Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 11.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. associate-/l/ 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-define 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-lft-identity 99.4%

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

   11. associate-+l+ 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

   2. metadata-eval 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. fma-undefine 99.4%

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

   5. +-commutative 99.4%

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

   6. associate-/l* 99.9%

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

   7. *-commutative 99.9%

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

   8. +-commutative 99.9%

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

   9. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0)
     (/ 6.0 (/ (+ t_0 1.0) (+ x -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 / ((t_0 + 1.0) / (x + -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 / ((t_0 + 1.0d0) / (x + (-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 / ((t_0 + 1.0) / (x + -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 / ((t_0 + 1.0) / (x + -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(t_0 + 1.0) / Float64(x + -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 / ((t_0 + 1.0) / (x + -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[(t$95$0 + 1.0), $MachinePrecision] / N[(x + -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. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. metadata-eval 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. fma-undefine 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/l* 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

   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 0 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

      1. fma-undefine 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. fma-undefine 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-/l* 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 97.5%

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

4. Final simplification 97.3%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. metadata-eval 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. fma-undefine 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/l* 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

   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 0 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

      1. fma-undefine 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. fma-undefine 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-/l* 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 97.5%

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

4. Final simplification 97.3%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-undefine 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

      1. fma-undefine 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. fma-undefine 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-/l* 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 97.5%

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

4. Final simplification 97.3%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-undefine 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 96.9%

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

   if 4 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.2%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.2%

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

Alternative 7: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ x (+ (* 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 / (x + ((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) / (x + ((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 / (x + ((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 / (x + ((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(x + 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 / (x + ((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[(x + 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. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.9%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. *-un-lft-identity 97.4%

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

      3. fma-define 97.4%

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

      4. sqrt-div 97.4%

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

      5. metadata-eval 97.4%

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

      6. un-div-inv 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. fma-undefine 97.4%

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

      2. *-lft-identity 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 97.2%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. associate-/l/ 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-define 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-lft-identity 99.4%

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

   11. associate-+l+ 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

   2. metadata-eval 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. fma-undefine 99.4%

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

   5. +-commutative 99.4%

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

   6. associate-/l* 99.9%

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

   7. *-commutative 99.9%

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

   8. +-commutative 99.9%

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

   9. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 9: 97.9% accurate, 1.0× speedup

Math

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

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. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.8%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. *-un-lft-identity 97.4%

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

      3. fma-define 97.4%

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

      4. sqrt-div 97.4%

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

      5. metadata-eval 97.4%

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

      6. un-div-inv 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. fma-undefine 97.4%

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

      2. *-lft-identity 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 97.2%

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

Alternative 10: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-1.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (* -1.5 (pow x -0.5)) (* (sqrt x) 1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 * pow(x, -0.5);
	} 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) * (x ** (-0.5d0))
    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.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 * math.pow(x, -0.5)
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 * (x ^ -0.5));
	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 * (x ^ -0.5);
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 * N[Power[x, -0.5], $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. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.8%

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

   6. Taylor expanded in x around inf 7.2%

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

      1. unpow-1 7.2%

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

      2. metadata-eval 7.2%

         \[\leadsto -1.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 7.2%

         \[\leadsto -1.5 \cdot \sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}} \]

      4. rem-sqrt-square 7.2%

         \[\leadsto -1.5 \cdot \color{blue}{\left|{x}^{-0.5}\right|} \]

      5. rem-square-sqrt 7.2%

         \[\leadsto -1.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

      6. fabs-sqr 7.2%

         \[\leadsto -1.5 \cdot \color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \]

      7. rem-square-sqrt 7.2%

         \[\leadsto -1.5 \cdot \color{blue}{{x}^{-0.5}} \]

   8. Simplified 7.2%

      \[\leadsto \color{blue}{-1.5 \cdot {x}^{-0.5}} \]

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

   6. Taylor expanded in x around 0 7.0%

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

      1. *-commutative 7.0%

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

   8. Simplified 7.0%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(sqrt(x) * -1.5);
	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 = sqrt(x) * -1.5;
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $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. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 1.9%

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

   6. Taylor expanded in x around 0 1.9%

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

      1. unpow-1 1.9%

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

      2. metadata-eval 1.9%

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

      3. pow-sqr 1.9%

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

      4. rem-sqrt-square 1.9%

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

      5. rem-square-sqrt 1.9%

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

      6. fabs-sqr 1.9%

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

      7. rem-square-sqrt 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around -inf 7.0%

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

       1. *-commutative 7.0%

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

   11. Simplified 7.0%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

   6. Taylor expanded in x around 0 7.0%

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

      1. *-commutative 7.0%

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

   8. Simplified 7.0%

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

Alternative 12: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-+l+ 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 1.9%

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

   6. Taylor expanded in x around 0 1.9%

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

      1. unpow-1 1.9%

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

      2. metadata-eval 1.9%

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

      3. pow-sqr 1.9%

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

      4. rem-sqrt-square 1.9%

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

      5. rem-square-sqrt 1.9%

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

      6. fabs-sqr 1.9%

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

      7. rem-square-sqrt 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around -inf 7.0%

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

       1. *-commutative 7.0%

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

   11. Simplified 7.0%

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

   if 1 < x

   1. Initial program 99.0%

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

      1. /-rgt-identity 99.0%

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

      2. associate-/l/ 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. fma-define 99.1%

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

      9. metadata-eval 99.1%

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

      10. *-lft-identity 99.1%

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

      11. associate-+l+ 99.1%

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

      12. +-commutative 99.1%

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

      13. fma-define 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around inf 97.4%

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

   6. Taylor expanded in x around 0 7.0%

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

      1. *-commutative 7.0%

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

   8. Simplified 7.0%

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

      1. add-sqr-sqrt 7.0%

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

      2. sqrt-unprod 7.0%

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

      3. swap-sqr 7.0%

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

      4. add-sqr-sqrt 7.0%

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

      5. metadata-eval 7.0%

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

   10. Applied egg-rr 7.0%

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

Alternative 13: 52.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. associate-/l/ 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-define 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-lft-identity 99.4%

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

   11. associate-+l+ 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 44.5%

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

   1. flip-+ 44.5%

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

   2. associate-/r/ 44.5%

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

   3. metadata-eval 44.5%

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

   4. *-commutative 44.5%

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

   5. *-commutative 44.5%

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

   6. swap-sqr 44.5%

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

   7. add-sqr-sqrt 44.5%

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

   8. metadata-eval 44.5%

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

   9. add-sqr-sqrt 44.5%

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

   10. sqrt-unprod 44.5%

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

   11. *-commutative 44.5%

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

   12. *-commutative 44.5%

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

   13. swap-sqr 44.5%

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

   14. add-sqr-sqrt 44.5%

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

   15. metadata-eval 44.5%

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

7. Applied egg-rr 44.5%

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

8. Taylor expanded in x around 0 47.1%

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

   1. cancel-sign-sub-inv 47.1%

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

   2. metadata-eval 47.1%

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

   3. *-commutative 47.1%

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

   4. distribute-rgt-in 47.1%

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

   5. metadata-eval 47.1%

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

   6. associate-*l* 47.1%

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

   7. metadata-eval 47.1%

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

10. Simplified 47.1%

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

Alternative 14: 4.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. associate-/l/ 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-define 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-lft-identity 99.4%

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

   11. associate-+l+ 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 54.5%

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

6. Taylor expanded in x around 0 4.7%

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

   1. *-commutative 4.7%

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

8. Simplified 4.7%

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

   1. add-sqr-sqrt 4.7%

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

   2. sqrt-unprod 4.7%

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

   3. swap-sqr 4.7%

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

   4. add-sqr-sqrt 4.7%

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

   5. metadata-eval 4.7%

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

10. Applied egg-rr 4.7%

    \[\leadsto \color{blue}{\sqrt{x \cdot 2.25}} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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)))))