Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

   11. sqrt-lowering-sqrt.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

4. Applied egg-rr 99.9%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[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. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(1 + \left(\mathsf{neg}\left(-4\right)\right) \cdot \sqrt{\color{blue}{x}}\right)\right) \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(1 + \left(\mathsf{neg}\left(-4 \cdot \sqrt{x}\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \left(1 - \color{blue}{-4 \cdot \sqrt{x}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(-4 \cdot \sqrt{x}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \left(\sqrt{x} \cdot \color{blue}{-4}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-4}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(6, \mathsf{\_.f64}\left(x, 1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   6. Step-by-step derivation

   7. Simplified 96.6%

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

4. Final simplification 97.6%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	} else {
		tmp = 6.0 * ((x + -1.0) / (x + (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 = (x + (-1.0d0)) / (0.16666666666666666d0 + (sqrt(x) * 0.6666666666666666d0))
    else
        tmp = 6.0d0 * ((x + (-1.0d0)) / (x + (4.0d0 * sqrt(x))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 100.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + -1\right), \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{6}\right)}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{6}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{6}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + 4 \cdot \sqrt{x}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right)} \cdot \frac{1}{6}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{6} \cdot \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{6} \cdot 4\right) \cdot \color{blue}{\sqrt{x}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\sqrt{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot 4\right)\right)\right)\right) \]

      9. metadata-eval 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{2}{3}\right)\right)\right) \]

   9. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   6. Step-by-step derivation

   7. Simplified 96.6%

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

4. Final simplification 97.6%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	} else {
		tmp = (x + -1.0) * (6.0 / (x + (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 = (x + (-1.0d0)) / (0.16666666666666666d0 + (sqrt(x) * 0.6666666666666666d0))
    else
        tmp = (x + (-1.0d0)) * (6.0d0 / (x + (4.0d0 * sqrt(x))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(x + N[(4.0 * N[Sqrt[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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 100.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + -1\right), \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{6}\right)}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{6}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{6}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + 4 \cdot \sqrt{x}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right)} \cdot \frac{1}{6}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{6} \cdot \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{6} \cdot 4\right) \cdot \color{blue}{\sqrt{x}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\sqrt{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot 4\right)\right)\right)\right) \]

      9. metadata-eval 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{2}{3}\right)\right)\right) \]

   9. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]
   6. Step-by-step derivation

   7. Simplified 96.6%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{6}{x + 4 \cdot \sqrt{x}}\right), \color{blue}{\left(x - 1\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \left(x + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + -1\right)\right) \]

      15. +-lowering-+.f64 96.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{-1}\right)\right) \]

   9. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	} 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 = (x + (-1.0d0)) / (0.16666666666666666d0 + (sqrt(x) * 0.6666666666666666d0))
    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 = (x + -1.0) / (0.16666666666666666 + (Math.sqrt(x) * 0.6666666666666666));
	} else {
		tmp = 6.0 / (1.0 + (4.0 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x + -1.0) / (0.16666666666666666 + (math.sqrt(x) * 0.6666666666666666))
	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(x + -1.0) / Float64(0.16666666666666666 + Float64(sqrt(x) * 0.6666666666666666)));
	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 = (x + -1.0) / (0.16666666666666666 + (sqrt(x) * 0.6666666666666666));
	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[(x + -1.0), $MachinePrecision] / N[(0.16666666666666666 + N[(N[Sqrt[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{6}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), 6\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right) \]

      11. sqrt-lowering-sqrt.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right) \]

   4. Applied egg-rr 100.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + -1\right), \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{6}\right)}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}}{6}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{6}\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 + 4 \cdot \sqrt{x}\right)\right), 6\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \left(4 \cdot \sqrt{x}\right)\right)\right), 6\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), 6\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), 6\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + 4 \cdot \sqrt{x}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(\frac{1}{6} + \color{blue}{\left(4 \cdot \sqrt{x}\right)} \cdot \frac{1}{6}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(4 \cdot \sqrt{x}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{6} \cdot \color{blue}{\left(4 \cdot \sqrt{x}\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{6} \cdot 4\right) \cdot \color{blue}{\sqrt{x}}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\sqrt{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{1}{6} \cdot 4\right)}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot 4\right)\right)\right)\right) \]

      9. metadata-eval 98.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{2}{3}\right)\right)\right) \]

   9. Simplified 98.8%

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

   if 4 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(x - 1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(4 \cdot \sqrt{\frac{1}{x}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 96.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \sqrt{\frac{1}{x}}\right), \color{blue}{1}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{1}{\sqrt{x}}\right), 1\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\frac{4}{\sqrt{x}}\right), 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \left(\sqrt{x}\right)\right), 1\right)\right) \]

      7. sqrt-lowering-sqrt.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), 1\right)\right) \]

   9. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

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

Alternative 6: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{-6}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(x - 1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(4 \cdot \sqrt{\frac{1}{x}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 96.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \sqrt{\frac{1}{x}}\right), \color{blue}{1}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{1}{\sqrt{x}}\right), 1\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\frac{4}{\sqrt{x}}\right), 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \left(\sqrt{x}\right)\right), 1\right)\right) \]

      7. sqrt-lowering-sqrt.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), 1\right)\right) \]

   9. Applied egg-rr 96.5%

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

4. Final simplification 97.6%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. clear-num N/A

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

   2. associate-/r* N/A

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

   3. associate-/r/ N/A

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

   4. clear-num N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right), \color{blue}{\left(x - 1\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right)\right), \left(x - 1\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right)\right), \left(x - 1\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x - 1\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\sqrt{x}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + -4 \cdot \sqrt{\color{blue}{x}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \color{blue}{\left(-4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \color{blue}{-4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-4}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 98.8%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right), \color{blue}{\left(x - 1\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x + 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(\color{blue}{x} - 1\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(4 \cdot \sqrt{x}\right)\right), \left(x - 1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \left(\sqrt{x}\right)\right)\right), \left(x - 1\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x - 1\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval 99.9%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(x, -1\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \color{blue}{\left(4 \cdot \sqrt{\frac{1}{x}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   7. Simplified 96.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \sqrt{\frac{1}{x}}\right), \color{blue}{1}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{\sqrt{1}}{\sqrt{x}}\right), 1\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(4 \cdot \frac{1}{\sqrt{x}}\right), 1\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\left(\frac{4}{\sqrt{x}}\right), 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \left(\sqrt{x}\right)\right), 1\right)\right) \]

      7. sqrt-lowering-sqrt.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right), 1\right)\right) \]

   9. Applied egg-rr 96.5%

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

4. Final simplification 97.5%

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

Alternative 9: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 6.0d0 / ((-1.0d0) + (sqrt(x) * (-4.0d0)))
    else
        tmp = sqrt(x) * 1.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(6.0 / N[(-1.0 + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\sqrt{x}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + -4 \cdot \sqrt{\color{blue}{x}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \color{blue}{\left(-4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \color{blue}{-4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-4}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 98.8%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 98.8%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(6 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

      2. *-lowering-*.f64 96.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   5. Simplified 96.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 96.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

       3. sqrt-lowering-sqrt.f64 7.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

   11. Simplified 7.4%

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

Alternative 10: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-neg-frac N/A

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

      3. distribute-neg-frac2 N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\sqrt{x}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(6, \left(-1 + -4 \cdot \sqrt{\color{blue}{x}}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \color{blue}{\left(-4 \cdot \sqrt{x}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \left(\sqrt{x} \cdot \color{blue}{-4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-4}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 98.8%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -4\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-3}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\frac{-3}{2}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \frac{-3}{2}\right) \]

      4. /-lowering-/.f64 7.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \frac{-3}{2}\right) \]

   8. Simplified 7.0%

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

      1. *-commutative N/A

         \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      2. sqrt-div N/A

         \[\leadsto \frac{-3}{2} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} \cdot \frac{1}{\sqrt{\color{blue}{x}}} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\frac{-3}{2}}{\color{blue}{\sqrt{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 7.0%

         \[\leadsto \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{sqrt.f64}\left(x\right)\right) \]

   10. Applied egg-rr 7.0%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(6 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

      2. *-lowering-*.f64 96.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   5. Simplified 96.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 96.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

       3. sqrt-lowering-sqrt.f64 7.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

   11. Simplified 7.4%

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

Alternative 11: 4.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt x) 1.5))

C

double code(double x) {
	return sqrt(x) * 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(x) * 1.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt(x) * 1.5;
}

Python

def code(x):
	return math.sqrt(x) * 1.5

Julia

function code(x)
	return Float64(sqrt(x) * 1.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(x) * 1.5;
end

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(6 \cdot x\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   2. *-lowering-*.f64 51.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, 1\right)}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

5. Simplified 51.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 6\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(4, \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 51.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{3}{2}} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\frac{3}{2}}\right) \]

    3. sqrt-lowering-sqrt.f64 4.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{3}{2}\right) \]

11. Simplified 4.8%

    \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
12. 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 2024192 
(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)))))