Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 9.9s

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;6 \cdot \frac{x + -1}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + t\_0\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 3.45)
     (* 6.0 (/ (+ x -1.0) (+ t_0 1.0)))
     (* 6.0 (/ x (+ (+ x t_0) 1.0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 3.45) {
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	} else {
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 3.45d0) then
        tmp = 6.0d0 * ((x + (-1.0d0)) / (t_0 + 1.0d0))
    else
        tmp = 6.0d0 * (x / ((x + t_0) + 1.0d0))
    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 <= 3.45) {
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	} else {
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 3.45:
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0))
	else:
		tmp = 6.0 * (x / ((x + t_0) + 1.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 3.45)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(t_0 + 1.0)));
	else
		tmp = Float64(6.0 * Float64(x / Float64(Float64(x + t_0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 3.45)
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	else
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.45], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(N[(x + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4500000000000002

   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. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.3%

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

   if 3.4500000000000002 < 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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-+r+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;6 \cdot \frac{x + -1}{4 \cdot \sqrt{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + 4 \cdot \sqrt{x}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.3%

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

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-+r+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.1%

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

4. Final simplification 98.2%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	} 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) / ((4.0d0 * sqrt(x)) + (x + 1.0d0))
    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 / ((4.0 * Math.sqrt(x)) + (x + 1.0));
	} 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 / ((4.0 * math.sqrt(x)) + (x + 1.0))
	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(4.0 * sqrt(x)) + Float64(x + 1.0)));
	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 / ((4.0 * sqrt(x)) + (x + 1.0));
	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[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.3%

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

   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 99.0%

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

      1. +-commutative 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. fma-define 99.0%

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

      4. sqrt-div 99.0%

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

      5. metadata-eval 99.0%

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

      6. un-div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. fma-undefine 99.0%

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

      2. *-lft-identity 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 98.1%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + 1.0);
	} 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) / ((4.0d0 * sqrt(x)) + 1.0d0)
    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 / ((4.0 * Math.sqrt(x)) + 1.0);
	} 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 / ((4.0 * math.sqrt(x)) + 1.0)
	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(4.0 * sqrt(x)) + 1.0));
	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 / ((4.0 * sqrt(x)) + 1.0);
	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[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $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. Add Preprocessing

   3. Taylor expanded in x around 0 97.2%

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

   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 99.0%

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

      1. +-commutative 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. fma-define 99.0%

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

      4. sqrt-div 99.0%

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

      5. metadata-eval 99.0%

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

      6. un-div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. fma-undefine 99.0%

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

      2. *-lft-identity 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 98.1%

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

Alternative 6: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 / 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(-1.5 / 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 = -1.5 / sqrt(x);
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.2%

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

   4. Taylor expanded in x around inf 7.1%

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

      1. *-un-lft-identity 7.1%

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

      2. associate-/r* 7.1%

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

      3. metadata-eval 7.1%

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

   6. Applied egg-rr 7.1%

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

      1. *-lft-identity 7.1%

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

   8. Simplified 7.1%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 7.0%

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

   4. Taylor expanded in x around -inf 1.3%

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

      1. *-commutative 1.3%

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

   6. Simplified 1.3%

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

      1. pow1 1.3%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 7.0%

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

      4. swap-sqr 7.0%

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

      5. add-sqr-sqrt 7.0%

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

      6. metadata-eval 7.0%

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

   8. Applied egg-rr 7.0%

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

      1. unpow1 7.0%

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

   10. Simplified 7.0%

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

Alternative 7: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.3%

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

   4. Taylor expanded in x around -inf 7.0%

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

      1. *-commutative 7.0%

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

   6. Simplified 7.0%

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

   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 0 7.0%

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

   4. Taylor expanded in x around -inf 1.3%

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

      1. *-commutative 1.3%

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

   6. Simplified 1.3%

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

      1. pow1 1.3%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 7.0%

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

      4. swap-sqr 7.0%

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

      5. add-sqr-sqrt 7.0%

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

      6. metadata-eval 7.0%

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

   8. Applied egg-rr 7.0%

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

      1. unpow1 7.0%

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

   10. Simplified 7.0%

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

Alternative 8: 52.1% 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. Add Preprocessing

3. Taylor expanded in x around 0 50.7%

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

   1. flip-+ 50.7%

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

   2. associate-/r/ 50.7%

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

   3. metadata-eval 50.7%

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

   4. *-commutative 50.7%

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

   5. *-commutative 50.7%

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

   6. swap-sqr 50.7%

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

   7. add-sqr-sqrt 50.7%

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

   8. metadata-eval 50.7%

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

   9. cancel-sign-sub-inv 50.7%

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

   10. metadata-eval 50.7%

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

5. Applied egg-rr 50.7%

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

   1. *-commutative 50.7%

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

7. Simplified 50.7%

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

8. Taylor expanded in x around 0 53.0%

   \[\leadsto \color{blue}{-6 \cdot \left(1 + -4 \cdot \sqrt{x}\right)} \]
9. 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%

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

10. Simplified 53.0%

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

11. Final simplification 53.0%

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

Alternative 9: 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 0 53.2%

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

4. Taylor expanded in x around -inf 4.2%

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

   1. *-commutative 4.2%

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

6. Simplified 4.2%

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

   1. pow1 4.2%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 4.4%

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

   4. swap-sqr 4.4%

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

   5. add-sqr-sqrt 4.4%

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

   6. metadata-eval 4.4%

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

8. Applied egg-rr 4.4%

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

   1. unpow1 4.4%

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

10. Simplified 4.4%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
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 2024118 
(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)))))