Asymptote C

Percentage Accurate: 54.5% → 100.0%

Time: 8.2s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{3 + \frac{1}{x}}{\frac{1}{x} - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ 3.0 (/ 1.0 x)) (- (/ 1.0 x) x)))

C

double code(double x) {
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 + (1.0d0 / x)) / ((1.0d0 / x) - x)
end function

Java

public static double code(double x) {
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x);
}

Python

def code(x):
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x)

Julia

function code(x)
	return Float64(Float64(3.0 + Float64(1.0 / x)) / Float64(Float64(1.0 / x) - x))
end

MATLAB

function tmp = code(x)
	tmp = (3.0 + (1.0 / x)) / ((1.0 / x) - x);
end

Wolfram

code[x_] := N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
2. Step-by-step derivation

   1. clear-num 61.8%

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

   2. frac-2neg 61.8%

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

   3. frac-sub 62.0%

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

   4. *-un-lft-identity 62.0%

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

   5. neg-sub0 62.0%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x - 1\right)\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   6. associate-+l- 62.0%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + 1\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   7. neg-sub0 62.0%

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

   8. +-commutative 62.0%

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

   9. sub-neg 62.0%

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

   10. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   11. +-commutative 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   12. associate--r+ 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   13. metadata-eval 62.0%

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

   14. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

   15. associate-+l- 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

   16. neg-sub0 62.0%

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

   17. +-commutative 62.0%

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

   18. sub-neg 62.0%

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

3. Applied egg-rr 62.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\frac{x + 1}{x} \cdot \left(1 - x\right)} \]
5. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\frac{x + 1}{x} \cdot \color{blue}{\left(1 + \left(-x\right)\right)}} \]

   2. distribute-lft-in 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{x + 1}{x} \cdot 1 + \frac{x + 1}{x} \cdot \left(-x\right)}} \]

   3. *-commutative 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{1 \cdot \frac{x + 1}{x}} + \frac{x + 1}{x} \cdot \left(-x\right)} \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{x + 1}{x}} + \frac{x + 1}{x} \cdot \left(-x\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{x + 1}{x} + \frac{x + 1}{x} \cdot \left(-x\right)}} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\frac{x + 1}{x} + \color{blue}{\left(-x\right) \cdot \frac{x + 1}{x}}} \]

   2. distribute-rgt1-in 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\left(\left(-x\right) + 1\right) \cdot \frac{x + 1}{x}}} \]

   3. +-commutative 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\left(1 + \left(-x\right)\right)} \cdot \frac{x + 1}{x}} \]

   4. sub-neg 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\left(1 - x\right)} \cdot \frac{x + 1}{x}} \]

   5. associate-*r/ 79.5%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x + 1\right)}{x}}} \]

   6. associate-*l/ 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1 - x}{x} \cdot \left(x + 1\right)}} \]

   7. *-commutative 100.0%

      \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\left(x + 1\right) \cdot \frac{1 - x}{x}}} \]

8. Simplified 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\left(x + 1\right) \cdot \frac{1 - x}{x}}} \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot x + \frac{1}{x}}} \]
10. Step-by-step derivation

    1. +-commutative 100.0%

       \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1}{x} + -1 \cdot x}} \]

    2. mul-1-neg 100.0%

       \[\leadsto \frac{3 + \frac{1}{x}}{\frac{1}{x} + \color{blue}{\left(-x\right)}} \]

    3. unsub-neg 100.0%

       \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1}{x} - x}} \]

11. Simplified 100.0%

    \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1}{x} - x}} \]

12. Final simplification 100.0%

    \[\leadsto \frac{3 + \frac{1}{x}}{\frac{1}{x} - x} \]

Alternative 2: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* x (+ 3.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (3.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * (3.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (3.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * (3.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(3.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * (3.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(3.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 10.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 99.2%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   4. Simplified 99.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (- (/ -1.0 x) 3.0) x)
   (+ 1.0 (* x (+ 3.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = 1.0 + (x * (3.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (((-1.0d0) / x) - 3.0d0) / x
    else
        tmp = 1.0d0 + (x * (3.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = 1.0 + (x * (3.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = ((-1.0 / x) - 3.0) / x
	else:
		tmp = 1.0 + (x * (3.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(3.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = ((-1.0 / x) - 3.0) / x;
	else
		tmp = 1.0 + (x * (3.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(x * N[(3.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 10.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Step-by-step derivation

      1. clear-num 10.3%

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

      2. frac-2neg 10.3%

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

      3. frac-sub 10.8%

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

      4. *-un-lft-identity 10.8%

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

      5. neg-sub0 10.8%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x - 1\right)\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

      6. associate-+l- 10.8%

         \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + 1\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

      7. neg-sub0 10.8%

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

      8. +-commutative 10.8%

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

      9. sub-neg 10.8%

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

      10. neg-sub0 10.8%

          \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

      11. +-commutative 10.8%

          \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

      12. associate--r+ 10.8%

          \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

      13. metadata-eval 10.8%

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

      14. neg-sub0 10.8%

          \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

      15. associate-+l- 10.8%

          \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

      16. neg-sub0 10.8%

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

      17. +-commutative 10.8%

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

      18. sub-neg 10.8%

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

   3. Applied egg-rr 10.8%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\frac{x + 1}{x} \cdot \left(1 - x\right)} \]

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto -\left(3 \cdot \frac{1}{x} + \frac{1}{\color{blue}{x \cdot x}}\right) \]

      2. associate-/l/ 97.2%

         \[\leadsto -\left(3 \cdot \frac{1}{x} + \color{blue}{\frac{\frac{1}{x}}{x}}\right) \]

      3. metadata-eval 97.2%

         \[\leadsto -\left(3 \cdot \frac{1}{x} + \frac{\frac{\color{blue}{1 \cdot 1}}{x}}{x}\right) \]

      4. associate-*r/ 97.2%

         \[\leadsto -\left(3 \cdot \frac{1}{x} + \frac{\color{blue}{1 \cdot \frac{1}{x}}}{x}\right) \]

      5. associate-*l/ 97.2%

         \[\leadsto -\left(3 \cdot \frac{1}{x} + \color{blue}{\frac{1}{x} \cdot \frac{1}{x}}\right) \]

      6. distribute-rgt-in 97.2%

         \[\leadsto -\color{blue}{\frac{1}{x} \cdot \left(3 + \frac{1}{x}\right)} \]

      7. associate-*l/ 97.5%

         \[\leadsto -\color{blue}{\frac{1 \cdot \left(3 + \frac{1}{x}\right)}{x}} \]

      8. *-lft-identity 97.5%

         \[\leadsto -\frac{\color{blue}{3 + \frac{1}{x}}}{x} \]

   7. Simplified 97.5%

      \[\leadsto \color{blue}{-\frac{3 + \frac{1}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 99.2%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   4. Simplified 99.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(3 + x\right)\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* 3.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (3.0 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (3.0d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (3.0 * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (3.0 * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(3.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (3.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(3.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 10.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot x\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 10.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 96.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 2.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 61.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
2. Step-by-step derivation

   1. clear-num 61.8%

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

   2. frac-2neg 61.8%

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

   3. frac-sub 62.0%

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

   4. *-un-lft-identity 62.0%

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

   5. neg-sub0 62.0%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x - 1\right)\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   6. associate-+l- 62.0%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + 1\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   7. neg-sub0 62.0%

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

   8. +-commutative 62.0%

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

   9. sub-neg 62.0%

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

   10. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   11. +-commutative 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   12. associate--r+ 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   13. metadata-eval 62.0%

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

   14. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

   15. associate-+l- 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

   16. neg-sub0 62.0%

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

   17. +-commutative 62.0%

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

   18. sub-neg 62.0%

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

3. Applied egg-rr 62.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\frac{x + 1}{x} \cdot \left(1 - x\right)} \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot x + \frac{1}{x}}} \]

7. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-0.5} \]

8. Final simplification 2.6%

   \[\leadsto -0.5 \]

Alternative 7: 4.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 61.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
2. Step-by-step derivation

   1. clear-num 61.8%

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

   2. frac-2neg 61.8%

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

   3. frac-sub 62.0%

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

   4. *-un-lft-identity 62.0%

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

   5. neg-sub0 62.0%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x - 1\right)\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   6. associate-+l- 62.0%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + 1\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   7. neg-sub0 62.0%

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

   8. +-commutative 62.0%

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

   9. sub-neg 62.0%

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

   10. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   11. +-commutative 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   12. associate--r+ 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   13. metadata-eval 62.0%

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

   14. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

   15. associate-+l- 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

   16. neg-sub0 62.0%

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

   17. +-commutative 62.0%

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

   18. sub-neg 62.0%

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

3. Applied egg-rr 62.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\frac{x + 1}{x} \cdot \left(1 - x\right)} \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot x + \frac{1}{x}}} \]

7. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 4.0%

   \[\leadsto 0 \]

Alternative 8: 11.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 61.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
2. Step-by-step derivation

   1. clear-num 61.8%

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

   2. frac-2neg 61.8%

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

   3. frac-sub 62.0%

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

   4. *-un-lft-identity 62.0%

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

   5. neg-sub0 62.0%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x - 1\right)\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   6. associate-+l- 62.0%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + 1\right)} - \frac{x + 1}{x} \cdot \left(-\left(x + 1\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   7. neg-sub0 62.0%

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

   8. +-commutative 62.0%

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

   9. sub-neg 62.0%

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

   10. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   11. +-commutative 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   12. associate--r+ 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)} \]

   13. metadata-eval 62.0%

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

   14. neg-sub0 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(0 - \left(x - 1\right)\right)}} \]

   15. associate-+l- 62.0%

       \[\leadsto \frac{\left(1 - x\right) - \frac{x + 1}{x} \cdot \left(-1 - x\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(\left(0 - x\right) + 1\right)}} \]

   16. neg-sub0 62.0%

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

   17. +-commutative 62.0%

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

   18. sub-neg 62.0%

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

3. Applied egg-rr 62.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\frac{x + 1}{x} \cdot \left(1 - x\right)} \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot x + \frac{1}{x}}} \]

7. Applied egg-rr 12.4%

   \[\leadsto \color{blue}{0.5} \]

8. Final simplification 12.4%

   \[\leadsto 0.5 \]

Alternative 9: 51.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 61.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

2. Taylor expanded in x around 0 57.2%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 57.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))