Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 9.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   6. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.9%

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

   if 4 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 97.6%

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

4. Final simplification 97.8%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 + N[(4.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $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 99.9%

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

   4. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 97.6%

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

Alternative 4: 95.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5) (- (* x 12.0) 6.0) (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = (x * 12.0d0) - 6.0d0
    else
        tmp = 6.0d0 - (12.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = (x * 12.0) - 6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = (x * 12.0) - 6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(Float64(x * 12.0) - 6.0);
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = (x * 12.0) - 6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[(x * 12.0), $MachinePrecision] - 6.0), $MachinePrecision], N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. distribute-rgt-out 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

   8. Taylor expanded in x around 0 95.2%

      \[\leadsto \color{blue}{12 \cdot x - 6} \]

   if 0.5 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-undefine 99.6%

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

      2. add-sqr-sqrt 99.1%

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

      3. distribute-rgt-out 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around inf 94.6%

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

   8. Taylor expanded in x around inf 94.9%

      \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 94.9%

         \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

      2. metadata-eval 94.9%

         \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   10. Simplified 94.9%

       \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;x \cdot 12 - 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 (- 6.0 (/ 12.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (12.0 / 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
    else
        tmp = 6.0d0 - (12.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (12.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0 - (12.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = Float64(6.0 - Float64(12.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0 - (12.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, N[(6.0 - N[(12.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. distribute-rgt-out 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

   8. Taylor expanded in x around 0 95.2%

      \[\leadsto \color{blue}{-6} \]

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-undefine 99.6%

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

      2. add-sqr-sqrt 99.1%

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

      3. distribute-rgt-out 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around inf 94.6%

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

   8. Taylor expanded in x around inf 94.9%

      \[\leadsto \color{blue}{6 - 12 \cdot \frac{1}{x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 94.9%

         \[\leadsto 6 - \color{blue}{\frac{12 \cdot 1}{x}} \]

      2. metadata-eval 94.9%

         \[\leadsto 6 - \frac{\color{blue}{12}}{x} \]

   10. Simplified 94.9%

       \[\leadsto \color{blue}{6 - \frac{12}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \frac{6 + x \cdot -6}{-1 - x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (6.0 + (x * -6.0)) / (-1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. metadata-eval 99.7%

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

   8. fma-define 99.7%

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

   9. metadata-eval 99.7%

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

   10. *-lft-identity 99.7%

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

   11. +-commutative 99.7%

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

   12. associate-+l+ 99.7%

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

   13. +-commutative 99.7%

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

   14. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. fma-undefine 99.7%

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

   2. add-sqr-sqrt 99.5%

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

   3. distribute-rgt-out 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around inf 94.9%

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

   1. frac-2neg 94.9%

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

   2. +-commutative 94.9%

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

   3. distribute-frac-neg2 94.9%

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

   4. fma-undefine 94.9%

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

   5. *-commutative 94.9%

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

   6. distribute-neg-in 94.9%

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

   7. distribute-rgt-neg-in 94.9%

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

   8. metadata-eval 94.9%

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

   9. metadata-eval 94.9%

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

9. Applied egg-rr 94.9%

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

10. Final simplification 94.9%

    \[\leadsto \frac{6 + x \cdot -6}{-1 - x} \]
11. Add Preprocessing

Alternative 7: 95.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, 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. Step-by-step derivation

      1. /-rgt-identity 99.9%

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

      2. associate-/l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-define 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. distribute-rgt-out 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 95.2%

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

   8. Taylor expanded in x around 0 95.2%

      \[\leadsto \color{blue}{-6} \]

   if 1 < x

   1. Initial program 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-/l/ 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-lft-identity 99.6%

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

      11. +-commutative 99.6%

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

      12. associate-+l+ 99.6%

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

      13. +-commutative 99.6%

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

      14. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. fma-undefine 99.6%

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

      2. add-sqr-sqrt 99.1%

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

      3. distribute-rgt-out 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around inf 94.9%

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

Alternative 8: 48.2% 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. Step-by-step derivation

   1. /-rgt-identity 99.8%

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

   2. associate-/l/ 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 99.7%

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

   5. metadata-eval 99.7%

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

   6. metadata-eval 99.7%

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

   7. metadata-eval 99.7%

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

   8. fma-define 99.7%

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

   9. metadata-eval 99.7%

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

   10. *-lft-identity 99.7%

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

   11. +-commutative 99.7%

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

   12. associate-+l+ 99.7%

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

   13. +-commutative 99.7%

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

   14. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. fma-undefine 99.7%

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

   2. add-sqr-sqrt 99.5%

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

   3. distribute-rgt-out 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around inf 94.9%

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

8. Taylor expanded in x around 0 46.6%

   \[\leadsto \color{blue}{-6} \]
9. 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 2024091 
(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)))))