Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 9.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% 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

\[\begin{array}{l} \\ \frac{x + -1}{\left(x + 1\right) + \sqrt{x \cdot 16}} \cdot 6 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ x -1.0) (+ (+ x 1.0) (sqrt (* x 16.0)))) 6.0))

C

double code(double x) {
	return ((x + -1.0) / ((x + 1.0) + sqrt((x * 16.0)))) * 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + (-1.0d0)) / ((x + 1.0d0) + sqrt((x * 16.0d0)))) * 6.0d0
end function

Java

public static double code(double x) {
	return ((x + -1.0) / ((x + 1.0) + Math.sqrt((x * 16.0)))) * 6.0;
}

Python

def code(x):
	return ((x + -1.0) / ((x + 1.0) + math.sqrt((x * 16.0)))) * 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x + -1.0) / Float64(Float64(x + 1.0) + sqrt(Float64(x * 16.0)))) * 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + -1.0) / ((x + 1.0) + sqrt((x * 16.0)))) * 6.0;
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. metadata-eval 99.9%

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

   5. sqrt-prod 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.078:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.078)
   (* (- (* x -78.0) 6.0) (+ (+ x 1.0) (* -4.0 (sqrt x))))
   (/ 6.0 (+ 1.0 (/ 4.0 (sqrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.078) {
		tmp = ((x * -78.0) - 6.0) * ((x + 1.0) + (-4.0 * sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.078d0) then
        tmp = ((x * (-78.0d0)) - 6.0d0) * ((x + 1.0d0) + ((-4.0d0) * sqrt(x)))
    else
        tmp = 6.0d0 / (1.0d0 + (4.0d0 / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.078) {
		tmp = ((x * -78.0) - 6.0) * ((x + 1.0) + (-4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.078:
		tmp = ((x * -78.0) - 6.0) * ((x + 1.0) + (-4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.078)
		tmp = Float64(Float64(Float64(x * -78.0) - 6.0) * Float64(Float64(x + 1.0) + Float64(-4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 / Float64(1.0 + Float64(4.0 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.078)
		tmp = ((x * -78.0) - 6.0) * ((x + 1.0) + (-4.0 * sqrt(x)));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.078], N[(N[(N[(x * -78.0), $MachinePrecision] - 6.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0779999999999999999

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-/r/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. fma-undefine 99.9%

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

      8. pow2 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. swap-sqr 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

      14. cancel-sign-sub-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\left(-78 \cdot x - 6\right)} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   if 0.0779999999999999999 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. add-exp-log 98.6%

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

      2. log-div 98.6%

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

      3. log1p-define 98.6%

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

      4. sqrt-div 98.6%

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

      5. metadata-eval 98.6%

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

      6. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. exp-diff 98.6%

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

      2. rem-exp-log 98.6%

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

      3. log1p-undefine 98.6%

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

      4. rem-exp-log 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.078:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 + \frac{4}{\sqrt{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (6.0 * (x + -1.0)) / (1.0 + (math.sqrt(x) * 4.0))
	else:
		tmp = 6.0 / (1.0 + (4.0 / math.sqrt(x)))
	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(sqrt(x) * 4.0)));
	else
		tmp = Float64(6.0 / Float64(1.0 + Float64(4.0 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (6.0 * (x + -1.0)) / (1.0 + (sqrt(x) * 4.0));
	else
		tmp = 6.0 / (1.0 + (4.0 / sqrt(x)));
	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[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 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 96.1%

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

   if 4 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. add-exp-log 99.2%

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

      2. log-div 99.2%

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

      3. log1p-define 99.2%

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

      4. sqrt-div 99.2%

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

      5. metadata-eval 99.2%

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

      6. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. exp-diff 99.2%

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

      2. rem-exp-log 99.2%

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

      3. log1p-undefine 99.2%

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

      4. rem-exp-log 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in x around 0 96.8%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. add-exp-log 98.6%

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

      2. log-div 98.6%

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

      3. log1p-define 98.6%

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

      4. sqrt-div 98.6%

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

      5. metadata-eval 98.6%

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

      6. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. exp-diff 98.6%

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

      2. rem-exp-log 98.6%

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

      3. log1p-undefine 98.6%

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

      4. rem-exp-log 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 97.7%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.6%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. add-exp-log 98.6%

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

      2. log-div 98.6%

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

      3. log1p-define 98.6%

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

      4. sqrt-div 98.6%

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

      5. metadata-eval 98.6%

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

      6. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. exp-diff 98.6%

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

      2. rem-exp-log 98.6%

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

      3. log1p-undefine 98.6%

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

      4. rem-exp-log 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 97.7%

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

Alternative 6: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.6%

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

   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%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 6.8%

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

   4. Taylor expanded in x around -inf 1.3%

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

      1. *-commutative 1.3%

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

   6. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.8%

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

      3. swap-sqr 6.8%

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

      4. add-sqr-sqrt 6.8%

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

      5. metadata-eval 6.8%

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

   8. Applied egg-rr 6.8%

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

Alternative 7: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 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 96.8%

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

   4. Taylor expanded in x around -inf 6.9%

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

      1. *-commutative 6.9%

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

   6. Simplified 6.9%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 6.8%

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

   4. Taylor expanded in x around -inf 1.3%

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

      1. *-commutative 1.3%

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

   6. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.8%

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

      3. swap-sqr 6.8%

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

      4. add-sqr-sqrt 6.8%

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

      5. metadata-eval 6.8%

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

   8. Applied egg-rr 6.8%

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

Alternative 8: 52.4% 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 99.7%

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

   1. flip-+ 73.7%

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

   2. associate-/r/ 73.7%

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

   3. sub-neg 73.7%

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

   4. distribute-lft-in 73.7%

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

   5. metadata-eval 73.7%

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

   6. metadata-eval 73.7%

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

   7. fma-undefine 73.7%

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

   8. pow2 73.7%

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

   9. *-commutative 73.7%

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

   10. *-commutative 73.7%

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

   11. swap-sqr 73.7%

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

   12. add-sqr-sqrt 73.7%

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

   13. metadata-eval 73.7%

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

   14. cancel-sign-sub-inv 73.7%

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

4. Applied egg-rr 73.7%

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

5. Taylor expanded in x around 0 46.9%

   \[\leadsto \color{blue}{\left(-78 \cdot x - 6\right)} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

6. Taylor expanded in x around 0 49.0%

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

   1. distribute-rgt-in 49.1%

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

   2. metadata-eval 49.1%

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

   3. *-commutative 49.1%

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

   4. associate-*l* 49.1%

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

   5. metadata-eval 49.1%

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

8. Simplified 49.1%

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

Alternative 9: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 49.3%

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

4. Taylor expanded in x around -inf 3.9%

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

   1. *-commutative 3.9%

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

6. Simplified 3.9%

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

   1. add-sqr-sqrt 0.0%

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

   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 2024085 
(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)))))