Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.8% → 99.9%

Time: 8.4s

Alternatives: 8

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. add-exp-log 96.3%

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

      2. log-rec 96.3%

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

      3. sqrt-div 96.3%

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

      4. metadata-eval 96.3%

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

      5. div-inv 96.3%

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

      6. log1p-define 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. exp-neg 96.3%

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

      2. log1p-undefine 96.3%

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

      3. rem-exp-log 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 97.3%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (+ x 1.0) (* 4.0 (sqrt x))))
   (* 6.0 (/ 1.0 (+ 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 / (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 / (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 / (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 / (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(Float64(x + 1.0) + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 * Float64(1.0 / Float64(1.0 + Float64(4.0 / sqrt(x)))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.6%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. add-exp-log 96.3%

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

      2. log-rec 96.3%

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

      3. sqrt-div 96.3%

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

      4. metadata-eval 96.3%

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

      5. div-inv 96.3%

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

      6. log1p-define 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. exp-neg 96.3%

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

      2. log1p-undefine 96.3%

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

      3. rem-exp-log 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 97.3%

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

Alternative 4: 99.8% 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]

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 5: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} 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 = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    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 = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = math.sqrt((x * 2.25))
	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 = sqrt(Float64(x * 2.25));
	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 = sqrt((x * 2.25));
	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[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. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf 96.2%

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

   4. Taylor expanded in x around 0 7.1%

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

      1. *-commutative 7.1%

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

   6. Simplified 7.1%

      \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
   7. 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%

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

   8. Applied egg-rr 7.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 1.0 / (0.16666666666666666 + (0.6666666666666666 / 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 = 1.0d0 / (0.16666666666666666d0 + (0.6666666666666666d0 / 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 = 1.0 / (0.16666666666666666 + (0.6666666666666666 / 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 = 1.0 / (0.16666666666666666 + (0.6666666666666666 / 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(1.0 / Float64(0.16666666666666666 + Float64(0.6666666666666666 / 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 = 1.0 / (0.16666666666666666 + (0.6666666666666666 / 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[(1.0 / N[(0.16666666666666666 + N[(0.6666666666666666 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf 96.2%

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

      1. add-exp-log 96.2%

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

      2. log-div 96.3%

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

      3. log1p-define 96.3%

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

      4. sqrt-div 96.3%

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

      5. metadata-eval 96.3%

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

      6. un-div-inv 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. exp-diff 96.2%

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

      2. rem-exp-log 96.2%

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

      3. log1p-undefine 96.2%

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

      4. rem-exp-log 96.2%

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

   7. Simplified 96.2%

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

      1. metadata-eval 96.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{0.16666666666666666}}}{1 + \frac{4}{\sqrt{x}}} \]

      2. +-commutative 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{\color{blue}{\frac{4}{\sqrt{x}} + 1}} \]

      3. div-inv 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{\color{blue}{4 \cdot \frac{1}{\sqrt{x}}} + 1} \]

      4. metadata-eval 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{4 \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}} + 1} \]

      5. sqrt-div 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{4 \cdot \color{blue}{\sqrt{\frac{1}{x}}} + 1} \]

      6. inv-pow 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{4 \cdot \sqrt{\color{blue}{{x}^{-1}}} + 1} \]

      7. sqrt-pow1 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{4 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} + 1} \]

      8. metadata-eval 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{4 \cdot {x}^{\color{blue}{-0.5}} + 1} \]

      9. fma-undefine 96.2%

         \[\leadsto \frac{\frac{1}{0.16666666666666666}}{\color{blue}{\mathsf{fma}\left(4, {x}^{-0.5}, 1\right)}} \]

      10. associate-/r* 96.2%

          \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot \mathsf{fma}\left(4, {x}^{-0.5}, 1\right)}} \]

      11. *-commutative 96.2%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, {x}^{-0.5}, 1\right) \cdot 0.16666666666666666}} \]

      12. inv-pow 96.3%

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(4, {x}^{-0.5}, 1\right) \cdot 0.16666666666666666\right)}^{-1}} \]

      13. pow-to-exp 96.3%

          \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(4, {x}^{-0.5}, 1\right) \cdot 0.16666666666666666\right) \cdot -1}} \]

   9. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{e^{\left(\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right) + \log 0.16666666666666666\right) \cdot -1}} \]
   10. Step-by-step derivation

       1. exp-prod 94.9%

          \[\leadsto \color{blue}{{\left(e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right) + \log 0.16666666666666666}\right)}^{-1}} \]

       2. unpow-1 94.9%

          \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right) + \log 0.16666666666666666}}} \]

       3. exp-sum 94.9%

          \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)} \cdot e^{\log 0.16666666666666666}}} \]

       4. rem-exp-log 96.2%

          \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{x}}\right)} \cdot \color{blue}{0.16666666666666666}} \]

       5. log1p-undefine 96.2%

          \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + \frac{4}{\sqrt{x}}\right)}} \cdot 0.16666666666666666} \]

       6. rem-exp-log 96.2%

          \[\leadsto \frac{1}{\color{blue}{\left(1 + \frac{4}{\sqrt{x}}\right)} \cdot 0.16666666666666666} \]

       7. +-commutative 96.2%

          \[\leadsto \frac{1}{\color{blue}{\left(\frac{4}{\sqrt{x}} + 1\right)} \cdot 0.16666666666666666} \]

       8. distribute-rgt1-in 96.3%

          \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + \frac{4}{\sqrt{x}} \cdot 0.16666666666666666}} \]

       9. associate-*l/ 96.3%

          \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{4 \cdot 0.16666666666666666}{\sqrt{x}}}} \]

       10. metadata-eval 96.3%

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

   11. Simplified 96.3%

       \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 + \frac{0.6666666666666666}{\sqrt{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

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

Alternative 7: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = sqrt((1.0 / 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((1.0d0 / 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((1.0 / 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((1.0 / 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(Float64(1.0 / 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((1.0 / 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[N[(1.0 / x), $MachinePrecision]], $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. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

   4. Taylor expanded in x around inf 6.9%

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

      1. *-commutative 6.9%

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

   6. Simplified 6.9%

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

   if 1 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf 96.2%

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

   4. Taylor expanded in x around 0 7.1%

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

      1. *-commutative 7.1%

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

   6. Simplified 7.1%

      \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
   7. 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%

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

   8. Applied egg-rr 7.1%

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

4. Final simplification 7.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot -1.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 99.4%

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

3. Taylor expanded in x around inf 53.1%

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

4. Taylor expanded in x around 0 4.7%

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

   1. *-commutative 4.7%

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

6. Simplified 4.7%

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

   1. add-sqr-sqrt 4.7%

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

   2. sqrt-unprod 4.7%

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

   3. swap-sqr 4.7%

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

   4. add-sqr-sqrt 4.7%

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

   5. metadata-eval 4.7%

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

8. Applied egg-rr 4.7%

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

9. Final simplification 4.7%

   \[\leadsto \sqrt{x \cdot 2.25} \]
10. 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 2024077 
(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)))))