Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.0s

Alternatives: 10

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

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

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(x + Float64(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[(x + N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $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-*r/ 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. metadata-eval N/A

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

   9. cancel-sign-sub-inv N/A

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

   10. *-commutative N/A

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

   11. associate-+r- N/A

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

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

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

   13. *-commutative N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. metadata-eval N/A

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

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

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

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

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

   18. sqrt-lowering-sqrt.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \frac{x + -1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x + t\_0}{x + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0)
     (* 6.0 (/ (+ x -1.0) (+ 1.0 t_0)))
     (/ 6.0 (/ (+ x t_0) (+ x -1.0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	} else {
		tmp = 6.0 / ((x + t_0) / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 6.0d0 * ((x + (-1.0d0)) / (1.0d0 + t_0))
    else
        tmp = 6.0d0 / ((x + t_0) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	} else {
		tmp = 6.0 / ((x + t_0) / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0))
	else:
		tmp = 6.0 / ((x + t_0) / (x + -1.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(1.0 + t_0)));
	else
		tmp = Float64(6.0 / Float64(Float64(x + t_0) / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	else
		tmp = 6.0 / ((x + t_0) / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(x + t$95$0), $MachinePrecision] / N[(x + -1.0), $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-*r/ 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. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-commutative N/A

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

      11. associate-+r- N/A

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

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

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

      13. *-commutative N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.6%

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

   7. Simplified 97.6%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-*r/ 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. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-commutative N/A

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

      11. associate-+r- N/A

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

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

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

      13. *-commutative N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 98.9%

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

   7. Simplified 98.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. +-lowering-+.f64 98.9%

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

   9. Applied egg-rr 98.9%

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

4. Final simplification 98.3%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;6 \cdot \frac{x + -1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x + -1}{x + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0)
     (* 6.0 (/ (+ x -1.0) (+ 1.0 t_0)))
     (* 6.0 (/ (+ x -1.0) (+ x t_0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	} else {
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 6.0d0 * ((x + (-1.0d0)) / (1.0d0 + t_0))
    else
        tmp = 6.0d0 * ((x + (-1.0d0)) / (x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	} else {
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0))
	else:
		tmp = 6.0 * ((x + -1.0) / (x + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(1.0 + t_0)));
	else
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(x + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 6.0 * ((x + -1.0) / (1.0 + t_0));
	else
		tmp = 6.0 * ((x + -1.0) / (x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + t$95$0), $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-*r/ 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. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-commutative N/A

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

      11. associate-+r- N/A

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

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

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

      13. *-commutative N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.6%

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

   7. Simplified 97.6%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-*r/ 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. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-commutative N/A

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

      11. associate-+r- N/A

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

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

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

      13. *-commutative N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.3%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      9. cancel-sign-sub-inv N/A

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

      10. *-commutative N/A

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

      11. associate-+r- N/A

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

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

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

      13. *-commutative N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.0%

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

   7. Simplified 97.0%

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

   if 4 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 99.5%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 99.5%

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 98.3%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 97.6%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 97.6%

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

   7. Applied egg-rr 97.6%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 98.9%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 98.2%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 / N[Sqrt[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 97.6%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 98.9%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 98.2%

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

Alternative 7: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 97.5%

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

   5. Simplified 97.5%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 98.9%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 98.9%

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

   7. Applied egg-rr 98.9%

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

Alternative 8: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + (4.0 * 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 = (-6.0d0) / (1.0d0 + (4.0d0 * 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 = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(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 + (4.0 * sqrt(x)));
	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[(4.0 * N[Sqrt[x], $MachinePrecision]), $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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 97.5%

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

   5. Simplified 97.5%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
   7. 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 6.5%

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

   8. Simplified 6.5%

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

Alternative 9: 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

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 6.9%

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

   8. Simplified 6.9%

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

      1. associate-/r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. sqrt-lowering-sqrt.f64 6.9%

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

   10. Applied egg-rr 6.9%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

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

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

      3. distribute-rgt-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
   7. 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 6.5%

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

   8. Simplified 6.5%

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

Alternative 10: 4.4% 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

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

   1. metadata-eval N/A

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

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

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

   3. distribute-rgt-neg-in N/A

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

   4. neg-mul-1 N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-in N/A

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

   13. +-commutative N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

5. Simplified 52.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{3}{2} \cdot \sqrt{x}} \]
7. 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 4.3%

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

8. Simplified 4.3%

   \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
9. 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 2024158 
(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)))))