Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.7s

Alternatives: 9

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.078:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot \left(1 + \left(x + \sqrt{x} \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + 4 \cdot {x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.078)
   (* (- (* x -78.0) 6.0) (+ 1.0 (+ x (* (sqrt x) -4.0))))
   (/ 6.0 (+ 1.0 (* 4.0 (pow x -0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.078) {
		tmp = ((x * -78.0) - 6.0) * (1.0 + (x + (sqrt(x) * -4.0)));
	} else {
		tmp = 6.0 / (1.0 + (4.0 * pow(x, -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.078d0) then
        tmp = ((x * (-78.0d0)) - 6.0d0) * (1.0d0 + (x + (sqrt(x) * (-4.0d0))))
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 * (x ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.078) {
		tmp = ((x * -78.0) - 6.0) * (1.0 + (x + (Math.sqrt(x) * -4.0)));
	} else {
		tmp = 6.0 / (1.0 + (4.0 * Math.pow(x, -0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.078:
		tmp = ((x * -78.0) - 6.0) * (1.0 + (x + (math.sqrt(x) * -4.0)))
	else:
		tmp = 6.0 / (1.0 + (4.0 * math.pow(x, -0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.078)
		tmp = Float64(Float64(Float64(x * -78.0) - 6.0) * Float64(1.0 + Float64(x + Float64(sqrt(x) * -4.0))));
	else
		tmp = Float64(6.0 / Float64(1.0 + Float64(4.0 * (x ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.078)
		tmp = ((x * -78.0) - 6.0) * (1.0 + (x + (sqrt(x) * -4.0)));
	else
		tmp = 6.0 / (1.0 + (4.0 * (x ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.078], N[(N[(N[(x * -78.0), $MachinePrecision] - 6.0), $MachinePrecision] * N[(1.0 + N[(x + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0779999999999999999

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{6 \cdot x + -6}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot 6} + -6}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      3. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\color{blue}{\left(1 + x\right)}}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(x + \left(1 + -4 \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)} \]

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{\left(-78 \cdot x - 6\right)} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right) \]

   8. Taylor expanded in x around 0 99.6%

      \[\leadsto \left(-78 \cdot x - 6\right) \cdot \color{blue}{\left(1 + \left(x + -4 \cdot \sqrt{x}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \left(-78 \cdot x - 6\right) \cdot \left(1 + \left(x + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \left(-78 \cdot x - 6\right) \cdot \color{blue}{\left(1 + \left(x + \sqrt{x} \cdot -4\right)\right)} \]

   if 0.0779999999999999999 < 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 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. inv-pow 98.4%

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

      3. sqrt-pow1 98.4%

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

      4. metadata-eval 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.078:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot \left(1 + \left(x + \sqrt{x} \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + 4 \cdot {x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

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 * pow(x, -0.5)));
	}
	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 * (x ** (-0.5d0))))
    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.pow(x, -0.5)));
	}
	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.pow(x, -0.5)))
	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(4.0 * (x ^ -0.5))));
	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 * (x ^ -0.5)));
	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[(4.0 * N[Power[x, -0.5], $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 98.8%

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

   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. Taylor expanded in x around inf 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. inv-pow 98.4%

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

      3. sqrt-pow1 98.4%

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

      4. metadata-eval 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

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 * pow(x, -0.5)));
	}
	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 * (x ** (-0.5d0))))
    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.pow(x, -0.5)));
	}
	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.pow(x, -0.5)))
	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 * (x ^ -0.5))));
	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 * (x ^ -0.5)));
	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[Power[x, -0.5], $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 98.7%

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

   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 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. inv-pow 98.4%

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

      3. sqrt-pow1 98.4%

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

      4. metadata-eval 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

   7. Simplified 98.4%

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

Alternative 5: 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]

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14.2:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-24}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 14.2) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (+ 6.0 (/ -24.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 14.2) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 14.2d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 + ((-24.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 14.2) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 14.2:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 + (-24.0 / math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 14.2)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 + Float64(-24.0 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 14.2)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 + (-24.0 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 14.2], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 14.199999999999999

   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 98.7%

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

   if 14.199999999999999 < 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. flip-+ 45.6%

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

      2. associate-/r/ 45.5%

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

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. fma-define 45.5%

         \[\leadsto \frac{\color{blue}{6 \cdot x + -6}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      2. *-commutative 45.5%

         \[\leadsto \frac{\color{blue}{x \cdot 6} + -6}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      3. fma-define 45.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      4. +-commutative 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\color{blue}{\left(1 + x\right)}}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      5. associate-+l+ 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(x + \left(1 + -4 \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   6. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)} \]

   7. Taylor expanded in x around inf 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r* 98.3%

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

      4. metadata-eval 98.3%

         \[\leadsto 6 + \color{blue}{-24} \cdot \sqrt{\frac{1}{x}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. sqrt-div 98.3%

          \[\leadsto 6 + -24 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

       2. metadata-eval 98.3%

          \[\leadsto 6 + -24 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

       3. un-div-inv 98.3%

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

   11. Applied egg-rr 98.3%

       \[\leadsto 6 + \color{blue}{\frac{-24}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 16.5:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-24}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 16.5) (+ -6.0 (* (sqrt x) 24.0)) (+ 6.0 (/ -24.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 16.5) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 16.5d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0 + ((-24.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 16.5) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 16.5:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0 + (-24.0 / math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 16.5)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(6.0 + Float64(-24.0 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 16.5)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0 + (-24.0 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 16.5], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 16.5

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{6 \cdot x + -6}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot 6} + -6}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      3. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\color{blue}{\left(1 + x\right)}}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      5. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(x + \left(1 + -4 \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)} \]

   7. Taylor expanded in x around 0 98.5%

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

      1. metadata-eval 98.5%

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

      2. distribute-lft-neg-in 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. distribute-lft-neg-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. associate-*r* 98.5%

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

      8. metadata-eval 98.5%

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

   9. Simplified 98.5%

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

   if 16.5 < 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. flip-+ 45.6%

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

      2. associate-/r/ 45.5%

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

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. fma-define 45.5%

         \[\leadsto \frac{\color{blue}{6 \cdot x + -6}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      2. *-commutative 45.5%

         \[\leadsto \frac{\color{blue}{x \cdot 6} + -6}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      3. fma-define 45.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      4. +-commutative 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\color{blue}{\left(1 + x\right)}}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

      5. associate-+l+ 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(x + \left(1 + -4 \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 45.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

   6. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)} \]

   7. Taylor expanded in x around inf 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r* 98.3%

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

      4. metadata-eval 98.3%

         \[\leadsto 6 + \color{blue}{-24} \cdot \sqrt{\frac{1}{x}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
   10. Step-by-step derivation

       1. sqrt-div 98.3%

          \[\leadsto 6 + -24 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

       2. metadata-eval 98.3%

          \[\leadsto 6 + -24 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

       3. un-div-inv 98.3%

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

   11. Applied egg-rr 98.3%

       \[\leadsto 6 + \color{blue}{\frac{-24}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 16.5:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-24}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. flip-+ 73.0%

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

   2. associate-/r/ 73.0%

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

4. Applied egg-rr 72.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]
5. Step-by-step derivation

   1. fma-define 73.0%

      \[\leadsto \frac{\color{blue}{6 \cdot x + -6}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   2. *-commutative 73.0%

      \[\leadsto \frac{\color{blue}{x \cdot 6} + -6}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   3. fma-define 72.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   4. +-commutative 72.9%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\color{blue}{\left(1 + x\right)}}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   5. associate-+l+ 72.9%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(x + \left(1 + -4 \cdot \sqrt{x}\right)\right)} \]

   6. *-commutative 72.9%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \color{blue}{\sqrt{x} \cdot -4}\right)\right) \]

6. Simplified 72.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{{\left(1 + x\right)}^{2} - x \cdot 16} \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)} \]

7. Taylor expanded in x around 0 53.0%

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

   1. metadata-eval 53.0%

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

   2. distribute-lft-neg-in 53.0%

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

   3. distribute-lft-in 53.0%

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

   4. metadata-eval 53.0%

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

   5. distribute-lft-neg-in 53.0%

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

   6. metadata-eval 53.0%

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

   7. associate-*r* 53.0%

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

   8. metadata-eval 53.0%

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

9. Simplified 53.0%

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

10. Final simplification 53.0%

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

Alternative 9: 4.5% 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 99.5%

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

3. Taylor expanded in x around inf 49.8%

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

4. Taylor expanded in x around 0 4.3%

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

   1. *-commutative 4.3%

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

6. Simplified 4.3%

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

   1. add-sqr-sqrt 4.3%

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

   2. sqrt-unprod 4.3%

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

   3. swap-sqr 4.3%

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

   4. add-sqr-sqrt 4.3%

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

   5. metadata-eval 4.3%

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

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
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 2024087 
(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)))))