Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.1s

Alternatives: 12

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. associate-+r+ 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-undefine 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 + (x + (4.0 * sqrt(x)));
	double tmp;
	if (x <= 1.0) {
		tmp = 6.0 * (-1.0 / t_0);
	} else {
		tmp = 6.0 * (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 = 1.0d0 + (x + (4.0d0 * sqrt(x)))
    if (x <= 1.0d0) then
        tmp = 6.0d0 * ((-1.0d0) / t_0)
    else
        tmp = 6.0d0 * (x / t_0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

   8. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. associate-+r+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-undefine 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

   8. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.3%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

   8. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

   6. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.3%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (x + (1.0 + t_0));
	} else {
		tmp = 6.0 / ((x + t_0) / x);
	}
	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 + t_0))
    else
        tmp = 6.0d0 / ((x + t_0) / x)
    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 + t_0));
	} else {
		tmp = 6.0 / ((x + t_0) / x);
	}
	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 + t_0))
	else:
		tmp = 6.0 / ((x + t_0) / x)
	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(x + Float64(1.0 + t_0)));
	else
		tmp = Float64(6.0 / Float64(Float64(x + t_0) / x));
	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 + t_0));
	else
		tmp = 6.0 / ((x + t_0) / x);
	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[(x + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(x + t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

   6. Taylor expanded in x around 0 98.8%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(x + 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 < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. fma-define 98.8%

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

      4. sqrt-div 98.8%

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

      5. metadata-eval 98.8%

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

      6. un-div-inv 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. fma-undefine 98.8%

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

      2. *-lft-identity 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 98.3%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (* 6.0 (/ -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 <= 1.0) {
		tmp = 6.0 * (-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 <= 1.0d0) then
        tmp = 6.0d0 * ((-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 <= 1.0) {
		tmp = 6.0 * (-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 <= 1.0:
		tmp = 6.0 * (-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 <= 1.0)
		tmp = Float64(6.0 * Float64(-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 <= 1.0)
		tmp = 6.0 * (-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, 1.0], N[(6.0 * N[(-1.0 / 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 < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. fma-define 98.8%

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

      4. sqrt-div 98.8%

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

      5. metadata-eval 98.8%

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

      6. un-div-inv 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. fma-undefine 98.8%

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

      2. *-lft-identity 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 98.3%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. fma-define 98.8%

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

      4. sqrt-div 98.8%

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

      5. metadata-eval 98.8%

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

      6. un-div-inv 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. fma-undefine 98.8%

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

      2. *-lft-identity 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 98.3%

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

Alternative 8: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14.2:\\ \;\;\;\;\frac{-6}{1 + 4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-24}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 14.2) (/ -6.0 (+ 1.0 (* 4.0 (sqrt x)))) (+ 6.0 (/ -24.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 14.2) {
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 14.2d0) then
        tmp = (-6.0d0) / (1.0d0 + (4.0d0 * sqrt(x)))
    else
        tmp = 6.0d0 + ((-24.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 14.2) {
		tmp = -6.0 / (1.0 + (4.0 * Math.sqrt(x)));
	} else {
		tmp = 6.0 + (-24.0 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 14.2:
		tmp = -6.0 / (1.0 + (4.0 * math.sqrt(x)))
	else:
		tmp = 6.0 + (-24.0 / math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 14.2)
		tmp = Float64(-6.0 / Float64(1.0 + Float64(4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 + Float64(-24.0 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 14.2)
		tmp = -6.0 / (1.0 + (4.0 * sqrt(x)));
	else
		tmp = 6.0 + (-24.0 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 14.2], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 + N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 14.199999999999999

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   if 14.199999999999999 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. flip-+ 98.8%

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

      2. associate-/r/ 98.8%

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

      3. metadata-eval 98.8%

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

      4. *-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. swap-sqr 98.8%

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

      7. add-sqr-sqrt 98.8%

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

      8. metadata-eval 98.8%

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

      9. sqrt-div 98.8%

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

      10. metadata-eval 98.8%

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

      11. un-div-inv 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{6}{1 - \frac{1}{x} \cdot 16} \cdot \left(1 - \frac{4}{\sqrt{x}}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. metadata-eval 98.8%

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

      7. associate-*l/ 98.8%

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

      8. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

   10. Taylor expanded in x around inf 98.7%

       \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 98.7%

          \[\leadsto 6 + \color{blue}{\sqrt{\frac{1}{x}} \cdot -24} \]

   12. Simplified 98.7%

       \[\leadsto \color{blue}{6 + \sqrt{\frac{1}{x}} \cdot -24} \]
   13. Step-by-step derivation

       1. *-commutative 98.7%

          \[\leadsto 6 + \color{blue}{-24 \cdot \sqrt{\frac{1}{x}}} \]

       2. sqrt-div 98.7%

          \[\leadsto 6 + -24 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

       3. metadata-eval 98.7%

          \[\leadsto 6 + -24 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

       4. un-div-inv 98.7%

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

   14. Applied egg-rr 98.7%

       \[\leadsto 6 + \color{blue}{\frac{-24}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.8%

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

   6. Taylor expanded in x around inf 7.3%

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

      1. *-commutative 7.3%

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

   8. Simplified 7.3%

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

      1. *-commutative 7.3%

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

      2. sqrt-div 7.3%

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

      3. metadata-eval 7.3%

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

      4. un-div-inv 7.3%

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

   10. Applied egg-rr 7.3%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

   6. Taylor expanded in x around 0 6.9%

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

      1. *-commutative 6.9%

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

   8. Simplified 6.9%

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

      1. pow1 6.9%

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

      2. add-sqr-sqrt 6.9%

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

      3. sqrt-unprod 6.9%

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

      4. swap-sqr 6.9%

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

      5. add-sqr-sqrt 6.9%

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

      6. metadata-eval 6.9%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   10. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 6.9%

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

   12. Simplified 6.9%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 1.8%

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

   6. Taylor expanded in x around 0 1.8%

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

      1. unpow-1 1.8%

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

      2. metadata-eval 1.8%

         \[\leadsto \frac{6}{4 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}}} \]

      3. pow-sqr 1.8%

         \[\leadsto \frac{6}{4 \cdot \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}}} \]

      4. rem-sqrt-square 1.8%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{\left|{x}^{-0.5}\right|}} \]

      5. rem-square-sqrt 1.8%

         \[\leadsto \frac{6}{4 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right|} \]

      6. fabs-sqr 1.8%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)}} \]

      7. rem-square-sqrt 1.8%

         \[\leadsto \frac{6}{4 \cdot \color{blue}{{x}^{-0.5}}} \]

   8. Simplified 1.8%

      \[\leadsto \frac{6}{\color{blue}{4 \cdot {x}^{-0.5}}} \]

   9. Taylor expanded in x around -inf 7.2%

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

       1. *-commutative 7.2%

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

   11. Simplified 7.2%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-/l/ 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. fma-define 99.7%

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

      9. metadata-eval 99.7%

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

      10. *-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

      13. +-commutative 99.7%

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

      14. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.8%

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

   6. Taylor expanded in x around 0 6.9%

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

      1. *-commutative 6.9%

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

   8. Simplified 6.9%

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

      1. pow1 6.9%

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

      2. add-sqr-sqrt 6.9%

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

      3. sqrt-unprod 6.9%

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

      4. swap-sqr 6.9%

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

      5. add-sqr-sqrt 6.9%

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

      6. metadata-eval 6.9%

         \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

   10. Applied egg-rr 6.9%

       \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 6.9%

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

   12. Simplified 6.9%

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

Alternative 11: 52.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 6 + \frac{-24}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 6.0 (/ -24.0 (sqrt x))))

C

double code(double x) {
	return 6.0 + (-24.0 / sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 + ((-24.0d0) / sqrt(x))
end function

Java

public static double code(double x) {
	return 6.0 + (-24.0 / Math.sqrt(x));
}

Python

def code(x):
	return 6.0 + (-24.0 / math.sqrt(x))

Julia

function code(x)
	return Float64(6.0 + Float64(-24.0 / sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 + (-24.0 / sqrt(x));
end

Wolfram

code[x_] := N[(6.0 + N[(-24.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 48.4%

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

   1. flip-+ 48.4%

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

   2. associate-/r/ 48.4%

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

   3. metadata-eval 48.4%

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

   4. *-commutative 48.4%

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

   5. *-commutative 48.4%

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

   6. swap-sqr 48.4%

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

   7. add-sqr-sqrt 48.4%

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

   8. metadata-eval 48.4%

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

   9. sqrt-div 48.4%

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

   10. metadata-eval 48.4%

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

   11. un-div-inv 48.4%

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

7. Applied egg-rr 48.4%

   \[\leadsto \color{blue}{\frac{6}{1 - \frac{1}{x} \cdot 16} \cdot \left(1 - \frac{4}{\sqrt{x}}\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 48.4%

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

   2. sub-neg 48.4%

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

   3. distribute-rgt-in 48.4%

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

   4. metadata-eval 48.4%

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

   5. distribute-neg-frac 48.4%

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

   6. metadata-eval 48.4%

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

   7. associate-*l/ 48.4%

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

   8. metadata-eval 48.4%

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

9. Simplified 48.4%

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

10. Taylor expanded in x around inf 51.1%

    \[\leadsto \color{blue}{6 + -24 \cdot \sqrt{\frac{1}{x}}} \]
11. Step-by-step derivation

    1. *-commutative 51.1%

       \[\leadsto 6 + \color{blue}{\sqrt{\frac{1}{x}} \cdot -24} \]

12. Simplified 51.1%

    \[\leadsto \color{blue}{6 + \sqrt{\frac{1}{x}} \cdot -24} \]
13. Step-by-step derivation

    1. *-commutative 51.1%

       \[\leadsto 6 + \color{blue}{-24 \cdot \sqrt{\frac{1}{x}}} \]

    2. sqrt-div 51.1%

       \[\leadsto 6 + -24 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

    3. metadata-eval 51.1%

       \[\leadsto 6 + -24 \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

    4. un-div-inv 51.1%

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

14. Applied egg-rr 51.1%

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

Alternative 12: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 48.4%

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

6. Taylor expanded in x around 0 4.3%

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

   1. *-commutative 4.3%

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

8. Simplified 4.3%

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

   1. pow1 4.3%

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

   2. add-sqr-sqrt 4.3%

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

   3. sqrt-unprod 4.3%

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

   4. swap-sqr 4.3%

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

   5. add-sqr-sqrt 4.3%

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

   6. metadata-eval 4.3%

      \[\leadsto {\left(\sqrt{x \cdot \color{blue}{2.25}}\right)}^{1} \]

10. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{{\left(\sqrt{x \cdot 2.25}\right)}^{1}} \]
11. Step-by-step derivation

    1. unpow1 4.3%

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

12. Simplified 4.3%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
13. 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 2024165 
(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)))))