Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 10.7s

Alternatives: 10

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(6.0 * x), $MachinePrecision] + -6.0), $MachinePrecision] / N[(1.0 + N[(x * N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 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. 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 inf 99.7%

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

   1. associate-+r+ 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

   5. rgt-mult-inverse 99.8%

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

7. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. sqrt-div 99.8%

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

   3. metadata-eval 99.8%

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

   4. un-div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in x around 0 96.6%

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

   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 97.0%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 97.2%

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

      5. unpow-1 97.2%

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

      6. metadata-eval 97.2%

         \[\leadsto \frac{-6}{4 \cdot \left(-1 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) + \left(-1\right)} \]

      7. pow-sqr 97.2%

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

      8. rem-sqrt-square 97.2%

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

      9. rem-square-sqrt 97.2%

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

      10. fabs-sqr 97.2%

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

      11. rem-square-sqrt 97.2%

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

      12. associate-*r* 97.2%

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

      13. metadata-eval 97.2%

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

      14. metadata-eval 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 96.8%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (x + (4.0 * sqrt(x))));
	} else {
		tmp = -6.0 / ((-4.0 * pow(x, -0.5)) + -1.0);
	}
	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 + (x + (4.0d0 * sqrt(x))))
    else
        tmp = (-6.0d0) / (((-4.0d0) * (x ** (-0.5d0))) + (-1.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 / N[(N[(-4.0 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + -1.0), $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. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 96.4%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 96.5%

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

      5. unpow-1 96.5%

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

      6. metadata-eval 96.5%

         \[\leadsto \frac{-6}{4 \cdot \left(-1 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) + \left(-1\right)} \]

      7. pow-sqr 96.5%

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

      8. rem-sqrt-square 96.5%

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

      9. rem-square-sqrt 96.5%

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

      10. fabs-sqr 96.5%

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

      11. rem-square-sqrt 96.5%

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

      12. associate-*r* 96.5%

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

      13. metadata-eval 96.5%

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

      14. metadata-eval 96.5%

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

   6. Simplified 96.5%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = -6.0 / ((-4.0 * pow(x, -0.5)) + -1.0);
	}
	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) / (((-4.0d0) * (x ** (-0.5d0))) + (-1.0d0))
    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 / ((-4.0 * Math.pow(x, -0.5)) + -1.0);
	}
	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 / ((-4.0 * math.pow(x, -0.5)) + -1.0)
	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(Float64(-4.0 * (x ^ -0.5)) + -1.0));
	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 / ((-4.0 * (x ^ -0.5)) + -1.0);
	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[(N[(-4.0 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] + -1.0), $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. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

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

   7. Simplified 97.0%

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

   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 96.4%

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

   4. Taylor expanded in x around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 96.5%

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

      5. unpow-1 96.5%

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

      6. metadata-eval 96.5%

         \[\leadsto \frac{-6}{4 \cdot \left(-1 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) + \left(-1\right)} \]

      7. pow-sqr 96.5%

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

      8. rem-sqrt-square 96.5%

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

      9. rem-square-sqrt 96.5%

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

      10. fabs-sqr 96.5%

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

      11. rem-square-sqrt 96.5%

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

      12. associate-*r* 96.5%

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

      13. metadata-eval 96.5%

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

      14. metadata-eval 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 96.8%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (* (sqrt x) 1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * 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 = (-6.0d0) / (1.0d0 + (4.0d0 * 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 = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(sqrt(x) * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

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

   7. Simplified 97.0%

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

   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 inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-define 99.7%

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

      5. rgt-mult-inverse 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 7.3%

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

   9. Taylor expanded in x around inf 7.3%

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

       1. *-commutative 7.3%

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

   11. Simplified 7.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 7.0% accurate, 1.0× speedup

Math

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

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. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around inf 7.3%

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

      1. unpow-1 7.3%

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

      2. metadata-eval 7.3%

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

      3. pow-sqr 7.3%

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

      4. rem-sqrt-square 7.3%

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

      5. rem-square-sqrt 7.3%

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

      6. fabs-sqr 7.3%

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

      7. rem-square-sqrt 7.3%

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

   10. Simplified 7.3%

       \[\leadsto \color{blue}{-1.5 \cdot {x}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 1.3%

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

       3. *-commutative 1.3%

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

       4. *-commutative 1.3%

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

       5. swap-sqr 1.3%

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

       6. pow-prod-up 1.3%

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

       7. metadata-eval 1.3%

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

       8. inv-pow 1.3%

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

       9. metadata-eval 1.3%

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

   12. Applied egg-rr 1.3%

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

       1. associate-*l/ 1.3%

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

       2. metadata-eval 1.3%

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

   14. Simplified 1.3%

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

       1. div-inv 1.3%

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

       2. metadata-eval 1.3%

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

       3. add-sqr-sqrt 1.3%

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

       4. swap-sqr 1.3%

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

       5. *-commutative 1.3%

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

       6. *-commutative 1.3%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 7.3%

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

       9. *-un-lft-identity 7.3%

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

       10. *-commutative 7.3%

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

       11. sqrt-div 7.3%

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

       12. metadata-eval 7.3%

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

       13. un-div-inv 7.3%

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

   16. Applied egg-rr 7.3%

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

       1. *-lft-identity 7.3%

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

   18. Simplified 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 inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-define 99.7%

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

      5. rgt-mult-inverse 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 7.3%

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

   9. Taylor expanded in x around inf 7.3%

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

       1. *-commutative 7.3%

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

   11. Simplified 7.3%

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

Alternative 8: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

   3. Taylor expanded in x around 0 97.1%

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

   4. Taylor expanded in x around -inf 7.2%

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

   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. 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 inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-define 99.7%

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

      5. rgt-mult-inverse 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 7.3%

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

   9. Taylor expanded in x around inf 7.3%

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

       1. *-commutative 7.3%

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

   11. Simplified 7.3%

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

4. Final simplification 7.3%

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

Alternative 9: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

   3. Taylor expanded in x around 0 97.1%

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

   4. Taylor expanded in x around -inf 7.2%

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

   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. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 1.9%

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

      1. *-commutative 1.9%

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

   7. Simplified 1.9%

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

   8. Taylor expanded in x around inf 1.9%

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

      1. unpow-1 1.9%

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

      2. metadata-eval 1.9%

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

      3. pow-sqr 1.9%

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

      4. rem-sqrt-square 1.9%

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

      5. rem-square-sqrt 1.9%

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

      6. fabs-sqr 1.9%

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

      7. rem-square-sqrt 1.9%

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

   10. Simplified 1.9%

       \[\leadsto \color{blue}{-1.5 \cdot {x}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 7.2%

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

       3. *-commutative 7.2%

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

       4. *-commutative 7.2%

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

       5. swap-sqr 7.2%

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

       6. pow-prod-up 7.2%

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

       7. metadata-eval 7.2%

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

       8. inv-pow 7.2%

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

       9. metadata-eval 7.2%

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

   12. Applied egg-rr 7.2%

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

       1. associate-*l/ 7.2%

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

       2. metadata-eval 7.2%

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

   14. Simplified 7.2%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -1.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.25}{x}}\\ \end{array} \]
5. 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. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 50.5%

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

   1. *-commutative 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in x around inf 4.7%

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

   1. unpow-1 4.7%

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

   2. metadata-eval 4.7%

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

   3. pow-sqr 4.7%

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

   4. rem-sqrt-square 4.7%

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

   5. rem-square-sqrt 4.7%

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

   6. fabs-sqr 4.7%

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

   7. rem-square-sqrt 4.7%

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

10. Simplified 4.7%

    \[\leadsto \color{blue}{-1.5 \cdot {x}^{-0.5}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 0.0%

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

    2. sqrt-unprod 4.2%

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

    3. *-commutative 4.2%

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

    4. *-commutative 4.2%

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

    5. swap-sqr 4.2%

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

    6. pow-prod-up 4.2%

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

    7. metadata-eval 4.2%

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

    8. inv-pow 4.2%

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

    9. metadata-eval 4.2%

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

12. Applied egg-rr 4.2%

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

    1. associate-*l/ 4.2%

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

    2. metadata-eval 4.2%

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

14. Simplified 4.2%

    \[\leadsto \color{blue}{\sqrt{\frac{2.25}{x}}} \]
15. 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 2024138 
(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)))))