. Add Preprocessing

Alternative 12: 48.5% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. flip-+ 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. swap-sqr 99.8%

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

   6. add-sqr-sqrt 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf 46.6%

   \[\leadsto \color{blue}{6} \]

8. Final simplification 46.6%

   \[\leadsto 6 \]
9. Add Preprocessing

Developer target: 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 2024067 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))