Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.7%

Time: 9.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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(x + Float64(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[(x + N[(1.0 + 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. metadata-eval 99.8%

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

   3. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 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(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(x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. metadata-eval 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 97.6%

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

   if 4 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. fma-define 98.9%

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

      4. sqrt-div 98.9%

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

      5. metadata-eval 98.9%

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

      6. un-div-inv 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-undefine 98.9%

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

      2. *-commutative 98.9%

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

      3. *-rgt-identity 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.2%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. metadata-eval 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 97.5%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. fma-define 98.9%

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

      4. sqrt-div 98.9%

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

      5. metadata-eval 98.9%

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

      6. un-div-inv 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-undefine 98.9%

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

      2. *-commutative 98.9%

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

      3. *-rgt-identity 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.2%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. fma-define 98.9%

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

      4. sqrt-div 98.9%

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

      5. metadata-eval 98.9%

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

      6. un-div-inv 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-undefine 98.9%

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

      2. *-commutative 98.9%

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

      3. *-rgt-identity 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.2%

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

Alternative 5: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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. Taylor expanded in x around inf 7.2%

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

      1. frac-2neg 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. *-commutative 7.2%

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

      5. distribute-rgt-neg-in 7.2%

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

      6. metadata-eval 7.2%

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

   8. Applied egg-rr 7.2%

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

      1. associate-*r/ 7.2%

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

      2. metadata-eval 7.2%

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

      3. *-commutative 7.2%

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

      4. associate-/r* 7.2%

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

      5. metadata-eval 7.2%

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

   10. Simplified 7.2%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.9%

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

   6. Taylor expanded in x around 0 6.8%

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

      1. *-commutative 6.8%

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

   8. Simplified 6.8%

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

      1. add-sqr-sqrt 6.8%

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

      2. sqrt-unprod 6.8%

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

      3. swap-sqr 6.8%

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

      4. add-sqr-sqrt 6.8%

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

      5. metadata-eval 6.8%

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

   10. Applied egg-rr 6.8%

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

Alternative 6: 52.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 50.4%

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

   1. +-commutative 50.4%

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

   2. flip-+ 50.4%

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

   3. *-commutative 50.4%

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

   4. *-commutative 50.4%

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

   5. swap-sqr 50.4%

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

   6. add-sqr-sqrt 50.4%

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

   7. metadata-eval 50.4%

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

   8. metadata-eval 50.4%

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

   9. add-sqr-sqrt 50.4%

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

   10. sqrt-unprod 50.4%

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

   11. *-commutative 50.4%

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

   12. *-commutative 50.4%

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

   13. swap-sqr 50.4%

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

   14. add-sqr-sqrt 50.4%

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

   15. metadata-eval 50.4%

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

7. Applied egg-rr 50.4%

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

8. Taylor expanded in x around 0 52.7%

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

9. Final simplification 52.7%

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

Alternative 7: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 49.6%

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

6. Taylor expanded in x around 0 4.3%

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

   1. *-commutative 4.3%

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

8. Simplified 4.3%

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

   1. add-sqr-sqrt 4.3%

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

   2. sqrt-unprod 4.3%

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

   3. swap-sqr 4.3%

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

   4. add-sqr-sqrt 4.3%

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

   5. metadata-eval 4.3%

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

10. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
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 2024185 
(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)))))