3frac (problem 3.3.3)

Percentage Accurate: 69.9% → 99.5%

Time: 9.3s

Alternatives: 13

Speedup: 1.0×

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: 69.9% 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: 99.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. div-inv 99.2%

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

   2. pow-flip 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. div-inv 100.0%

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

   6. fma-define 100.0%

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

   7. pow-flip 100.0%

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

   8. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]
10. 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 98.7%

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

   1. div-inv 98.7%

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

   2. pow-flip 99.5%

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

   3. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 3: 69.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	t_0 = Float64(Float64(x + -1.0) * 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(x - Float64(x + -1.0)) / Float64(x * Float64(x + -1.0)));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(x + -1.0) + Float64(x - -1.0))) + Float64(-2.0 * t_0)) / Float64(x * t_0));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + -1.0), $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[(x - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(x + -1.0), $MachinePrecision] + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$0), $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 65.5%

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

      1. +-commutative 65.5%

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

      2. associate-+r- 65.5%

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

      3. sub-neg 65.5%

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

      4. remove-double-neg 65.5%

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

      5. neg-sub0 65.5%

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

      6. associate-+l- 65.5%

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

      7. neg-sub0 65.5%

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

      8. distribute-neg-frac2 65.5%

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

      9. distribute-frac-neg2 65.5%

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

      10. associate-+r+ 65.5%

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

      11. +-commutative 65.5%

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

      12. remove-double-neg 65.5%

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

      13. distribute-neg-frac2 65.5%

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

      14. sub0-neg 65.5%

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

      15. associate-+l- 65.5%

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

      16. neg-sub0 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around inf 65.2%

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

      1. div-inv 65.2%

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

      2. mul-1-neg 65.2%

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

      3. add-sqr-sqrt 17.3%

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

      4. sqrt-prod 14.6%

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

      5. frac-times 13.8%

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

      6. metadata-eval 13.8%

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

      7. metadata-eval 13.8%

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

      8. frac-times 14.6%

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

      9. sqrt-unprod 2.8%

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

      10. add-sqr-sqrt 6.1%

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

      11. sub-neg 6.1%

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

      12. clear-num 6.1%

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

      13. frac-sub 51.0%

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

   7. Applied egg-rr 65.2%

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

   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 37.9%

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

      1. +-commutative 37.9%

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

      2. associate-+r- 37.9%

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

      3. sub-neg 37.9%

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

      4. remove-double-neg 37.9%

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

      5. neg-sub0 37.9%

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

      6. associate-+l- 37.9%

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

      7. neg-sub0 37.9%

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

      8. distribute-neg-frac2 37.9%

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

      9. distribute-frac-neg2 37.9%

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

      10. associate-+r+ 37.9%

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

      11. +-commutative 37.9%

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

      12. remove-double-neg 37.9%

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

      13. distribute-neg-frac2 37.9%

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

      14. sub0-neg 37.9%

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

      15. associate-+l- 37.9%

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

      16. neg-sub0 37.9%

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

   3. Simplified 37.9%

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

      1. +-commutative 37.9%

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

      2. associate-+l- 37.9%

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

   6. Applied egg-rr 37.9%

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

      1. frac-sub 37.4%

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

      2. frac-sub 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 65.9%

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

Alternative 4: 69.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 / (x + -1.0)) + ((-2.0 / x) + (-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))) + (((-2.0d0) / x) + ((-1.0d0) / (x * ((-1.0d0) + ((-1.0d0) / x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 65.0%

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

6. Final simplification 65.0%

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

Alternative 5: 69.9% 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]

Derivation

1. Initial program 65.0%

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

3. Final simplification 65.0%

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

Alternative 6: 68.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + \frac{-1}{x}}{x}} \]
8. Add Preprocessing

Alternative 7: 68.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.2%

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

   1. div-inv 64.2%

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

   2. mul-1-neg 64.2%

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

   3. add-sqr-sqrt 17.2%

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

   4. sqrt-prod 14.6%

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

   5. frac-times 13.7%

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

   6. metadata-eval 13.7%

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

   7. metadata-eval 13.7%

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

   8. frac-times 14.6%

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

   9. sqrt-unprod 2.8%

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

   10. add-sqr-sqrt 6.2%

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

   11. sub-neg 6.2%

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

   12. clear-num 6.2%

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

   13. frac-sub 50.2%

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

7. Applied egg-rr 64.2%

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

8. Final simplification 64.2%

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

Alternative 8: 68.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.2%

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

6. Taylor expanded in x around inf 64.2%

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

7. Final simplification 64.2%

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

Alternative 9: 68.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.2%

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

6. Final simplification 64.2%

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

Alternative 10: 68.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.2%

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

6. Taylor expanded in x around inf 63.9%

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

Alternative 11: 6.3% 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

   1. clear-num 65.0%

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

   2. frac-sub 18.5%

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

   3. *-un-lft-identity 18.5%

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

   4. div-inv 18.5%

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

   5. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot \color{blue}{-0.5}\right) \cdot 1}{\frac{x}{-2} \cdot \left(-1 - x\right)} \]

   6. div-inv 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-1 - x\right)} \]

   7. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-1 - x\right)} \]

6. Applied egg-rr 18.5%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
7. Step-by-step derivation

   1. sub-neg 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 - x\right) + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   2. sub-neg 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 + \left(-x\right)\right)} + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   3. distribute-rgt-neg-in 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{\left(x \cdot -0.5\right) \cdot \left(-1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   4. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot -0.5\right) \cdot \color{blue}{-1}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   5. associate-+l+ 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1 + \left(\left(-x\right) + \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   6. neg-mul-1 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{-1 \cdot x} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   7. *-commutative 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{x \cdot -1} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   8. associate-*l* 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(x \cdot -1 + \color{blue}{x \cdot \left(-0.5 \cdot -1\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   9. distribute-lft-out 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{x \cdot \left(-1 + -0.5 \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   10. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \left(-1 + \color{blue}{0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   11. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \color{blue}{-0.5}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   12. associate-*l* 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{x \cdot \left(-0.5 \cdot \left(-1 - x\right)\right)}} \]

   13. sub-neg 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(-0.5 \cdot \color{blue}{\left(-1 + \left(-x\right)\right)}\right)} \]

   14. +-commutative 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(-0.5 \cdot \color{blue}{\left(\left(-x\right) + -1\right)}\right)} \]

   15. distribute-rgt-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \color{blue}{\left(\left(-x\right) \cdot -0.5 + -1 \cdot -0.5\right)}} \]

   16. distribute-lft-neg-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(\color{blue}{\left(-x \cdot -0.5\right)} + -1 \cdot -0.5\right)} \]

   17. distribute-rgt-neg-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(\color{blue}{x \cdot \left(--0.5\right)} + -1 \cdot -0.5\right)} \]

   18. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot \color{blue}{0.5} + -1 \cdot -0.5\right)} \]

   19. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + \color{blue}{0.5}\right)} \]

8. Simplified 18.5%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + 0.5\right)}} \]

9. Taylor expanded in x around inf 17.5%

   \[\leadsto \frac{1}{\color{blue}{x}} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + 0.5\right)} \]

10. Taylor expanded in x around 0 4.9%

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

    1. div-inv 4.9%

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

    2. frac-2neg 4.9%

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

    3. metadata-eval 4.9%

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

    4. pow1 4.9%

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

    5. metadata-eval 4.9%

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

    6. sqrt-pow1 49.7%

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

    7. pow2 49.7%

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

    8. sqr-neg 49.7%

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

    9. sqrt-prod 3.5%

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

    10. add-sqr-sqrt 6.2%

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

12. Applied egg-rr 6.2%

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

    1. associate-*r/ 6.2%

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

    2. metadata-eval 6.2%

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

14. Simplified 6.2%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
15. Add Preprocessing

Alternative 12: 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

   1. clear-num 65.0%

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

   2. frac-sub 18.5%

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

   3. *-un-lft-identity 18.5%

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

   4. div-inv 18.5%

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

   5. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot \color{blue}{-0.5}\right) \cdot 1}{\frac{x}{-2} \cdot \left(-1 - x\right)} \]

   6. div-inv 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-1 - x\right)} \]

   7. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-1 - x\right)} \]

6. Applied egg-rr 18.5%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
7. Step-by-step derivation

   1. sub-neg 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 - x\right) + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   2. sub-neg 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 + \left(-x\right)\right)} + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   3. distribute-rgt-neg-in 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{\left(x \cdot -0.5\right) \cdot \left(-1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   4. metadata-eval 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot -0.5\right) \cdot \color{blue}{-1}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   5. associate-+l+ 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1 + \left(\left(-x\right) + \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   6. neg-mul-1 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{-1 \cdot x} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   7. *-commutative 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{x \cdot -1} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   8. associate-*l* 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(x \cdot -1 + \color{blue}{x \cdot \left(-0.5 \cdot -1\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   9. distribute-lft-out 18.5%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{x \cdot \left(-1 + -0.5 \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   10. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \left(-1 + \color{blue}{0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   11. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \color{blue}{-0.5}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   12. associate-*l* 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{x \cdot \left(-0.5 \cdot \left(-1 - x\right)\right)}} \]

   13. sub-neg 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(-0.5 \cdot \color{blue}{\left(-1 + \left(-x\right)\right)}\right)} \]

   14. +-commutative 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(-0.5 \cdot \color{blue}{\left(\left(-x\right) + -1\right)}\right)} \]

   15. distribute-rgt-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \color{blue}{\left(\left(-x\right) \cdot -0.5 + -1 \cdot -0.5\right)}} \]

   16. distribute-lft-neg-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(\color{blue}{\left(-x \cdot -0.5\right)} + -1 \cdot -0.5\right)} \]

   17. distribute-rgt-neg-in 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(\color{blue}{x \cdot \left(--0.5\right)} + -1 \cdot -0.5\right)} \]

   18. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot \color{blue}{0.5} + -1 \cdot -0.5\right)} \]

   19. metadata-eval 18.5%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + \color{blue}{0.5}\right)} \]

8. Simplified 18.5%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + 0.5\right)}} \]

9. Taylor expanded in x around inf 17.5%

   \[\leadsto \frac{1}{\color{blue}{x}} + \frac{-1 + x \cdot -0.5}{x \cdot \left(x \cdot 0.5 + 0.5\right)} \]

10. Taylor expanded in x around 0 4.9%

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

Alternative 13: 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around 0 4.9%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. Add Preprocessing

Developer Target 1: 99.1% 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 2024123 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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