3frac (problem 3.3.3)

Percentage Accurate: 70.1% → 99.7%

Time: 11.6s

Alternatives: 11

Speedup: 1.7×

Specification

\[\left|x\right| > 1\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma 2.0 (pow x -2.0) (fma 2.0 (pow x -4.0) 2.0)) (pow x -3.0)))

C

double code(double x) {
	return fma(2.0, pow(x, -2.0), fma(2.0, pow(x, -4.0), 2.0)) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(fma(2.0, (x ^ -2.0), fma(2.0, (x ^ -4.0), 2.0)) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 * N[Power[x, -4.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 98.6%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-+r+ 98.6%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 98.6%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 98.6%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 98.6%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 98.6%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. div-inv 98.6%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   3. fma-define 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 + \frac{2}{{x}^{4}}\right)} \cdot \frac{1}{{x}^{3}} \]

   4. pow-flip 98.6%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   5. metadata-eval 98.6%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   6. +-commutative 98.6%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\frac{2}{{x}^{4}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

   7. div-inv 98.6%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   8. fma-define 98.6%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{4}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 98.6%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-4\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 98.6%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-4}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   11. pow-flip 99.7%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   12. metadata-eval 99.7%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3}} \]

10. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3} \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (pow x -3.0)))

C

double code(double x) {
	return 2.0 * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(2.0 * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

code[x_] := N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 97.9%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
6. Step-by-step derivation

   1. div-inv 97.9%

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

   2. pow-flip 99.0%

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

   3. metadata-eval 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

   \[\leadsto 2 \cdot {x}^{-3} \]
9. Add Preprocessing

Alternative 3: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) \cdot \left(x + 1\right)\\ \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot t\_0 - x \cdot \left(x + x\right)}{x \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x) (+ x 1.0))))
   (if (<= (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))) 5e-25)
     (/ (/ (/ 1.0 x) (+ x -1.0)) x)
     (/ (- (* -2.0 t_0) (* x (+ x x))) (* x t_0)))))

C

double code(double x) {
	double t_0 = (1.0 - x) * (x + 1.0);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-25) {
		tmp = ((1.0 / x) / (x + -1.0)) / x;
	} else {
		tmp = ((-2.0 * t_0) - (x * (x + x))) / (x * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) * (x + 1.0d0)
    if ((((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))) <= 5d-25) then
        tmp = ((1.0d0 / x) / (x + (-1.0d0))) / x
    else
        tmp = (((-2.0d0) * t_0) - (x * (x + x))) / (x * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 - x) * (x + 1.0);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-25) {
		tmp = ((1.0 / x) / (x + -1.0)) / x;
	} else {
		tmp = ((-2.0 * t_0) - (x * (x + x))) / (x * t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 - x) * (x + 1.0)
	tmp = 0
	if (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-25:
		tmp = ((1.0 / x) / (x + -1.0)) / x
	else:
		tmp = ((-2.0 * t_0) - (x * (x + x))) / (x * t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 - x) * Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0))) <= 5e-25)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(x + -1.0)) / x);
	else
		tmp = Float64(Float64(Float64(-2.0 * t_0) - Float64(x * Float64(x + x))) / Float64(x * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 - x) * (x + 1.0);
	tmp = 0.0;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-25)
		tmp = ((1.0 / x) / (x + -1.0)) / x;
	else
		tmp = ((-2.0 * t_0) - (x * (x + x))) / (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] - N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 4.99999999999999962e-25

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. associate-+r- 70.6%

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

      3. sub-neg 70.6%

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

      4. remove-double-neg 70.6%

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

      5. neg-sub0 70.6%

         \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

      6. associate-+l- 70.6%

         \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

      7. neg-sub0 70.6%

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

      8. distribute-neg-frac2 70.6%

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

      9. distribute-frac-neg2 70.6%

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

      10. associate-+r+ 70.7%

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

      11. +-commutative 70.7%

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

      12. remove-double-neg 70.7%

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

      13. distribute-neg-frac2 70.7%

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

      14. sub0-neg 70.7%

          \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

      15. associate-+l- 70.7%

          \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

      16. neg-sub0 70.7%

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

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

      3. distribute-neg-in 70.4%

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

      4. metadata-eval 70.4%

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

      5. distribute-neg-frac 70.4%

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

      6. metadata-eval 70.4%

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

   7. Simplified 70.4%

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

      1. frac-2neg 70.4%

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

      2. div-inv 70.5%

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

      3. +-commutative 70.5%

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

      4. add-sqr-sqrt 35.0%

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

      5. sqrt-unprod 70.6%

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

      6. frac-times 70.6%

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

      7. metadata-eval 70.6%

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

      8. metadata-eval 70.6%

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

      9. frac-times 70.6%

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

      10. sqrt-prod 35.6%

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

      11. add-sqr-sqrt 70.4%

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

      12. metadata-eval 70.4%

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

      13. frac-2neg 70.4%

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

      14. add-sqr-sqrt 17.6%

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

      15. sqrt-unprod 16.2%

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

      16. frac-times 13.1%

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

      17. metadata-eval 13.1%

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

      18. metadata-eval 13.1%

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

      19. frac-times 16.2%

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

      20. sqrt-prod 3.2%

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

      21. add-sqr-sqrt 6.3%

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

   9. Applied egg-rr 6.3%

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

       1. add-sqr-sqrt 6.3%

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

       2. sqrt-unprod 6.3%

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

       3. sqr-neg 6.3%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 70.4%

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

       6. cancel-sign-sub 70.4%

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

       7. un-div-inv 70.4%

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

       8. frac-sub 70.4%

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

   11. Applied egg-rr 70.5%

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

       1. associate-/r* 70.5%

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

   13. Simplified 74.4%

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

   if 4.99999999999999962e-25 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. associate-+r- 66.2%

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

      3. sub-neg 66.2%

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

      4. remove-double-neg 66.2%

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

      5. neg-sub0 66.2%

         \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

      6. associate-+l- 66.2%

         \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

      7. neg-sub0 66.2%

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

      8. distribute-neg-frac2 66.2%

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

      9. distribute-frac-neg2 66.2%

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

      10. associate-+r+ 66.7%

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

      11. +-commutative 66.7%

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

      12. remove-double-neg 66.7%

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

      13. distribute-neg-frac2 66.7%

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

      14. sub0-neg 66.7%

          \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

      15. associate-+l- 66.7%

          \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

      16. neg-sub0 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 66.7%

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

      2. associate-+l- 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. frac-sub 65.7%

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

      2. frac-sub 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. metadata-eval 99.5%

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

      5. div-inv 99.5%

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

      6. associate--l+ 99.5%

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

      7. div-inv 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-rgt-identity 99.5%

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

   8. Applied egg-rr 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(1 - x\right) \cdot \left(x + 1\right)\right) - x \cdot \left(x + x\right)}{x \cdot \left(\left(1 - x\right) \cdot \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.3%

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

6. Final simplification 69.3%

   \[\leadsto \frac{1}{x + -1} + \frac{-1}{x} \]
7. Add Preprocessing

Alternative 5: 73.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.4%

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

   1. associate-*r/ 69.4%

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

   2. neg-mul-1 69.4%

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

   3. distribute-neg-in 69.4%

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

   4. metadata-eval 69.4%

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

   5. distribute-neg-frac 69.4%

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

   6. metadata-eval 69.4%

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

7. Simplified 69.4%

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

   1. frac-2neg 69.4%

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

   2. div-inv 69.5%

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

   3. +-commutative 69.5%

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

   4. add-sqr-sqrt 34.3%

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

   5. sqrt-unprod 69.5%

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

   6. frac-times 69.5%

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

   7. metadata-eval 69.5%

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

   8. metadata-eval 69.5%

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

   9. frac-times 69.5%

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

   10. sqrt-prod 35.2%

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

   11. add-sqr-sqrt 69.3%

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

   12. metadata-eval 69.3%

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

   13. frac-2neg 69.3%

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

   14. add-sqr-sqrt 17.2%

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

   15. sqrt-unprod 16.1%

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

   16. frac-times 13.1%

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

   17. metadata-eval 13.1%

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

   18. metadata-eval 13.1%

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

   19. frac-times 16.1%

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

   20. sqrt-prod 3.4%

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

   21. add-sqr-sqrt 6.5%

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

9. Applied egg-rr 6.5%

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

    1. add-sqr-sqrt 6.5%

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

    2. sqrt-unprod 6.5%

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

    3. sqr-neg 6.5%

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

    4. sqrt-unprod 0.0%

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

    5. add-sqr-sqrt 69.3%

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

    6. cancel-sign-sub 69.3%

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

    7. un-div-inv 69.3%

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

    8. frac-sub 69.3%

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

11. Applied egg-rr 69.5%

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

    1. associate-/r* 69.5%

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

13. Simplified 73.4%

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

14. Final simplification 73.4%

    \[\leadsto \frac{\frac{\frac{1}{x}}{x + -1}}{x} \]
15. Add Preprocessing

Alternative 6: 68.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + \frac{-2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (+ 1.0 (/ -2.0 x))))

C

double code(double x) {
	return -1.0 + (1.0 + (-2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (1.0d0 + ((-2.0d0) / x))
end function

Java

public static double code(double x) {
	return -1.0 + (1.0 + (-2.0 / x));
}

Python

def code(x):
	return -1.0 + (1.0 + (-2.0 / x))

Julia

function code(x)
	return Float64(-1.0 + Float64(1.0 + Float64(-2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (1.0 + (-2.0 / x));
end

Wolfram

code[x_] := N[(-1.0 + N[(1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 3.5%

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

   1. div-sub 3.5%

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

   2. sub-neg 3.5%

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

   3. *-inverses 3.5%

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

   4. metadata-eval 3.5%

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

   5. distribute-neg-frac 3.5%

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

   6. remove-double-neg 3.5%

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

7. Simplified 3.5%

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

8. Taylor expanded in x around 0 68.9%

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

9. Final simplification 68.9%

   \[\leadsto -1 + \left(1 + \frac{-2}{x}\right) \]
10. Add Preprocessing

Alternative 7: 68.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} - \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ -2.0 x) (/ -2.0 x)))

C

double code(double x) {
	return (-2.0 / x) - (-2.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / x) - ((-2.0d0) / x)
end function

Java

public static double code(double x) {
	return (-2.0 / x) - (-2.0 / x);
}

Python

def code(x):
	return (-2.0 / x) - (-2.0 / x)

Julia

function code(x)
	return Float64(Float64(-2.0 / x) - Float64(-2.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (-2.0 / x) - (-2.0 / x);
end

Wolfram

code[x_] := N[(N[(-2.0 / x), $MachinePrecision] - N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 70.6%

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

   2. associate-+l- 70.5%

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

6. Applied egg-rr 70.5%

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

   1. frac-sub 19.0%

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

   2. div-inv 19.8%

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

   3. *-un-lft-identity 19.8%

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

   4. metadata-eval 19.8%

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

   5. div-inv 19.8%

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

   6. associate--l+ 19.7%

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

   7. div-inv 19.7%

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

   8. metadata-eval 19.7%

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

   9. *-rgt-identity 19.7%

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

8. Applied egg-rr 19.7%

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

   1. associate--r- 19.8%

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

   2. metadata-eval 19.8%

      \[\leadsto \frac{-2}{x} - \left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{1}{\left(-1 - x\right) \cdot \left(x + -1\right)} \]

   3. *-commutative 19.8%

      \[\leadsto \frac{-2}{x} - \left(x + \left(0 + x\right)\right) \cdot \frac{1}{\color{blue}{\left(x + -1\right) \cdot \left(-1 - x\right)}} \]

10. Simplified 19.8%

    \[\leadsto \frac{-2}{x} - \color{blue}{\left(x + \left(0 + x\right)\right) \cdot \frac{1}{\left(x + -1\right) \cdot \left(-1 - x\right)}} \]

11. Taylor expanded in x around inf 69.0%

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

12. Final simplification 69.0%

    \[\leadsto \frac{-2}{x} - \frac{-2}{x} \]
13. Add Preprocessing

Alternative 8: 5.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / x
end function

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / x;
end

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 9: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 3.5%

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

   1. div-sub 3.5%

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

   2. sub-neg 3.5%

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

   3. *-inverses 3.5%

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

   4. metadata-eval 3.5%

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

   5. distribute-neg-frac 3.5%

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

   6. remove-double-neg 3.5%

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

7. Simplified 3.5%

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

8. Taylor expanded in x around inf 3.5%

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

9. Taylor expanded in x around 0 5.2%

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

10. Final simplification 5.2%

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

Alternative 10: 6.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 x))

C

double code(double x) {
	return 2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / x
end function

Java

public static double code(double x) {
	return 2.0 / x;
}

Python

def code(x):
	return 2.0 / x

Julia

function code(x)
	return Float64(2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / x;
end

Wolfram

code[x_] := N[(2.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.4%

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

   1. associate-*r/ 69.4%

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

   2. neg-mul-1 69.4%

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

   3. distribute-neg-in 69.4%

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

   4. metadata-eval 69.4%

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

   5. distribute-neg-frac 69.4%

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

   6. metadata-eval 69.4%

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

7. Simplified 69.4%

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

   1. frac-2neg 69.4%

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

   2. div-inv 69.5%

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

   3. +-commutative 69.5%

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

   4. add-sqr-sqrt 34.3%

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

   5. sqrt-unprod 69.5%

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

   6. frac-times 69.5%

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

   7. metadata-eval 69.5%

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

   8. metadata-eval 69.5%

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

   9. frac-times 69.5%

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

   10. sqrt-prod 35.2%

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

   11. add-sqr-sqrt 69.3%

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

   12. metadata-eval 69.3%

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

   13. frac-2neg 69.3%

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

   14. add-sqr-sqrt 17.2%

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

   15. sqrt-unprod 16.1%

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

   16. frac-times 13.1%

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

   17. metadata-eval 13.1%

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

   18. metadata-eval 13.1%

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

   19. frac-times 16.1%

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

   20. sqrt-prod 3.4%

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

   21. add-sqr-sqrt 6.5%

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

9. Applied egg-rr 6.5%

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

10. Taylor expanded in x around inf 6.5%

    \[\leadsto \color{blue}{\frac{2}{x}} \]

11. Final simplification 6.5%

    \[\leadsto \frac{2}{x} \]
12. Add Preprocessing

Alternative 11: 3.4% accurate, 15.0× 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 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r- 70.5%

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

   3. sub-neg 70.5%

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

   4. remove-double-neg 70.5%

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

   5. neg-sub0 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.5%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.5%

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

   8. distribute-neg-frac2 70.5%

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

   9. distribute-frac-neg2 70.5%

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

   10. associate-+r+ 70.6%

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

   11. +-commutative 70.6%

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

   12. remove-double-neg 70.6%

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

   13. distribute-neg-frac2 70.6%

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

   14. sub0-neg 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.6%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.6%

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

3. Simplified 70.6%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 3.5%

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

   1. div-sub 3.5%

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

   2. sub-neg 3.5%

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

   3. *-inverses 3.5%

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

   4. metadata-eval 3.5%

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

   5. distribute-neg-frac 3.5%

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

   6. remove-double-neg 3.5%

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

7. Simplified 3.5%

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

8. Taylor expanded in x around inf 3.5%

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

9. Final simplification 3.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024079 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))