Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.8%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + -1}{x + \left(\sqrt{x \cdot 16} + 1\right)} \cdot 6 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ x -1.0) (+ x (+ (sqrt (* x 16.0)) 1.0))) 6.0))

C

double code(double x) {
	return ((x + -1.0) / (x + (sqrt((x * 16.0)) + 1.0))) * 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + (-1.0d0)) / (x + (sqrt((x * 16.0d0)) + 1.0d0))) * 6.0d0
end function

Java

public static double code(double x) {
	return ((x + -1.0) / (x + (Math.sqrt((x * 16.0)) + 1.0))) * 6.0;
}

Python

def code(x):
	return ((x + -1.0) / (x + (math.sqrt((x * 16.0)) + 1.0))) * 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x + -1.0) / Float64(x + Float64(sqrt(Float64(x * 16.0)) + 1.0))) * 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + -1.0) / (x + (sqrt((x * 16.0)) + 1.0))) * 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-undefine 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. sqrt-unprod 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. swap-sqr 99.9%

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

   7. add-sqr-sqrt 99.9%

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

   8. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 98.2%

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

4. Final simplification 97.7%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0)
     (* (+ x -1.0) (/ 6.0 (+ 1.0 t_0)))
     (* (+ x -1.0) (/ 6.0 (+ 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 / (1.0 + t_0));
	} else {
		tmp = (x + -1.0) * (6.0 / (x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * sqrt(x)
    if (x <= 1.0d0) then
        tmp = (x + (-1.0d0)) * (6.0d0 / (1.0d0 + t_0))
    else
        tmp = (x + (-1.0d0)) * (6.0d0 / (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 / (1.0 + t_0));
	} else {
		tmp = (x + -1.0) * (6.0 / (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 / (1.0 + t_0))
	else:
		tmp = (x + -1.0) * (6.0 / (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(1.0 + t_0)));
	else
		tmp = Float64(Float64(x + -1.0) * Float64(6.0 / 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 / (1.0 + t_0));
	else
		tmp = (x + -1.0) * (6.0 / (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[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * N[(6.0 / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 98.2%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.1%

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

   if 4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.1%

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

   4. Taylor expanded in x around 0 98.1%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= x 1.0) (/ -6.0 (+ t_0 (+ x 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 / (t_0 + (x + 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) / (t_0 + (x + 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 / (t_0 + (x + 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 / (t_0 + (x + 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(t_0 + Float64(x + 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 / (t_0 + (x + 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[(t$95$0 + N[(x + 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. Add Preprocessing

   3. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.1%

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

   4. Taylor expanded in x around 0 98.1%

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

4. Final simplification 97.7%

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

Alternative 6: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.1%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. fma-define 98.1%

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

      4. sqrt-div 98.1%

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

      5. metadata-eval 98.1%

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

      6. un-div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. fma-undefine 98.1%

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

      2. *-lft-identity 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 97.7%

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

Alternative 7: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.0%

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

   if 1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. fma-define 98.1%

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

      4. sqrt-div 98.1%

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

      5. metadata-eval 98.1%

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

      6. un-div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. fma-undefine 98.1%

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

      2. *-lft-identity 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 97.6%

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

Alternative 8: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.0%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. add-sqr-sqrt 99.9%

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

      10. sqrt-unprod 99.9%

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

      11. *-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. swap-sqr 99.9%

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

      14. add-sqr-sqrt 99.9%

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

      15. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

Alternative 9: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.25) (+ -6.0 (* (sqrt x) 24.0)) 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = -6.0 + (sqrt(x) * 24.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = (-6.0d0) + (sqrt(x) * 24.0d0)
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = -6.0 + (Math.sqrt(x) * 24.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = -6.0 + (math.sqrt(x) * 24.0)
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = Float64(-6.0 + Float64(sqrt(x) * 24.0));
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = -6.0 + (sqrt(x) * 24.0);
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], N[(-6.0 + N[(N[Sqrt[x], $MachinePrecision] * 24.0), $MachinePrecision]), $MachinePrecision], 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.0%

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

      1. flip-+ 97.0%

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

      2. associate-/r/ 97.0%

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

      3. metadata-eval 97.0%

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

      4. *-commutative 97.0%

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

      5. *-commutative 97.0%

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

      6. swap-sqr 97.0%

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

      7. add-sqr-sqrt 97.0%

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

      8. metadata-eval 97.0%

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

      9. cancel-sign-sub-inv 97.0%

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

      10. metadata-eval 97.0%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in x around 0 96.7%

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

      1. distribute-lft-in 96.7%

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

      2. metadata-eval 96.7%

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

      3. associate-*r* 96.7%

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

      4. metadata-eval 96.7%

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

   8. Simplified 96.7%

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

   if 0.25 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. add-sqr-sqrt 99.9%

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

      10. sqrt-unprod 99.9%

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

      11. *-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. swap-sqr 99.9%

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

      14. add-sqr-sqrt 99.9%

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

      15. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

      \[\leadsto \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;-6 + \sqrt{x} \cdot 24\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.0%

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

   4. Taylor expanded in x around inf 6.8%

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

      1. *-commutative 6.8%

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

   6. Simplified 6.8%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. add-sqr-sqrt 99.9%

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

      10. sqrt-unprod 99.9%

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

      11. *-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. swap-sqr 99.9%

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

      14. add-sqr-sqrt 99.9%

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

      15. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

Alternative 11: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.1%

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

   6. Taylor expanded in x around -inf 6.7%

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

      1. *-commutative 6.7%

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

   8. Simplified 6.7%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. associate-/l* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. flip-+ 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. swap-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. add-sqr-sqrt 99.9%

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

      10. sqrt-unprod 99.9%

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

      11. *-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. swap-sqr 99.9%

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

      14. add-sqr-sqrt 99.9%

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

      15. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

Alternative 12: 48.0% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.9%

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

   2. *-commutative 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-undefine 99.9%

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

4. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

   2. flip-+ 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. swap-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

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

   9. add-sqr-sqrt 99.9%

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

   10. sqrt-unprod 99.9%

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

   11. *-commutative 99.9%

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

   12. *-commutative 99.9%

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

   13. swap-sqr 99.9%

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

   14. add-sqr-sqrt 99.9%

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

   15. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 50.2%

   \[\leadsto \color{blue}{6} \]
8. Add Preprocessing

Developer target: 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 2024084 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))