Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.6% → 99.9%

Time: 10.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-define 99.8%

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. sub-neg 99.8%

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

   6. +-commutative 99.8%

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

   7. fma-undefine 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-/l* 99.9%

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

   11. *-commutative 99.9%

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

   12. sub-neg 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-commutative 99.9%

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

   15. associate-+r+ 99.9%

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

   16. fma-undefine 99.9%

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

   17. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   if 4 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.2%

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

4. Final simplification 98.4%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(1.0 + Float64(4.0 * sqrt(x)))));
	else
		tmp = Float64(6.0 / Float64(1.0 + Float64(4.0 * sqrt(Float64(1.0 / 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 + (4.0 * sqrt(x))));
	else
		tmp = 6.0 / (1.0 + (4.0 * sqrt((1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-define 99.9%

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+r+ 99.9%

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

      16. fma-undefine 99.9%

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

      17. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 97.6%

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

   if 4 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.2%

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

4. Final simplification 98.4%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. add-sqr-sqrt 99.9%

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

      2. difference-of-sqr-1 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. metadata-eval 0.0%

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

   7. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{6}{x} - \frac{\frac{6}{x} + 0}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   8. Taylor expanded in x around 0 97.6%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.2%

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

4. Final simplification 98.4%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 + \sqrt{\frac{1}{x}} \cdot 0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (* 4.0 (sqrt x)) (+ x 1.0)))
   (/ 1.0 (+ 0.16666666666666666 (* (sqrt (/ 1.0 x)) 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	} else {
		tmp = 1.0 / (0.16666666666666666 + (sqrt((1.0 / x)) * 0.6666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((4.0d0 * sqrt(x)) + (x + 1.0d0))
    else
        tmp = 1.0d0 / (0.16666666666666666d0 + (sqrt((1.0d0 / x)) * 0.6666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * Math.sqrt(x)) + (x + 1.0));
	} else {
		tmp = 1.0 / (0.16666666666666666 + (Math.sqrt((1.0 / x)) * 0.6666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((4.0 * math.sqrt(x)) + (x + 1.0))
	else:
		tmp = 1.0 / (0.16666666666666666 + (math.sqrt((1.0 / x)) * 0.6666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0)));
	else
		tmp = Float64(1.0 / Float64(0.16666666666666666 + Float64(sqrt(Float64(1.0 / x)) * 0.6666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	else
		tmp = 1.0 / (0.16666666666666666 + (sqrt((1.0 / x)) * 0.6666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.16666666666666666 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.6666666666666666), $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. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. difference-of-sqr-1 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. metadata-eval 0.0%

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

   7. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{6}{x} - \frac{\frac{6}{x} + 0}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   8. Taylor expanded in x around 0 97.6%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-define 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. unpow-1 99.6%

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

       2. sub-neg 99.6%

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

       3. distribute-neg-frac 99.6%

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

       4. metadata-eval 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in x around inf 99.2%

       \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot \left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}} \]
   14. Step-by-step derivation

       1. distribute-lft-in 99.2%

          \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot 1 + 0.16666666666666666 \cdot \left(4 \cdot \sqrt{\frac{1}{x}}\right)}} \]

       2. metadata-eval 99.2%

          \[\leadsto \frac{1}{\color{blue}{0.16666666666666666} + 0.16666666666666666 \cdot \left(4 \cdot \sqrt{\frac{1}{x}}\right)} \]

       3. associate-*r* 99.2%

          \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\left(0.16666666666666666 \cdot 4\right) \cdot \sqrt{\frac{1}{x}}}} \]

       4. metadata-eval 99.2%

          \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{0.6666666666666666} \cdot \sqrt{\frac{1}{x}}} \]

   15. Simplified 99.2%

       \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + 0.6666666666666666 \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 + \sqrt{\frac{1}{x}} \cdot 0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ (* 4.0 (sqrt x)) (+ x 1.0))) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((4.0d0 * sqrt(x)) + (x + 1.0d0))
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * Math.sqrt(x)) + (x + 1.0));
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((4.0 * math.sqrt(x)) + (x + 1.0))
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0)));
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

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

      1. add-sqr-sqrt 99.9%

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

      2. difference-of-sqr-1 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

      3. associate-*r/ 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 0.0%

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

      6. metadata-eval 0.0%

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

   7. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{6}{x} - \frac{\frac{6}{x} + 0}{x}\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   8. Taylor expanded in x around 0 97.6%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-define 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-+r+ 100.0%

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

      16. fma-undefine 100.0%

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

      17. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 7.0%

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

   8. Taylor expanded in x around -inf 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 7.0%

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

      3. *-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. swap-sqr 7.0%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + \left(x + 4 \cdot \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ 1.0 (+ x (* 4.0 (sqrt x))))) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (x + (4.0 * sqrt(x))));
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (x + (4.0d0 * sqrt(x))))
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (x + (4.0 * Math.sqrt(x))));
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (x + (4.0 * math.sqrt(x))))
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(x + Float64(4.0 * sqrt(x)))));
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (x + (4.0 * sqrt(x))));
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x * 2.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 97.6%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-define 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-+r+ 100.0%

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

      16. fma-undefine 100.0%

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

      17. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 7.0%

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

   8. Taylor expanded in x around -inf 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 7.0%

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

      3. *-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. swap-sqr 7.0%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-lft-in 99.8%

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

   3. metadata-eval 99.8%

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

   4. metadata-eval 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

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(Float64(6.0 * Float64(x + -1.0)) / Float64(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[(N[(6.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 10: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (sqrt (* x 2.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = sqrt((x * 2.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = sqrt((x * 2.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = Math.sqrt((x * 2.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = math.sqrt((x * 2.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = sqrt(Float64(x * 2.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = sqrt((x * 2.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x * 2.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-define 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-+r+ 100.0%

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

      16. fma-undefine 100.0%

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

      17. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 7.0%

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

   8. Taylor expanded in x around -inf 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 7.0%

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

      3. *-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. swap-sqr 7.0%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Power[x, -0.5], $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. Step-by-step derivation

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in x around inf 7.3%

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

      1. *-commutative 7.3%

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

   10. Simplified 7.3%

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

       1. *-un-lft-identity 7.3%

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

       2. inv-pow 7.3%

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

       3. sqrt-pow1 7.3%

          \[\leadsto \left(1 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot -1.5 \]

       4. metadata-eval 7.3%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-0.5}}\right) \cdot -1.5 \]

   12. Applied egg-rr 7.3%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot -1.5 \]
   13. Step-by-step derivation

       1. *-lft-identity 7.3%

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

   14. Simplified 7.3%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-define 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-+r+ 100.0%

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

      16. fma-undefine 100.0%

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

      17. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 7.0%

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

   8. Taylor expanded in x around -inf 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 7.0%

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

      3. *-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. swap-sqr 7.0%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

Alternative 12: 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. Step-by-step derivation

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-define 99.9%

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+r+ 99.9%

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

      16. fma-undefine 99.9%

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

      17. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 97.6%

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

   8. Taylor expanded in x around -inf 7.3%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-define 99.6%

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

      2. metadata-eval 99.6%

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

      3. metadata-eval 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-undefine 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-/l* 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-+r+ 100.0%

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

      16. fma-undefine 100.0%

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

      17. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 7.0%

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

   8. Taylor expanded in x around -inf 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 7.0%

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

      3. *-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. swap-sqr 7.0%

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

      6. add-sqr-sqrt 7.0%

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

      7. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

4. Final simplification 7.1%

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

Alternative 13: 4.5% 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. Step-by-step derivation

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-define 99.8%

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. sub-neg 99.8%

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

   6. +-commutative 99.8%

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

   7. fma-undefine 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-/l* 99.9%

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

   11. *-commutative 99.9%

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

   12. sub-neg 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-commutative 99.9%

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

   15. associate-+r+ 99.9%

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

   16. fma-undefine 99.9%

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

   17. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 52.3%

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

8. Taylor expanded in x around -inf 4.3%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 4.4%

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

   3. *-commutative 4.4%

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

   4. *-commutative 4.4%

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

   5. swap-sqr 4.4%

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

   6. add-sqr-sqrt 4.4%

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

   7. metadata-eval 4.4%

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

10. Applied egg-rr 4.4%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

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

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