Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.8%

Time: 8.3s

Alternatives: 11

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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in x around inf 99.5%

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

   1. associate-/l* 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. un-div-inv 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-define 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r/ 99.9%

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

7. Simplified 99.9%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 4.0) (/ (- (* 6.0 x) 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 = ((6.0 * x) - 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 = ((6.0d0 * x) - 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 = ((6.0 * x) - 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 = ((6.0 * x) - 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(6.0 * x) - 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 = ((6.0 * x) - 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[(6.0 * x), $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 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   4. Taylor expanded in x around 0 98.5%

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

   if 4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 4.0) (/ (* 6.0 (+ x -1.0)) (+ 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 = (6.0 * (x + -1.0)) / (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 = (6.0d0 * (x + (-1.0d0))) / (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 = (6.0 * (x + -1.0)) / (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 = (6.0 * (x + -1.0)) / (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(6.0 * Float64(x + -1.0)) / 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 = (6.0 * (x + -1.0)) / (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[(6.0 * N[(x + -1.0), $MachinePrecision]), $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 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

4. Final simplification 98.8%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 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(Float64(x + 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[(N[(x + 1.0), $MachinePrecision] + 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 < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

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

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(Float64(x + 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[(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[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. sqrt-div 99.2%

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

      2. metadata-eval 99.2%

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

      3. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

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(x + -1.0) * Float64(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[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. associate-/l* 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

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 100.0%

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

   3. Taylor expanded in x around 0 98.4%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. sqrt-div 99.2%

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

      2. metadata-eval 99.2%

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

      3. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

Alternative 8: 52.2% 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 100.0%

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

   3. Taylor expanded in x around 0 98.4%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 6.9%

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

      1. *-commutative 6.9%

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

   6. Simplified 6.9%

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

      1. pow1 6.9%

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

      2. add-sqr-sqrt 6.9%

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

      3. sqrt-unprod 6.9%

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

      4. swap-sqr 6.9%

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

      5. add-sqr-sqrt 6.9%

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

      6. metadata-eval 6.9%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   8. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 6.9%

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

   10. Simplified 6.9%

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

Alternative 9: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 * pow(x, -0.5);
	} 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 = (-1.5d0) * (x ** (-0.5d0))
    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 = -1.5 * Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 * (x ^ -0.5));
	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 = -1.5 * (x ^ -0.5);
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.4%

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

   4. Taylor expanded in x around inf 6.8%

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

      1. unpow-1 6.8%

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

      2. metadata-eval 6.8%

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

      3. pow-sqr 6.8%

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

      4. rem-sqrt-square 6.8%

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

      5. rem-square-sqrt 6.8%

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

      6. fabs-sqr 6.8%

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

      7. rem-square-sqrt 6.8%

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

   6. Simplified 6.8%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 6.9%

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

      1. *-commutative 6.9%

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

   6. Simplified 6.9%

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

      1. pow1 6.9%

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

      2. add-sqr-sqrt 6.9%

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

      3. sqrt-unprod 6.9%

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

      4. swap-sqr 6.9%

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

      5. add-sqr-sqrt 6.9%

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

      6. metadata-eval 6.9%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   8. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 6.9%

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

   10. Simplified 6.9%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 1.9%

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

   4. Taylor expanded in x around 0 1.9%

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

      1. unpow-1 1.9%

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

      2. metadata-eval 1.9%

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

      3. pow-sqr 1.9%

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

      4. rem-sqrt-square 1.9%

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

      5. rem-square-sqrt 1.9%

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

      6. fabs-sqr 1.9%

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

      7. rem-square-sqrt 1.9%

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

   6. Simplified 1.9%

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

   7. Taylor expanded in x around -inf 6.7%

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

      1. *-commutative 6.7%

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

   9. Simplified 6.7%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.2%

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

   4. Taylor expanded in x around 0 6.9%

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

      1. *-commutative 6.9%

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

   6. Simplified 6.9%

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

      1. pow1 6.9%

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

      2. add-sqr-sqrt 6.9%

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

      3. sqrt-unprod 6.9%

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

      4. swap-sqr 6.9%

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

      5. add-sqr-sqrt 6.9%

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

      6. metadata-eval 6.9%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   8. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 6.9%

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

   10. Simplified 6.9%

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

Alternative 11: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 48.2%

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

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. pow1 4.3%

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

   2. add-sqr-sqrt 4.3%

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

   3. sqrt-unprod 4.3%

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

   4. swap-sqr 4.3%

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

   5. add-sqr-sqrt 4.3%

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

   6. metadata-eval 4.3%

      \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
9. Step-by-step derivation

   1. unpow1 4.3%

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

10. Simplified 4.3%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1))))

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