Asymptote C

Percentage Accurate: 53.3% → 99.6%

Time: 5.3s

Alternatives: 8

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: 53.3% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t_0 \leq 10^{-10}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_0 1e-10)
     (- (+ (/ -1.0 (* x x)) (/ -3.0 x)) (/ 3.0 (pow x 3.0)))
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))
    if (t_0 <= 1d-10) then
        tmp = (((-1.0d0) / (x * x)) + ((-3.0d0) / x)) - (3.0d0 / (x ** 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / Math.pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 1e-10:
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / math.pow(x, 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 1e-10)
		tmp = Float64(Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / x)) - Float64(3.0 / (x ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 1e-10)
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / (x ^ 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-10], N[(N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] - N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000004e-10

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) + 3 \cdot \frac{1}{{x}^{3}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}} \]

      4. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(\left(-\frac{1}{{x}^{2}}\right) + \left(-3 \cdot \frac{1}{x}\right)\right)} - 3 \cdot \frac{1}{{x}^{3}} \]

      5. distribute-neg-frac 99.5%

         \[\leadsto \left(\color{blue}{\frac{-1}{{x}^{2}}} + \left(-3 \cdot \frac{1}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(\frac{\color{blue}{-1}}{{x}^{2}} + \left(-3 \cdot \frac{1}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}} \]

      7. unpow2 99.5%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-neg-frac 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 10^{-10}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t_0 \leq 10^{-10}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_0 1e-10) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))
    if (t_0 <= 1d-10) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 1e-10:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 1e-10)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 1e-10)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-10], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000004e-10

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\left(x \cdot -0.3333333333333333\right) \cdot \left(-x\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{-\left(x \cdot -0.3333333333333333\right) \cdot x}} \]

      2. *-commutative 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{-\color{blue}{x \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      3. distribute-rgt-neg-in 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{x \cdot \left(-x \cdot -0.3333333333333333\right)}} \]

      4. mul-1-neg 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x \cdot \color{blue}{\left(-1 \cdot \left(x \cdot -0.3333333333333333\right)\right)}} \]

      5. associate-/r* 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      6. +-commutative 99.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x} + \left(-x\right)}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      7. *-commutative 99.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.3333333333333333 \cdot x\right)} \cdot \frac{1}{x} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      8. associate-*l* 99.5%

         \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 \cdot \left(x \cdot \frac{1}{x}\right)} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      9. rgt-mult-inverse 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333 \cdot \color{blue}{1} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      10. metadata-eval 99.5%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      11. sub-neg 99.5%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 - x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      12. *-commutative 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot -1}} \]

      13. associate-*l* 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{x \cdot \left(-0.3333333333333333 \cdot -1\right)}} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot \color{blue}{0.3333333333333333}} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot 0.3333333333333333}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\frac{-0.3333333333333333 - x}{x}}{x}}{0.3333333333333333}} \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x} \cdot \frac{1}{0.3333333333333333}} \]

      3. div-sub 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} - \frac{x}{x}}}{x} \cdot \frac{1}{0.3333333333333333} \]

      4. *-inverses 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} - \color{blue}{1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} + \left(-1\right)}}{x} \cdot \frac{1}{0.3333333333333333} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + \color{blue}{-1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      7. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot \color{blue}{3} \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot 3} \]

   11. Taylor expanded in x around 0 99.5%

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

       1. associate-*r/ 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \color{blue}{\frac{3 \cdot 1}{x}}\right) \]

       2. metadata-eval 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \frac{\color{blue}{3}}{x}\right) \]

       3. distribute-neg-in 100.0%

          \[\leadsto \color{blue}{\left(-\frac{1}{{x}^{2}}\right) + \left(-\frac{3}{x}\right)} \]

       4. unpow2 100.0%

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

       5. associate-/r* 100.0%

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

       6. distribute-frac-neg 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{0.3333333333333333 \cdot 3}}{x}}{x} + \left(-\frac{3}{x}\right) \]

       8. associate-*l/ 100.0%

          \[\leadsto \frac{-\color{blue}{\frac{0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{--0.3333333333333333}}{x} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x}\right)} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       11. distribute-lft-neg-out 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x} \cdot 3\right)}}{x} + \left(-\frac{3}{x}\right) \]

       12. remove-double-neg 100.0%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       13. associate-*r/ 100.0%

           \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x} \cdot \frac{3}{x}} + \left(-\frac{3}{x}\right) \]

       14. neg-mul-1 100.0%

           \[\leadsto \frac{-0.3333333333333333}{x} \cdot \frac{3}{x} + \color{blue}{-1 \cdot \frac{3}{x}} \]

       15. distribute-rgt-in 99.9%

           \[\leadsto \color{blue}{\frac{3}{x} \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)} \]

       16. associate-*l/ 100.0%

           \[\leadsto \color{blue}{\frac{3 \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)}{x}} \]

   13. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 10^{-10}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - \left(-1 + -2 \cdot \left(x + x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (if (<= x 1.0)
     (- x (+ -1.0 (* -2.0 (+ x (* x x)))))
     (+ (/ -3.0 x) (/ (/ -1.0 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else if (x <= 1.0) {
		tmp = x - (-1.0 + (-2.0 * (x + (x * x))));
	} else {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.9d0)) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else if (x <= 1.0d0) then
        tmp = x - ((-1.0d0) + ((-2.0d0) * (x + (x * x))))
    else
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else if (x <= 1.0) {
		tmp = x - (-1.0 + (-2.0 * (x + (x * x))));
	} else {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9:
		tmp = (-3.0 + (-1.0 / x)) / x
	elif x <= 1.0:
		tmp = x - (-1.0 + (-2.0 * (x + (x * x))))
	else:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	elseif (x <= 1.0)
		tmp = Float64(x - Float64(-1.0 + Float64(-2.0 * Float64(x + Float64(x * x)))));
	else
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = (-3.0 + (-1.0 / x)) / x;
	elseif (x <= 1.0)
		tmp = x - (-1.0 + (-2.0 * (x + (x * x))));
	else
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(x - N[(-1.0 + N[(-2.0 * N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999

   1. Initial program 7.5%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   6. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\left(x \cdot -0.3333333333333333\right) \cdot \left(-x\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{-\left(x \cdot -0.3333333333333333\right) \cdot x}} \]

      2. *-commutative 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{-\color{blue}{x \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      3. distribute-rgt-neg-in 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{x \cdot \left(-x \cdot -0.3333333333333333\right)}} \]

      4. mul-1-neg 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x \cdot \color{blue}{\left(-1 \cdot \left(x \cdot -0.3333333333333333\right)\right)}} \]

      5. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      6. +-commutative 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x} + \left(-x\right)}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      7. *-commutative 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.3333333333333333 \cdot x\right)} \cdot \frac{1}{x} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 \cdot \left(x \cdot \frac{1}{x}\right)} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      9. rgt-mult-inverse 99.6%

         \[\leadsto \frac{\frac{-0.3333333333333333 \cdot \color{blue}{1} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      11. sub-neg 99.6%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 - x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      12. *-commutative 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot -1}} \]

      13. associate-*l* 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{x \cdot \left(-0.3333333333333333 \cdot -1\right)}} \]

      14. metadata-eval 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot \color{blue}{0.3333333333333333}} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot 0.3333333333333333}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\frac{-0.3333333333333333 - x}{x}}{x}}{0.3333333333333333}} \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x} \cdot \frac{1}{0.3333333333333333}} \]

      3. div-sub 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} - \frac{x}{x}}}{x} \cdot \frac{1}{0.3333333333333333} \]

      4. *-inverses 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} - \color{blue}{1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} + \left(-1\right)}}{x} \cdot \frac{1}{0.3333333333333333} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + \color{blue}{-1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      7. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot \color{blue}{3} \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot 3} \]

   11. Taylor expanded in x around 0 99.5%

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

       1. associate-*r/ 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \color{blue}{\frac{3 \cdot 1}{x}}\right) \]

       2. metadata-eval 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \frac{\color{blue}{3}}{x}\right) \]

       3. distribute-neg-in 100.0%

          \[\leadsto \color{blue}{\left(-\frac{1}{{x}^{2}}\right) + \left(-\frac{3}{x}\right)} \]

       4. unpow2 100.0%

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

       5. associate-/r* 100.0%

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

       6. distribute-frac-neg 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{0.3333333333333333 \cdot 3}}{x}}{x} + \left(-\frac{3}{x}\right) \]

       8. associate-*l/ 100.0%

          \[\leadsto \frac{-\color{blue}{\frac{0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{--0.3333333333333333}}{x} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x}\right)} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       11. distribute-lft-neg-out 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x} \cdot 3\right)}}{x} + \left(-\frac{3}{x}\right) \]

       12. remove-double-neg 100.0%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       13. associate-*r/ 100.0%

           \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x} \cdot \frac{3}{x}} + \left(-\frac{3}{x}\right) \]

       14. neg-mul-1 100.0%

           \[\leadsto \frac{-0.3333333333333333}{x} \cdot \frac{3}{x} + \color{blue}{-1 \cdot \frac{3}{x}} \]

       15. distribute-rgt-in 99.9%

           \[\leadsto \color{blue}{\frac{3}{x} \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)} \]

       16. associate-*l/ 100.0%

           \[\leadsto \color{blue}{\frac{3 \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)}{x}} \]

   13. Simplified 100.0%

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

   if -1.8999999999999999 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-lft-out 99.4%

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

      3. unpow2 99.4%

         \[\leadsto x - \left(-2 \cdot \left(\color{blue}{x \cdot x} + x\right) + \left(-1\right)\right) \]

      4. metadata-eval 99.4%

         \[\leadsto x - \left(-2 \cdot \left(x \cdot x + x\right) + \color{blue}{-1}\right) \]

   5. Simplified 99.4%

      \[\leadsto x - \color{blue}{\left(-2 \cdot \left(x \cdot x + x\right) + -1\right)} \]

   if 1 < x

   1. Initial program 8.6%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 99.9%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 99.9%

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

      9. associate-/r* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - \left(-1 + -2 \cdot \left(x + x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (if (<= x 1.0) (+ 1.0 (* x 3.0)) (+ (/ -3.0 x) (/ (/ -1.0 x) x)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 3.0d0)
    else
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-3.0 + (-1.0 / x)) / x
	elif x <= 1.0:
		tmp = 1.0 + (x * 3.0)
	else:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 7.5%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   6. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\left(x \cdot -0.3333333333333333\right) \cdot \left(-x\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{-\left(x \cdot -0.3333333333333333\right) \cdot x}} \]

      2. *-commutative 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{-\color{blue}{x \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      3. distribute-rgt-neg-in 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{x \cdot \left(-x \cdot -0.3333333333333333\right)}} \]

      4. mul-1-neg 50.0%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x \cdot \color{blue}{\left(-1 \cdot \left(x \cdot -0.3333333333333333\right)\right)}} \]

      5. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      6. +-commutative 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x} + \left(-x\right)}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      7. *-commutative 99.6%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.3333333333333333 \cdot x\right)} \cdot \frac{1}{x} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 \cdot \left(x \cdot \frac{1}{x}\right)} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      9. rgt-mult-inverse 99.6%

         \[\leadsto \frac{\frac{-0.3333333333333333 \cdot \color{blue}{1} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      11. sub-neg 99.6%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 - x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      12. *-commutative 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot -1}} \]

      13. associate-*l* 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{x \cdot \left(-0.3333333333333333 \cdot -1\right)}} \]

      14. metadata-eval 99.6%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot \color{blue}{0.3333333333333333}} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot 0.3333333333333333}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\frac{-0.3333333333333333 - x}{x}}{x}}{0.3333333333333333}} \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x} \cdot \frac{1}{0.3333333333333333}} \]

      3. div-sub 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} - \frac{x}{x}}}{x} \cdot \frac{1}{0.3333333333333333} \]

      4. *-inverses 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} - \color{blue}{1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} + \left(-1\right)}}{x} \cdot \frac{1}{0.3333333333333333} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + \color{blue}{-1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      7. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot \color{blue}{3} \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot 3} \]

   11. Taylor expanded in x around 0 99.5%

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

       1. associate-*r/ 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \color{blue}{\frac{3 \cdot 1}{x}}\right) \]

       2. metadata-eval 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \frac{\color{blue}{3}}{x}\right) \]

       3. distribute-neg-in 100.0%

          \[\leadsto \color{blue}{\left(-\frac{1}{{x}^{2}}\right) + \left(-\frac{3}{x}\right)} \]

       4. unpow2 100.0%

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

       5. associate-/r* 100.0%

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

       6. distribute-frac-neg 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{0.3333333333333333 \cdot 3}}{x}}{x} + \left(-\frac{3}{x}\right) \]

       8. associate-*l/ 100.0%

          \[\leadsto \frac{-\color{blue}{\frac{0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{--0.3333333333333333}}{x} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x}\right)} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       11. distribute-lft-neg-out 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x} \cdot 3\right)}}{x} + \left(-\frac{3}{x}\right) \]

       12. remove-double-neg 100.0%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       13. associate-*r/ 100.0%

           \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x} \cdot \frac{3}{x}} + \left(-\frac{3}{x}\right) \]

       14. neg-mul-1 100.0%

           \[\leadsto \frac{-0.3333333333333333}{x} \cdot \frac{3}{x} + \color{blue}{-1 \cdot \frac{3}{x}} \]

       15. distribute-rgt-in 99.9%

           \[\leadsto \color{blue}{\frac{3}{x} \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)} \]

       16. associate-*l/ 100.0%

           \[\leadsto \color{blue}{\frac{3 \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)}{x}} \]

   13. Simplified 100.0%

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

   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 99.4%

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

   if 1 < x

   1. Initial program 8.6%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 99.9%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 99.9%

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

      9. associate-/r* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	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) + ((-1.0d0) / x)) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    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 + (-1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\left(x \cdot -0.3333333333333333\right) \cdot \left(-x\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{-\left(x \cdot -0.3333333333333333\right) \cdot x}} \]

      2. *-commutative 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{-\color{blue}{x \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      3. distribute-rgt-neg-in 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{\color{blue}{x \cdot \left(-x \cdot -0.3333333333333333\right)}} \]

      4. mul-1-neg 50.2%

         \[\leadsto \frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x \cdot \color{blue}{\left(-1 \cdot \left(x \cdot -0.3333333333333333\right)\right)}} \]

      5. associate-/r* 99.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) + \left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)}} \]

      6. +-commutative 99.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \frac{1}{x} + \left(-x\right)}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      7. *-commutative 99.5%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.3333333333333333 \cdot x\right)} \cdot \frac{1}{x} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      8. associate-*l* 99.5%

         \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 \cdot \left(x \cdot \frac{1}{x}\right)} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      9. rgt-mult-inverse 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333 \cdot \color{blue}{1} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      10. metadata-eval 99.5%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333} + \left(-x\right)}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      11. sub-neg 99.5%

          \[\leadsto \frac{\frac{\color{blue}{-0.3333333333333333 - x}}{x}}{-1 \cdot \left(x \cdot -0.3333333333333333\right)} \]

      12. *-commutative 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot -1}} \]

      13. associate-*l* 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{\color{blue}{x \cdot \left(-0.3333333333333333 \cdot -1\right)}} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot \color{blue}{0.3333333333333333}} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x \cdot 0.3333333333333333}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{\frac{-0.3333333333333333 - x}{x}}{x}}{0.3333333333333333}} \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 - x}{x}}{x} \cdot \frac{1}{0.3333333333333333}} \]

      3. div-sub 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} - \frac{x}{x}}}{x} \cdot \frac{1}{0.3333333333333333} \]

      4. *-inverses 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} - \color{blue}{1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} + \left(-1\right)}}{x} \cdot \frac{1}{0.3333333333333333} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + \color{blue}{-1}}{x} \cdot \frac{1}{0.3333333333333333} \]

      7. metadata-eval 99.5%

         \[\leadsto \frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot \color{blue}{3} \]

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{x} + -1}{x} \cdot 3} \]

   11. Taylor expanded in x around 0 99.5%

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

       1. associate-*r/ 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \color{blue}{\frac{3 \cdot 1}{x}}\right) \]

       2. metadata-eval 100.0%

          \[\leadsto -\left(\frac{1}{{x}^{2}} + \frac{\color{blue}{3}}{x}\right) \]

       3. distribute-neg-in 100.0%

          \[\leadsto \color{blue}{\left(-\frac{1}{{x}^{2}}\right) + \left(-\frac{3}{x}\right)} \]

       4. unpow2 100.0%

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

       5. associate-/r* 100.0%

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

       6. distribute-frac-neg 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{0.3333333333333333 \cdot 3}}{x}}{x} + \left(-\frac{3}{x}\right) \]

       8. associate-*l/ 100.0%

          \[\leadsto \frac{-\color{blue}{\frac{0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{-\frac{\color{blue}{--0.3333333333333333}}{x} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       10. distribute-neg-frac 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x}\right)} \cdot 3}{x} + \left(-\frac{3}{x}\right) \]

       11. distribute-lft-neg-out 100.0%

           \[\leadsto \frac{-\color{blue}{\left(-\frac{-0.3333333333333333}{x} \cdot 3\right)}}{x} + \left(-\frac{3}{x}\right) \]

       12. remove-double-neg 100.0%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{x} \cdot 3}}{x} + \left(-\frac{3}{x}\right) \]

       13. associate-*r/ 100.0%

           \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x} \cdot \frac{3}{x}} + \left(-\frac{3}{x}\right) \]

       14. neg-mul-1 100.0%

           \[\leadsto \frac{-0.3333333333333333}{x} \cdot \frac{3}{x} + \color{blue}{-1 \cdot \frac{3}{x}} \]

       15. distribute-rgt-in 99.9%

           \[\leadsto \color{blue}{\frac{3}{x} \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)} \]

       16. associate-*l/ 100.0%

           \[\leadsto \color{blue}{\frac{3 \cdot \left(\frac{-0.3333333333333333}{x} + -1\right)}{x}} \]

   13. Simplified 100.0%

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

   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 99.4%

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

4. Final simplification 99.7%

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

Alternative 6: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (+ 1.0 (* x 3.0)) (/ -3.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 3.0d0)
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * 3.0)
	else:
		tmp = -3.0 / x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 98.7%

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

   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 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (- x -1.0) (/ -3.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = x - (-1.0d0)
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = x - -1.0
	else:
		tmp = -3.0 / x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 98.7%

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

   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 99.4%

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

   3. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 8: 49.9% 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 53.2%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))