Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 8.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{1 - \frac{16}{x}}{1 - \frac{4}{\sqrt{x}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (* (+ x -1.0) (/ 6.0 (+ 1.0 (* 4.0 (sqrt x)))))
   (/ 6.0 (/ (- 1.0 (/ 16.0 x)) (- 1.0 (/ 4.0 (sqrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	} else {
		tmp = 6.0 / ((1.0 - (16.0 / x)) / (1.0 - (4.0 / sqrt(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0 / ((1.0d0 - (16.0d0 / x)) / (1.0d0 - (4.0d0 / sqrt(x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 / ((1.0 - (16.0 / x)) / (1.0 - (4.0 / Math.sqrt(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * math.sqrt(x))))
	else:
		tmp = 6.0 / ((1.0 - (16.0 / x)) / (1.0 - (4.0 / math.sqrt(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 / Float64(Float64(1.0 - Float64(16.0 / x)) / Float64(1.0 - Float64(4.0 / sqrt(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 / ((1.0 - (16.0 / x)) / (1.0 - (4.0 / sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(1.0 - N[(16.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 96.5%

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

      1. flip-+ 96.5%

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

      2. metadata-eval 96.5%

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

      3. div-sub 96.5%

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

      4. sqrt-div 96.5%

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

      5. metadata-eval 96.5%

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

      6. un-div-inv 96.5%

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

      7. *-commutative 96.5%

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

      8. *-commutative 96.5%

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

      9. swap-sqr 96.5%

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

      10. add-sqr-sqrt 96.5%

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

      11. metadata-eval 96.5%

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

      12. sqrt-div 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. div-sub 96.5%

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

      2. associate-*l/ 96.5%

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

      3. metadata-eval 96.5%

         \[\leadsto \frac{6}{\frac{1 - \frac{\color{blue}{16}}{x}}{1 - \frac{4}{\sqrt{x}}}} \]

   7. Simplified 96.5%

      \[\leadsto \frac{6}{\color{blue}{\frac{1 - \frac{16}{x}}{1 - \frac{4}{\sqrt{x}}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{-1 + \left(\frac{4}{\sqrt{x}} + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (* (+ x -1.0) (/ 6.0 (+ 1.0 (* 4.0 (sqrt x)))))
   (/ 6.0 (+ -1.0 (+ (/ 4.0 (sqrt x)) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	} else {
		tmp = 6.0 / (-1.0 + ((4.0 / sqrt(x)) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + (4.0d0 * sqrt(x))))
    else
        tmp = 6.0d0 / ((-1.0d0) + ((4.0d0 / sqrt(x)) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * Math.sqrt(x))));
	} else {
		tmp = 6.0 / (-1.0 + ((4.0 / Math.sqrt(x)) + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * math.sqrt(x))))
	else:
		tmp = 6.0 / (-1.0 + ((4.0 / math.sqrt(x)) + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 / Float64(-1.0 + Float64(Float64(4.0 / sqrt(x)) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x + -1.0) * (6.0 / (1.0 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 / (-1.0 + ((4.0 / sqrt(x)) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(-1.0 + N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 96.5%

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

      1. +-commutative 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. fma-define 96.5%

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

      4. sqrt-div 96.5%

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

      5. metadata-eval 96.5%

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

      6. un-div-inv 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. fma-undefine 96.5%

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

      2. *-lft-identity 96.5%

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

   7. Simplified 96.5%

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

      1. add-sqr-sqrt 96.5%

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

      2. sqrt-unprod 96.5%

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

      3. frac-times 96.5%

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

      4. metadata-eval 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{\color{blue}{16}}{\sqrt{x} \cdot \sqrt{x}}} + 1} \]

      5. add-sqr-sqrt 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{16}{\color{blue}{x}}} + 1} \]

   9. Applied egg-rr 96.5%

      \[\leadsto \frac{6}{\color{blue}{\sqrt{\frac{16}{x}}} + 1} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 96.5%

          \[\leadsto \frac{6}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{16}{x}} + 1\right)\right)}} \]

       2. expm1-undefine 96.5%

          \[\leadsto \frac{6}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{16}{x}} + 1\right)} - 1}} \]

       3. +-commutative 96.5%

          \[\leadsto \frac{6}{e^{\mathsf{log1p}\left(\color{blue}{1 + \sqrt{\frac{16}{x}}}\right)} - 1} \]

       4. sqrt-div 96.5%

          \[\leadsto \frac{6}{e^{\mathsf{log1p}\left(1 + \color{blue}{\frac{\sqrt{16}}{\sqrt{x}}}\right)} - 1} \]

       5. metadata-eval 96.5%

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

   11. Applied egg-rr 96.5%

       \[\leadsto \frac{6}{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{4}{\sqrt{x}}\right)} - 1}} \]
   12. Step-by-step derivation

       1. sub-neg 96.5%

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

       2. log1p-undefine 96.5%

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

       3. rem-exp-log 96.5%

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

       4. associate-+r+ 96.5%

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

       5. metadata-eval 96.5%

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

       6. metadata-eval 96.5%

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

   13. Simplified 96.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{-1 + \left(\frac{4}{\sqrt{x}} + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{-1 + \left(\frac{4}{\sqrt{x}} + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (+ x 1.0) (* 4.0 (sqrt x))))
   (/ 6.0 (+ -1.0 (+ (/ 4.0 (sqrt x)) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((x + 1.0) + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 / (-1.0 + ((4.0 / sqrt(x)) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 / ((-1.0d0) + ((4.0d0 / sqrt(x)) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((x + 1.0) + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 / (-1.0 + ((4.0 / Math.sqrt(x)) + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((x + 1.0) + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 / (-1.0 + ((4.0 / math.sqrt(x)) + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 / Float64(-1.0 + Float64(Float64(4.0 / sqrt(x)) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((x + 1.0) + (4.0 * sqrt(x)));
	else
		tmp = 6.0 / (-1.0 + ((4.0 / sqrt(x)) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(-1.0 + N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   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 96.5%

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

      1. +-commutative 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. fma-define 96.5%

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

      4. sqrt-div 96.5%

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

      5. metadata-eval 96.5%

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

      6. un-div-inv 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. fma-undefine 96.5%

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

      2. *-lft-identity 96.5%

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

   7. Simplified 96.5%

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

      1. add-sqr-sqrt 96.5%

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

      2. sqrt-unprod 96.5%

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

      3. frac-times 96.5%

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

      4. metadata-eval 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{\color{blue}{16}}{\sqrt{x} \cdot \sqrt{x}}} + 1} \]

      5. add-sqr-sqrt 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{16}{\color{blue}{x}}} + 1} \]

   9. Applied egg-rr 96.5%

      \[\leadsto \frac{6}{\color{blue}{\sqrt{\frac{16}{x}}} + 1} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 96.5%

          \[\leadsto \frac{6}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{16}{x}} + 1\right)\right)}} \]

       2. expm1-undefine 96.5%

          \[\leadsto \frac{6}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{16}{x}} + 1\right)} - 1}} \]

       3. +-commutative 96.5%

          \[\leadsto \frac{6}{e^{\mathsf{log1p}\left(\color{blue}{1 + \sqrt{\frac{16}{x}}}\right)} - 1} \]

       4. sqrt-div 96.5%

          \[\leadsto \frac{6}{e^{\mathsf{log1p}\left(1 + \color{blue}{\frac{\sqrt{16}}{\sqrt{x}}}\right)} - 1} \]

       5. metadata-eval 96.5%

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

   11. Applied egg-rr 96.5%

       \[\leadsto \frac{6}{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{4}{\sqrt{x}}\right)} - 1}} \]
   12. Step-by-step derivation

       1. sub-neg 96.5%

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

       2. log1p-undefine 96.5%

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

       3. rem-exp-log 96.5%

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

       4. associate-+r+ 96.5%

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

       5. metadata-eval 96.5%

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

       6. metadata-eval 96.5%

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

   13. Simplified 96.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{-1 + \left(\frac{4}{\sqrt{x}} + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (+ x 1.0) (* 4.0 (sqrt x))))
   (/ 6.0 (+ 1.0 (sqrt (/ 16.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((x + 1.0) + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + sqrt((16.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 / (1.0d0 + sqrt((16.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((x + 1.0) + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + Math.sqrt((16.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((x + 1.0) + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 / (1.0 + math.sqrt((16.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 / Float64(1.0 + sqrt(Float64(16.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((x + 1.0) + (4.0 * sqrt(x)));
	else
		tmp = 6.0 / (1.0 + sqrt((16.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[Sqrt[N[(16.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   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 96.5%

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

      1. +-commutative 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. fma-define 96.5%

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

      4. sqrt-div 96.5%

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

      5. metadata-eval 96.5%

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

      6. un-div-inv 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. fma-undefine 96.5%

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

      2. *-lft-identity 96.5%

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

   7. Simplified 96.5%

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

      1. add-sqr-sqrt 96.5%

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

      2. sqrt-unprod 96.5%

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

      3. frac-times 96.5%

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

      4. metadata-eval 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{\color{blue}{16}}{\sqrt{x} \cdot \sqrt{x}}} + 1} \]

      5. add-sqr-sqrt 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{16}{\color{blue}{x}}} + 1} \]

   9. Applied egg-rr 96.5%

      \[\leadsto \frac{6}{\color{blue}{\sqrt{\frac{16}{x}}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ 1.0 (* 4.0 (sqrt x))))
   (/ 6.0 (+ 1.0 (sqrt (/ 16.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + sqrt((16.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 / (1.0d0 + sqrt((16.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + Math.sqrt((16.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 / (1.0 + math.sqrt((16.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 / Float64(1.0 + sqrt(Float64(16.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 / (1.0 + sqrt((16.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[Sqrt[N[(16.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   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 96.5%

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

      1. +-commutative 96.5%

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

      2. *-un-lft-identity 96.5%

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

      3. fma-define 96.5%

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

      4. sqrt-div 96.5%

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

      5. metadata-eval 96.5%

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

      6. un-div-inv 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. fma-undefine 96.5%

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

      2. *-lft-identity 96.5%

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

   7. Simplified 96.5%

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

      1. add-sqr-sqrt 96.5%

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

      2. sqrt-unprod 96.5%

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

      3. frac-times 96.5%

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

      4. metadata-eval 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{\color{blue}{16}}{\sqrt{x} \cdot \sqrt{x}}} + 1} \]

      5. add-sqr-sqrt 96.5%

         \[\leadsto \frac{6}{\sqrt{\frac{16}{\color{blue}{x}}} + 1} \]

   9. Applied egg-rr 96.5%

      \[\leadsto \frac{6}{\color{blue}{\sqrt{\frac{16}{x}}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \sqrt{\frac{16}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 + -24 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 14.0)
   (/ -6.0 (+ 1.0 (* 4.0 (sqrt x))))
   (+ 6.0 (* -24.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 14.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 14.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 + ((-24.0d0) * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 14.0) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 14.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 + (-24.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 14.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 + Float64(-24.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 14.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 + (-24.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 14.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.8%

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

   if 14 < 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 99.7%

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

   5. Applied egg-rr 47.9%

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

   6. Taylor expanded in x around inf 96.8%

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

      1. distribute-lft-in 96.8%

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

      2. metadata-eval 96.8%

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

      3. associate-*r* 96.8%

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

      4. metadata-eval 96.8%

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

   8. Simplified 96.8%

      \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + -24 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17.0)
   (+ -6.0 (* (sqrt x) 24.0))
   (+ 6.0 (* -24.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 17.0) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 17.0d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0 + ((-24.0d0) * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 17.0) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0 + (-24.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 17.0:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0 + (-24.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 17.0)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = Float64(6.0 + Float64(-24.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 17.0)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0 + (-24.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 17.0], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. distribute-lft-in 97.7%

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

      2. metadata-eval 97.7%

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

      3. associate-*r* 97.7%

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

      4. metadata-eval 97.7%

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

   8. Simplified 97.7%

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

   if 17 < 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 99.7%

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

   5. Applied egg-rr 47.9%

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

   6. Taylor expanded in x around inf 96.8%

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

      1. distribute-lft-in 96.8%

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

      2. metadata-eval 96.8%

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

      3. associate-*r* 96.8%

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

      4. metadata-eval 96.8%

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

   8. Simplified 96.8%

      \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6 + -24 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   4. Taylor expanded in x around inf 7.0%

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

      1. *-commutative 7.0%

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

   6. Simplified 7.0%

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

      1. *-commutative 7.0%

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

      2. sqrt-div 7.0%

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

      3. metadata-eval 7.0%

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

      4. un-div-inv 7.0%

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

   8. Applied egg-rr 7.0%

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

   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 inf 96.5%

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

   4. Taylor expanded in x around 0 7.1%

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

      1. *-commutative 7.1%

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

   6. Simplified 7.1%

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

      1. add-sqr-sqrt 7.1%

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

      2. sqrt-unprod 7.1%

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

      3. swap-sqr 7.1%

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

      4. add-sqr-sqrt 7.1%

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

      5. metadata-eval 7.1%

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

   8. Applied egg-rr 7.1%

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

Alternative 11: 52.2% 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 99.8%

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

5. Applied egg-rr 73.7%

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

6. Taylor expanded in x around 0 52.0%

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

   1. distribute-lft-in 52.0%

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

   2. metadata-eval 52.0%

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

   3. associate-*r* 52.0%

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

   4. metadata-eval 52.0%

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

8. Simplified 52.0%

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

9. Final simplification 52.0%

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

Alternative 12: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 49.9%

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

4. Taylor expanded in x around 0 4.5%

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

   1. *-commutative 4.5%

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

6. Simplified 4.5%

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

   1. add-sqr-sqrt 4.5%

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

   2. sqrt-unprod 4.5%

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

   3. swap-sqr 4.5%

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

   4. add-sqr-sqrt 4.5%

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

   5. metadata-eval 4.5%

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

8. Applied egg-rr 4.5%

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

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024101 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

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

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