Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.9s

Alternatives: 12

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 9.670419858984021e+246

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. sub-neg 99.8%

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

   6. +-commutative 99.8%

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

   7. fma-undefine 99.8%

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

   8. associate-+r+ 99.8%

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

   9. +-commutative 99.8%

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

   10. associate-/l* 99.9%

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

   11. *-commutative 99.9%

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

   12. sub-neg 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-commutative 99.9%

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

   15. fma-define 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. associate-+r+ 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.3%

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

   if 1 < x

   1. Initial program 99.7%

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

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

      2. metadata-eval 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. associate-+r+ 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. associate-*l/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. fma-undefine 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-define 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 98.0%

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

4. Final simplification 98.5%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;6 \cdot \frac{x + -1}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + t\_0\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 3.4)
     (* 6.0 (/ (+ x -1.0) (+ t_0 1.0)))
     (* 6.0 (/ x (+ (+ x t_0) 1.0))))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if (x <= 3.4) {
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	} else {
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 3.4d0) then
        tmp = 6.0d0 * ((x + (-1.0d0)) / (t_0 + 1.0d0))
    else
        tmp = 6.0d0 * (x / ((x + t_0) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 4.0 * Math.sqrt(x);
	double tmp;
	if (x <= 3.4) {
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	} else {
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 4.0 * math.sqrt(x)
	tmp = 0
	if x <= 3.4:
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0))
	else:
		tmp = 6.0 * (x / ((x + t_0) + 1.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / Float64(t_0 + 1.0)));
	else
		tmp = Float64(6.0 * Float64(x / Float64(Float64(x + t_0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 4.0 * sqrt(x);
	tmp = 0.0;
	if (x <= 3.4)
		tmp = 6.0 * ((x + -1.0) / (t_0 + 1.0));
	else
		tmp = 6.0 * (x / ((x + t_0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.4], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x / N[(N[(x + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   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%

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.6%

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

   if 3.39999999999999991 < 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%

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

      2. metadata-eval 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. associate-+r+ 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;6 \cdot \frac{x + -1}{4 \cdot \sqrt{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{\left(x + 4 \cdot \sqrt{x}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 1 < x

   1. Initial program 99.7%

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

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

      2. metadata-eval 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. associate-+r+ 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 97.9%

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

4. Final simplification 98.5%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{-1}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ (* 4.0 (sqrt x)) (+ x 1.0)))
   (* 6.0 (/ -1.0 (+ -1.0 (* (pow x -0.5) -4.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	} else {
		tmp = 6.0 * (-1.0 / (-1.0 + (pow(x, -0.5) * -4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-6.0d0) / ((4.0d0 * sqrt(x)) + (x + 1.0d0))
    else
        tmp = 6.0d0 * ((-1.0d0) / ((-1.0d0) + ((x ** (-0.5d0)) * (-4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / ((4.0 * Math.sqrt(x)) + (x + 1.0));
	} else {
		tmp = 6.0 * (-1.0 / (-1.0 + (Math.pow(x, -0.5) * -4.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0 / ((4.0 * math.sqrt(x)) + (x + 1.0))
	else:
		tmp = 6.0 * (-1.0 / (-1.0 + (math.pow(x, -0.5) * -4.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(Float64(4.0 * sqrt(x)) + Float64(x + 1.0)));
	else
		tmp = Float64(6.0 * Float64(-1.0 / Float64(-1.0 + Float64((x ^ -0.5) * -4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0 / ((4.0 * sqrt(x)) + (x + 1.0));
	else
		tmp = 6.0 * (-1.0 / (-1.0 + ((x ^ -0.5) * -4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(-1.0 / N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 1 < x

   1. Initial program 99.7%

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

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

      2. metadata-eval 99.7%

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

      3. metadata-eval 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. associate-+r+ 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. sub-neg 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 97.8%

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

       5. associate-*l* 97.8%

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

       6. unpow-1 97.8%

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

       7. metadata-eval 97.8%

          \[\leadsto \frac{-1}{\sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       8. pow-sqr 97.8%

          \[\leadsto \frac{-1}{\sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       9. rem-sqrt-square 97.8%

          \[\leadsto \frac{-1}{\color{blue}{\left|{x}^{-0.5}\right|} \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       10. rem-square-sqrt 97.8%

           \[\leadsto \frac{-1}{\left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       11. fabs-sqr 97.8%

           \[\leadsto \frac{-1}{\color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       12. rem-square-sqrt 97.8%

           \[\leadsto \frac{-1}{\color{blue}{{x}^{-0.5}} \cdot \left(-1 \cdot 4\right) + \left(-1\right)} \cdot 6 \]

       13. metadata-eval 97.8%

           \[\leadsto \frac{-1}{{x}^{-0.5} \cdot \color{blue}{-4} + \left(-1\right)} \cdot 6 \]

       14. metadata-eval 97.8%

           \[\leadsto \frac{-1}{{x}^{-0.5} \cdot -4 + \color{blue}{-1}} \cdot 6 \]

   11. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{-1}{{x}^{-0.5} \cdot -4 + -1}} \cdot 6 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{4 \cdot \sqrt{x} + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{-1}{-1 + {x}^{-0.5} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 97.8%

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

      1. +-commutative 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. fma-define 97.8%

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

      4. sqrt-div 97.8%

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

      5. metadata-eval 97.8%

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

      6. un-div-inv 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. fma-undefine 97.8%

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

      2. *-lft-identity 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 7: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. associate-/l* 99.9%

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

      11. *-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

      15. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.3%

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

   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 97.8%

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

      1. +-commutative 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. fma-define 97.8%

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

      4. sqrt-div 97.8%

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

      5. metadata-eval 97.8%

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

      6. un-div-inv 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. fma-undefine 97.8%

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

      2. *-lft-identity 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 8: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   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 97.8%

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

      1. +-commutative 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. fma-define 97.8%

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

      4. sqrt-div 97.8%

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

      5. metadata-eval 97.8%

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

      6. un-div-inv 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. fma-undefine 97.8%

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

      2. *-lft-identity 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 98.4%

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

Alternative 9: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   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 97.8%

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

   6. Taylor expanded in x around 0 7.2%

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

      1. *-commutative 7.2%

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

   8. Simplified 7.2%

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

4. Final simplification 46.0%

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

Alternative 10: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. 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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around inf 6.8%

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

      1. *-commutative 6.8%

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

   10. Simplified 6.8%

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

       1. *-commutative 6.8%

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

       2. sqrt-div 6.8%

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

       3. metadata-eval 6.8%

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

       4. un-div-inv 6.8%

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

   12. Applied egg-rr 6.8%

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

   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 97.8%

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

   6. Taylor expanded in x around 0 7.2%

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

      1. *-commutative 7.2%

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

   8. Simplified 7.2%

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

Alternative 11: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. 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 57.3%

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

6. Taylor expanded in x around 0 4.9%

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

   1. *-commutative 4.9%

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

8. Simplified 4.9%

   \[\leadsto \color{blue}{\sqrt{x} \cdot 1.5} \]
9. Add Preprocessing

Alternative 12: 4.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. fma-define 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-lft-identity 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 42.9%

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

   1. *-commutative 42.9%

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

7. Simplified 42.9%

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

8. Taylor expanded in x around inf 4.0%

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

   1. *-commutative 4.0%

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

10. Simplified 4.0%

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

    1. add-sqr-sqrt 0.0%

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

    2. sqrt-unprod 4.6%

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

    3. swap-sqr 4.6%

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

    4. add-sqr-sqrt 4.6%

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

    5. metadata-eval 4.6%

       \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{2.25}} \]

12. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 2.25}} \]
13. Step-by-step derivation

    1. associate-*l/ 4.6%

       \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 2.25}{x}}} \]

    2. metadata-eval 4.6%

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

14. Simplified 4.6%

    \[\leadsto \color{blue}{\sqrt{\frac{2.25}{x}}} \]
15. 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 2024160 
(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)))))