Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 8.9s

Alternatives: 10

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: 99.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $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. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. *-un-lft-identity 99.9%

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

   5. +-commutative 99.9%

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

   6. 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. Step-by-step derivation

   1. associate-+r+ 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. +-commutative 99.9%

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

      6. 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. Step-by-step derivation

      1. associate-+r+ 99.9%

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

   9. Simplified 99.9%

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

       1. *-commutative 99.9%

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

       2. clear-num 99.9%

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

       3. un-div-inv 99.9%

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

       4. associate-+l+ 99.9%

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

       5. +-commutative 99.9%

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

       6. associate-+r+ 99.9%

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

       7. fma-define 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. 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 0 100.0%

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

      1. associate-+r+ 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf 97.0%

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

4. Final simplification 97.4%

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

Alternative 3: 97.8% 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 97.9%

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

   if 0.28999999999999998 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. 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 0 100.0%

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

      1. associate-+r+ 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf 97.0%

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

4. Final simplification 97.4%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{-1}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ x (+ 1.0 (* 4.0 (sqrt x)))))
   (* 6.0 (/ -1.0 (+ -1.0 (* (pow x -0.5) -4.0))))))

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 / (-1.0 + (pow(x, -0.5) * -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 + (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0 * ((-1.0d0) / ((-1.0d0) + ((x ** (-0.5d0)) * (-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 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 * (-1.0 / (-1.0 + (Math.pow(x, -0.5) * -4.0)));
	}
	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 / (-1.0 + (math.pow(x, -0.5) * -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(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 * Float64(-1.0 / Float64(-1.0 + Float64((x ^ -0.5) * -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 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 * (-1.0 / (-1.0 + ((x ^ -0.5) * -4.0)));
	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[(-1.0 + N[(N[Power[x, -0.5], $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. 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 97.9%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. 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 0.0%

      \[\leadsto \color{blue}{\frac{-1}{4 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) - 1}} \cdot 6 \]
   8. Step-by-step derivation

      1. sub-neg 0.0%

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

   9. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{-1}{{x}^{-0.5} \cdot -4 + -1}} \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{-1}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{x + \left(1 + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ x (+ 1.0 (* 4.0 (sqrt x)))))
   (/ 6.0 (+ 1.0 (sqrt (/ 16.0 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 + sqrt((16.0 / 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 + (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0 / (1.0d0 + sqrt((16.0d0 / 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 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 / (1.0 + Math.sqrt((16.0 / 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 + math.sqrt((16.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(x + Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 / Float64(1.0 + sqrt(Float64(16.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (x + (1.0 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 / (1.0 + sqrt((16.0 / 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[Sqrt[N[(16.0 / 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 97.9%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. inv-pow 96.8%

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

      3. sqrt-pow1 96.8%

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

      4. metadata-eval 96.8%

         \[\leadsto \frac{6}{1 + 4 \cdot \left(1 \cdot {x}^{\color{blue}{-0.5}}\right)} \]

   7. Applied egg-rr 96.8%

      \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 96.8%

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{{x}^{-0.5}}} \]

   9. Simplified 96.8%

      \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{{x}^{-0.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 96.8%

          \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{4 \cdot {x}^{-0.5}} \cdot \sqrt{4 \cdot {x}^{-0.5}}}} \]

       2. sqrt-unprod 96.8%

          \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\left(4 \cdot {x}^{-0.5}\right) \cdot \left(4 \cdot {x}^{-0.5}\right)}}} \]

       3. *-commutative 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{\left({x}^{-0.5} \cdot 4\right)} \cdot \left(4 \cdot {x}^{-0.5}\right)}} \]

       4. *-commutative 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\left({x}^{-0.5} \cdot 4\right) \cdot \color{blue}{\left({x}^{-0.5} \cdot 4\right)}}} \]

       5. swap-sqr 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot \left(4 \cdot 4\right)}}} \]

       6. pow-prod-up 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot \left(4 \cdot 4\right)}} \]

       7. metadata-eval 96.8%

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

       8. inv-pow 96.8%

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

       9. metadata-eval 96.8%

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

   11. Applied egg-rr 96.8%

       \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{1}{x} \cdot 16}}} \]
   12. Step-by-step derivation

       1. associate-*l/ 96.8%

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

       2. metadata-eval 96.8%

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

   13. Simplified 96.8%

       \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{16}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ 1.0 (* 4.0 (sqrt x))))
   (/ 6.0 (+ 1.0 (sqrt (/ 16.0 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 + sqrt((16.0 / 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) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 / (1.0d0 + sqrt((16.0d0 / 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 + Math.sqrt((16.0 / 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 + math.sqrt((16.0 / 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 + sqrt(Float64(16.0 / 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 + sqrt((16.0 / 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[Sqrt[N[(16.0 / 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 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. inv-pow 96.8%

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

      3. sqrt-pow1 96.8%

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

      4. metadata-eval 96.8%

         \[\leadsto \frac{6}{1 + 4 \cdot \left(1 \cdot {x}^{\color{blue}{-0.5}}\right)} \]

   7. Applied egg-rr 96.8%

      \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 96.8%

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{{x}^{-0.5}}} \]

   9. Simplified 96.8%

      \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{{x}^{-0.5}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 96.8%

          \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{4 \cdot {x}^{-0.5}} \cdot \sqrt{4 \cdot {x}^{-0.5}}}} \]

       2. sqrt-unprod 96.8%

          \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\left(4 \cdot {x}^{-0.5}\right) \cdot \left(4 \cdot {x}^{-0.5}\right)}}} \]

       3. *-commutative 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{\left({x}^{-0.5} \cdot 4\right)} \cdot \left(4 \cdot {x}^{-0.5}\right)}} \]

       4. *-commutative 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\left({x}^{-0.5} \cdot 4\right) \cdot \color{blue}{\left({x}^{-0.5} \cdot 4\right)}}} \]

       5. swap-sqr 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot \left(4 \cdot 4\right)}}} \]

       6. pow-prod-up 96.8%

          \[\leadsto \frac{6}{1 + \sqrt{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot \left(4 \cdot 4\right)}} \]

       7. metadata-eval 96.8%

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

       8. inv-pow 96.8%

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

       9. metadata-eval 96.8%

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

   11. Applied egg-rr 96.8%

       \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{1}{x} \cdot 16}}} \]
   12. Step-by-step derivation

       1. associate-*l/ 96.8%

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

       2. metadata-eval 96.8%

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

   13. Simplified 96.8%

       \[\leadsto \frac{6}{1 + \color{blue}{\sqrt{\frac{16}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} \cdot -1.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* (pow x -0.5) -1.5) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = pow(x, -0.5) * -1.5;
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (x ** (-0.5d0)) * (-1.5d0)
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = Math.pow(x, -0.5) * -1.5;
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = math.pow(x, -0.5) * -1.5
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64((x ^ -0.5) * -1.5);
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x ^ -0.5) * -1.5;
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Power[x, -0.5], $MachinePrecision] * -1.5), $MachinePrecision], N[Sqrt[N[(x * 2.25), $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%

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

   6. Taylor expanded in x around inf 6.9%

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

      1. *-commutative 6.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot -1.5} \]

      2. unpow-1 6.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{-1}}} \cdot -1.5 \]

      3. metadata-eval 6.9%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot -1.5 \]

      4. pow-sqr 6.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \cdot -1.5 \]

      5. rem-sqrt-square 6.9%

         \[\leadsto \color{blue}{\left|{x}^{-0.5}\right|} \cdot -1.5 \]

      6. rem-square-sqrt 6.9%

         \[\leadsto \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot -1.5 \]

      7. fabs-sqr 6.9%

         \[\leadsto \color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot -1.5 \]

      8. rem-square-sqrt 6.9%

         \[\leadsto \color{blue}{{x}^{-0.5}} \cdot -1.5 \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot -1.5} \]

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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 96.8%

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

   6. Taylor expanded in x around 0 7.1%

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

      1. *-commutative 7.1%

         \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]

   8. Simplified 7.1%

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

      1. add-sqr-sqrt 7.1%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}} \]

      2. sqrt-unprod 7.1%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}} \]

      3. swap-sqr 7.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}} \]

      4. add-sqr-sqrt 7.1%

         \[\leadsto \sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)} \]

      5. metadata-eval 7.1%

         \[\leadsto \sqrt{x \cdot \color{blue}{2.25}} \]

   10. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -1.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* (sqrt x) -1.5) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = sqrt(x) * -1.5;
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = sqrt(x) * (-1.5d0)
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = Math.sqrt(x) * -1.5;
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = math.sqrt(x) * -1.5
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(sqrt(x) * -1.5);
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = sqrt(x) * -1.5;
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $MachinePrecision], N[Sqrt[N[(x * 2.25), $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 inf 1.9%

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

   6. Taylor expanded in x around 0 1.9%

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

      1. unpow-1 1.9%

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

      2. metadata-eval 1.9%

         \[\leadsto \frac{6}{4 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}}} \]

      3. pow-sqr 1.9%

         \[\leadsto \frac{6}{4 \cdot \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}}} \]

      4. rem-sqrt-square 1.9%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{\left|{x}^{-0.5}\right|}} \]

      5. rem-square-sqrt 1.9%

         \[\leadsto \frac{6}{4 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right|} \]

      6. fabs-sqr 1.9%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)}} \]

      7. rem-square-sqrt 1.9%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{{x}^{-0.5}}} \]

   8. Simplified 1.9%

      \[\leadsto \frac{6}{\color{blue}{4 \cdot {x}^{-0.5}}} \]

   9. Taylor expanded in x around -inf 6.8%

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

       1. *-commutative 6.8%

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

   11. Simplified 6.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      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 96.8%

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

   6. Taylor expanded in x around 0 7.1%

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

      1. *-commutative 7.1%

         \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]

   8. Simplified 7.1%

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

      1. add-sqr-sqrt 7.1%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}} \]

      2. sqrt-unprod 7.1%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}} \]

      3. swap-sqr 7.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}} \]

      4. add-sqr-sqrt 7.1%

         \[\leadsto \sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)} \]

      5. metadata-eval 7.1%

         \[\leadsto \sqrt{x \cdot \color{blue}{2.25}} \]

   10. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (math.sqrt(x) * 24.0) - 6.0

Julia

function code(x)
	return Float64(Float64(sqrt(x) * 24.0) - 6.0)
end

MATLAB

function tmp = code(x)
	tmp = (sqrt(x) * 24.0) - 6.0;
end

Wolfram

code[x_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 24.0), $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. associate-+r+ 99.8%

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

   2. flip-+ 76.4%

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

   3. *-commutative 76.4%

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

   4. *-commutative 76.4%

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

   5. swap-sqr 76.4%

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

   6. add-sqr-sqrt 76.4%

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

   7. metadata-eval 76.4%

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

6. Applied egg-rr 76.4%

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

7. Taylor expanded in x around 0 52.3%

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

   1. cancel-sign-sub-inv 52.3%

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

   2. metadata-eval 52.3%

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

   3. distribute-lft-in 52.3%

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

   4. metadata-eval 52.3%

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

   5. *-commutative 52.3%

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

9. Simplified 52.3%

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

10. Taylor expanded in x around 0 52.3%

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

11. Final simplification 52.3%

    \[\leadsto \sqrt{x} \cdot 24 - 6 \]
12. Add Preprocessing

Alternative 10: 4.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 2.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* x 2.25)))

C

double code(double x) {
	return sqrt((x * 2.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * 2.25d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((x * 2.25));
}

Python

def code(x):
	return math.sqrt((x * 2.25))

Julia

function code(x)
	return sqrt(Float64(x * 2.25))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x * 2.25));
end

Wolfram

code[x_] := N[Sqrt[N[(x * 2.25), $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. Taylor expanded in x around inf 49.3%

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

6. Taylor expanded in x around 0 4.5%

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

   1. *-commutative 4.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]

8. Simplified 4.5%

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

   1. add-sqr-sqrt 4.5%

      \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot 1.5} \cdot \sqrt{\sqrt{x} \cdot 1.5}} \]

   2. sqrt-unprod 4.5%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot 1.5\right) \cdot \left(\sqrt{x} \cdot 1.5\right)}} \]

   3. swap-sqr 4.5%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1.5 \cdot 1.5\right)}} \]

   4. add-sqr-sqrt 4.5%

      \[\leadsto \sqrt{\color{blue}{x} \cdot \left(1.5 \cdot 1.5\right)} \]

   5. metadata-eval 4.5%

      \[\leadsto \sqrt{x \cdot \color{blue}{2.25}} \]

10. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
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 2024135 
(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)))))