Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 9.4s

Alternatives: 10

Speedup: 0.5×

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 99.8%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   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. distribute-lft-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. metadata-eval 95.3%

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

      2. distribute-lft-in 95.3%

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

      3. *-commutative 95.3%

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

   7. Applied egg-rr 95.3%

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

   if 4 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.1%

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

   6. Taylor expanded in x around 0 98.1%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (x + (1.0 + t_0))
	else:
		tmp = 6.0 / ((x + t_0) / x)
	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(x + Float64(1.0 + t_0)));
	else
		tmp = Float64(6.0 / Float64(Float64(x + t_0) / x));
	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 / (x + (1.0 + t_0));
	else
		tmp = 6.0 / ((x + t_0) / x);
	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[(x + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(x + t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. 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 95.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. add-exp-log 97.5%

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

      2. log-div 97.5%

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

      3. log1p-define 97.5%

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

      4. sqrt-div 97.5%

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

      5. metadata-eval 97.5%

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

      6. un-div-inv 97.5%

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

   7. Applied egg-rr 97.5%

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

      1. exp-diff 97.5%

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

      2. rem-exp-log 97.5%

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

      3. log1p-undefine 97.5%

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

      4. rem-exp-log 97.5%

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

   9. Simplified 97.5%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Final simplification 99.8%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. add-exp-log 97.5%

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

      2. log-div 97.5%

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

      3. log1p-define 97.5%

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

      4. sqrt-div 97.5%

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

      5. metadata-eval 97.5%

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

      6. un-div-inv 97.5%

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

   7. Applied egg-rr 97.5%

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

      1. exp-diff 97.5%

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

      2. rem-exp-log 97.5%

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

      3. log1p-undefine 97.5%

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

      4. rem-exp-log 97.5%

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

   9. Simplified 97.5%

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

Alternative 7: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. 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 95.8%

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

   6. Taylor expanded in x around inf 7.3%

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

      1. *-commutative 7.3%

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

   8. Simplified 7.3%

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

      1. *-commutative 7.3%

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

      2. sqrt-div 7.3%

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

      3. metadata-eval 7.3%

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

      4. un-div-inv 7.3%

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

   10. Applied egg-rr 7.3%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 6.9%

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

   6. Taylor expanded in x around inf 6.9%

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

      1. *-commutative 6.9%

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

   8. Simplified 6.9%

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

Alternative 8: 53.0% 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 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. distribute-lft-in 99.8%

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

   3. metadata-eval 99.8%

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

   4. metadata-eval 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. +-commutative 99.8%

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

   3. flip-+ 72.9%

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

   4. pow2 72.9%

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

   5. *-commutative 72.9%

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

   6. *-commutative 72.9%

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

   7. swap-sqr 72.9%

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

   8. add-sqr-sqrt 72.9%

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

   9. metadata-eval 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in x around 0 49.7%

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

   1. sub-neg 49.7%

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

   2. distribute-lft-in 49.7%

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

   3. metadata-eval 49.7%

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

   4. *-commutative 49.7%

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

   5. distribute-rgt-neg-in 49.7%

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

   6. metadata-eval 49.7%

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

9. Simplified 49.7%

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

10. Taylor expanded in x around 0 49.7%

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

    1. *-commutative 49.7%

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

12. Simplified 49.7%

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

Alternative 9: 4.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt x) 1.5))

C

double code(double x) {
	return sqrt(x) * 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(x) * 1.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt(x) * 1.5;
}

Python

def code(x):
	return math.sqrt(x) * 1.5

Julia

function code(x)
	return Float64(sqrt(x) * 1.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(x) * 1.5;
end

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. distribute-lft-in 99.8%

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

   3. metadata-eval 99.8%

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

   4. metadata-eval 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 50.0%

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

6. Taylor expanded in x around inf 4.4%

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

   1. *-commutative 4.4%

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

8. Simplified 4.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
9. Add Preprocessing

Alternative 10: 4.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.25 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 47.4%

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

6. Taylor expanded in x around inf 4.5%

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

   1. *-commutative 4.5%

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

8. Simplified 4.5%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 4.1%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{\frac{1}{x}} \cdot -1.5\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot -1.5\right)}} \]

   3. swap-sqr 4.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(-1.5 \cdot -1.5\right)}} \]

   4. add-sqr-sqrt 4.1%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{x}} \cdot \left(-1.5 \cdot -1.5\right)} \]

   5. metadata-eval 4.1%

      \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{2.25}} \]

10. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 2.25}} \]
11. Step-by-step derivation

    1. associate-*l/ 4.1%

       \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 2.25}{x}}} \]

    2. metadata-eval 4.1%

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

12. Simplified 4.1%

    \[\leadsto \color{blue}{\sqrt{\frac{2.25}{x}}} \]
13. 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 2024100 
(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)))))