3frac (problem 3.3.3)

Percentage Accurate: 69.1% → 98.9%

Time: 8.7s

Alternatives: 10

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: 69.1% 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: 98.9% 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.4%

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

   11. +-commutative 68.4%

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

   12. neg-mul-1 68.4%

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

   13. metadata-eval 68.4%

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

   14. associate-/r* 68.4%

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

   15. metadata-eval 68.4%

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

   16. metadata-eval 68.4%

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

   17. +-commutative 68.4%

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

   18. +-commutative 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around inf 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   2. expm1-udef 67.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1} \]

   3. div-inv 67.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1 \]

   4. pow-flip 67.2%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1 \]

   5. metadata-eval 67.2%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1 \]

7. Applied egg-rr 67.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} \]

   2. expm1-log1p 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 2: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -1\right) \cdot \left(x + 1\right)\\ \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x} + \frac{-1}{1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + x\right) + t_0 \cdot -2}{x \cdot t_0}\\ \end{array} \end{array} \]

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))) 4e-28)
     (+ (/ -2.0 x) (+ (/ 1.0 x) (/ -1.0 (- 1.0 x))))
     (/ (+ (* x (+ x x)) (* t_0 -2.0)) (* 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))) <= 4e-28) {
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	} else {
		tmp = ((x * (x + x)) + (t_0 * -2.0)) / (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)))) <= 4d-28) then
        tmp = ((-2.0d0) / x) + ((1.0d0 / x) + ((-1.0d0) / (1.0d0 - x)))
    else
        tmp = ((x * (x + x)) + (t_0 * (-2.0d0))) / (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))) <= 4e-28) {
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	} else {
		tmp = ((x * (x + x)) + (t_0 * -2.0)) / (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))) <= 4e-28:
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)))
	else:
		tmp = ((x * (x + x)) + (t_0 * -2.0)) / (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))) <= 4e-28)
		tmp = Float64(Float64(-2.0 / x) + Float64(Float64(1.0 / x) + Float64(-1.0 / Float64(1.0 - x))));
	else
		tmp = Float64(Float64(Float64(x * Float64(x + x)) + Float64(t_0 * -2.0)) / 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))) <= 4e-28)
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	else
		tmp = ((x * (x + x)) + (t_0 * -2.0)) / (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], 4e-28], N[(N[(-2.0 / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 3.99999999999999988e-28

   1. Initial program 68.9%

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

      1. sub-neg 68.9%

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

      2. distribute-neg-frac 68.9%

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

      3. metadata-eval 68.9%

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

      4. metadata-eval 68.9%

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

      5. metadata-eval 68.9%

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

      6. associate-/r* 68.9%

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

      7. metadata-eval 68.9%

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

      8. neg-mul-1 68.9%

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

      9. +-commutative 68.9%

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

      10. associate-+l+ 68.8%

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

      11. +-commutative 68.8%

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

      12. neg-mul-1 68.8%

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

      13. metadata-eval 68.8%

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

      14. associate-/r* 68.8%

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

      15. metadata-eval 68.8%

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

      16. metadata-eval 68.8%

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

      17. +-commutative 68.8%

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

      18. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around inf 67.8%

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

   if 3.99999999999999988e-28 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 45.4%

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

      1. sub-neg 45.4%

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

      2. distribute-neg-frac 45.4%

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

      3. metadata-eval 45.4%

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

      4. metadata-eval 45.4%

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

      5. metadata-eval 45.4%

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

      6. associate-/r* 45.4%

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

      7. metadata-eval 45.4%

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

      8. neg-mul-1 45.4%

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

      9. +-commutative 45.4%

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

      10. associate-+l+ 45.2%

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

      11. +-commutative 45.2%

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

      12. neg-mul-1 45.2%

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

      13. metadata-eval 45.2%

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

      14. associate-/r* 45.2%

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

      15. metadata-eval 45.2%

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

      16. metadata-eval 45.2%

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

      17. +-commutative 45.2%

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

      18. +-commutative 45.2%

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

   3. Simplified 45.2%

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

      1. frac-2neg 45.2%

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

      2. metadata-eval 45.2%

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

      3. frac-add 41.2%

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

      4. *-un-lft-identity 41.2%

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

      5. *-commutative 41.2%

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

      6. +-commutative 41.2%

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

      7. *-un-lft-identity 41.2%

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

      8. +-commutative 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. associate-/r* 41.7%

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

      2. neg-sub0 41.7%

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

      3. associate--r- 41.7%

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

      4. metadata-eval 41.7%

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

      5. +-commutative 41.7%

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

      6. neg-sub0 41.7%

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

      7. associate--r- 41.7%

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

      8. metadata-eval 41.7%

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

      9. +-commutative 41.7%

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

      10. associate-/r* 41.2%

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

      11. +-commutative 41.2%

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

      12. associate-+l+ 41.2%

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

      13. +-commutative 41.2%

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

      14. associate-+r+ 41.2%

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

      15. metadata-eval 41.2%

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

   8. Simplified 41.2%

      \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 41.2%

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

      2. frac-add 80.0%

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

      3. +-lft-identity 80.0%

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

      4. +-commutative 80.0%

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

      5. *-commutative 80.0%

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

      6. +-commutative 80.0%

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

      7. +-commutative 80.0%

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

      8. *-commutative 80.0%

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

      9. +-commutative 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x} + \frac{-1}{1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + x\right) + \left(\left(x + -1\right) \cdot \left(x + 1\right)\right) \cdot -2}{x \cdot \left(\left(x + -1\right) \cdot \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;x \leq 140000000:\\ \;\;\;\;\frac{x \cdot \left(x + x\right) + \left(\left(x + -1\right) \cdot \left(x + 1\right)\right) \cdot -2}{t_0 + x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x} + \frac{-1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ x -1.0))))
   (if (<= x 140000000.0)
     (/ (+ (* x (+ x x)) (* (* (+ x -1.0) (+ x 1.0)) -2.0)) (+ t_0 (* x t_0)))
     (+ (/ -2.0 x) (+ (/ 1.0 x) (/ -1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x * (x + -1.0);
	double tmp;
	if (x <= 140000000.0) {
		tmp = ((x * (x + x)) + (((x + -1.0) * (x + 1.0)) * -2.0)) / (t_0 + (x * t_0));
	} else {
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + (-1.0d0))
    if (x <= 140000000.0d0) then
        tmp = ((x * (x + x)) + (((x + (-1.0d0)) * (x + 1.0d0)) * (-2.0d0))) / (t_0 + (x * t_0))
    else
        tmp = ((-2.0d0) / x) + ((1.0d0 / x) + ((-1.0d0) / (1.0d0 - x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x + -1.0);
	double tmp;
	if (x <= 140000000.0) {
		tmp = ((x * (x + x)) + (((x + -1.0) * (x + 1.0)) * -2.0)) / (t_0 + (x * t_0));
	} else {
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x + -1.0)
	tmp = 0
	if x <= 140000000.0:
		tmp = ((x * (x + x)) + (((x + -1.0) * (x + 1.0)) * -2.0)) / (t_0 + (x * t_0))
	else:
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x + -1.0))
	tmp = 0.0
	if (x <= 140000000.0)
		tmp = Float64(Float64(Float64(x * Float64(x + x)) + Float64(Float64(Float64(x + -1.0) * Float64(x + 1.0)) * -2.0)) / Float64(t_0 + Float64(x * t_0)));
	else
		tmp = Float64(Float64(-2.0 / x) + Float64(Float64(1.0 / x) + Float64(-1.0 / Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x + -1.0);
	tmp = 0.0;
	if (x <= 140000000.0)
		tmp = ((x * (x + x)) + (((x + -1.0) * (x + 1.0)) * -2.0)) / (t_0 + (x * t_0));
	else
		tmp = (-2.0 / x) + ((1.0 / x) + (-1.0 / (1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 140000000.0], N[(N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e8

   1. Initial program 69.8%

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

      1. sub-neg 69.8%

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

      2. distribute-neg-frac 69.8%

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

      3. metadata-eval 69.8%

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

      4. metadata-eval 69.8%

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

      5. metadata-eval 69.8%

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

      6. associate-/r* 69.8%

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

      7. metadata-eval 69.8%

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

      8. neg-mul-1 69.8%

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

      9. +-commutative 69.8%

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

      10. associate-+l+ 69.7%

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

      11. +-commutative 69.7%

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

      12. neg-mul-1 69.7%

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

      13. metadata-eval 69.7%

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

      14. associate-/r* 69.7%

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

      15. metadata-eval 69.7%

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

      16. metadata-eval 69.7%

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

      17. +-commutative 69.7%

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

      18. +-commutative 69.7%

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

   3. Simplified 69.7%

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

      1. frac-2neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. frac-add 16.9%

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

      4. *-un-lft-identity 16.9%

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

      5. *-commutative 16.9%

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

      6. +-commutative 16.9%

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

      7. *-un-lft-identity 16.9%

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

      8. +-commutative 16.9%

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

   6. Applied egg-rr 16.9%

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

      1. associate-/r* 68.8%

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

      2. neg-sub0 68.8%

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

      3. associate--r- 68.8%

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

      4. metadata-eval 68.8%

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

      5. +-commutative 68.8%

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

      6. neg-sub0 68.8%

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

      7. associate--r- 68.8%

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

      8. metadata-eval 68.8%

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

      9. +-commutative 68.8%

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

      10. associate-/r* 16.9%

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

      11. +-commutative 16.9%

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

      12. associate-+l+ 16.9%

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

      13. +-commutative 16.9%

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

      14. associate-+r+ 16.9%

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

      15. metadata-eval 16.9%

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

   8. Simplified 16.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 16.9%

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

      2. frac-add 17.1%

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

      3. +-lft-identity 17.1%

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

      4. +-commutative 17.1%

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

      5. *-commutative 17.1%

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

      6. +-commutative 17.1%

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

      7. +-commutative 17.1%

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

      8. *-commutative 17.1%

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

      9. +-commutative 17.1%

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

   10. Applied egg-rr 17.1%

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

       1. *-commutative 17.1%

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

       2. distribute-lft-in 17.1%

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

       3. *-rgt-identity 17.1%

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

       4. distribute-rgt-in 17.1%

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

   12. Applied egg-rr 17.1%

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

   if 1.4e8 < x

   1. Initial program 66.9%

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

      1. sub-neg 66.9%

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

      2. distribute-neg-frac 66.9%

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

      3. metadata-eval 66.9%

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

      4. metadata-eval 66.9%

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

      5. metadata-eval 66.9%

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

      6. associate-/r* 66.9%

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

      7. metadata-eval 66.9%

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

      8. neg-mul-1 66.9%

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

      9. +-commutative 66.9%

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

      10. associate-+l+ 66.8%

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

      11. +-commutative 66.8%

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

      12. neg-mul-1 66.8%

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

      13. metadata-eval 66.8%

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

      14. associate-/r* 66.8%

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

      15. metadata-eval 66.8%

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

      16. metadata-eval 66.8%

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

      17. +-commutative 66.8%

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

      18. +-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in x around inf 67.0%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 140000000:\\ \;\;\;\;\frac{x \cdot \left(x + x\right) + \left(\left(x + -1\right) \cdot \left(x + 1\right)\right) \cdot -2}{x \cdot \left(x + -1\right) + x \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x} + \frac{-1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \frac{\frac{-1 + x \cdot 0.5}{x + -1}}{x \cdot -0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ x 1.0)) (/ (/ (+ -1.0 (* x 0.5)) (+ x -1.0)) (* x -0.5))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) + (((-1.0 + (x * 0.5)) / (x + -1.0)) / (x * -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) + ((((-1.0d0) + (x * 0.5d0)) / (x + (-1.0d0))) / (x * (-0.5d0)))
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) + (((-1.0 + (x * 0.5)) / (x + -1.0)) / (x * -0.5));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) + (((-1.0 + (x * 0.5)) / (x + -1.0)) / (x * -0.5))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(Float64(Float64(-1.0 + Float64(x * 0.5)) / Float64(x + -1.0)) / Float64(x * -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) + (((-1.0 + (x * 0.5)) / (x + -1.0)) / (x * -0.5));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. associate-+l- 68.4%

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

   2. sub-neg 68.4%

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

   3. +-commutative 68.4%

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

   4. sub-neg 68.4%

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

   5. distribute-neg-in 68.4%

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

   6. distribute-neg-frac 68.4%

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

   7. metadata-eval 68.4%

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

   8. remove-double-neg 68.4%

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

   9. sub-neg 68.4%

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

   10. metadata-eval 68.4%

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

3. Simplified 68.4%

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

   1. clear-num 68.4%

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

   2. frac-add 18.4%

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

   3. *-un-lft-identity 18.4%

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

   4. div-inv 18.4%

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

   5. metadata-eval 18.4%

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

   6. div-inv 18.4%

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

   7. metadata-eval 18.4%

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

6. Applied egg-rr 18.4%

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

   1. *-rgt-identity 18.4%

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

   2. +-commutative 18.4%

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

   3. associate-/l/ 68.4%

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

   4. associate-+r+ 68.4%

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

   5. *-commutative 68.4%

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

   6. distribute-lft1-in 68.4%

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

   7. metadata-eval 68.4%

      \[\leadsto \frac{1}{1 + x} + \frac{\frac{\color{blue}{0.5} \cdot x + -1}{x + -1}}{x \cdot -0.5} \]

   8. metadata-eval 68.4%

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

   9. associate-*r* 68.4%

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

   10. *-commutative 68.4%

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

   11. +-commutative 68.4%

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

   12. *-commutative 68.4%

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

   13. associate-*l* 68.4%

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

   14. metadata-eval 68.4%

       \[\leadsto \frac{1}{1 + x} + \frac{\frac{-1 + x \cdot \color{blue}{0.5}}{x + -1}}{x \cdot -0.5} \]

8. Simplified 68.4%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\frac{-1 + x \cdot 0.5}{x + -1}}{x \cdot -0.5}} \]

9. Final simplification 68.4%

   \[\leadsto \frac{1}{x + 1} + \frac{\frac{-1 + x \cdot 0.5}{x + -1}}{x \cdot -0.5} \]
10. Add Preprocessing

Alternative 5: 69.1% 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]

Derivation

1. Initial program 68.4%

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

3. Final simplification 68.4%

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

Alternative 6: 67.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.4%

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

   11. +-commutative 68.4%

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

   12. neg-mul-1 68.4%

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

   13. metadata-eval 68.4%

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

   14. associate-/r* 68.4%

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

   15. metadata-eval 68.4%

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

   16. metadata-eval 68.4%

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

   17. +-commutative 68.4%

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

   18. +-commutative 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around inf 66.7%

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

6. Final simplification 66.7%

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

Alternative 7: 67.5% 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.4%

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

   11. +-commutative 68.4%

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

   12. neg-mul-1 68.4%

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

   13. metadata-eval 68.4%

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

   14. associate-/r* 68.4%

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

   15. metadata-eval 68.4%

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

   16. metadata-eval 68.4%

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

   17. +-commutative 68.4%

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

   18. +-commutative 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around inf 66.5%

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

6. Final simplification 66.5%

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

Alternative 8: 5.0% 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.4%

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

   11. +-commutative 68.4%

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

   12. neg-mul-1 68.4%

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

   13. metadata-eval 68.4%

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

   14. associate-/r* 68.4%

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

   15. metadata-eval 68.4%

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

   16. metadata-eval 68.4%

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

   17. +-commutative 68.4%

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

   18. +-commutative 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

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

Alternative 9: 5.0% 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 68.4%

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

   1. associate-+l- 68.4%

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

   2. sub-neg 68.4%

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

   3. +-commutative 68.4%

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

   4. sub-neg 68.4%

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

   5. distribute-neg-in 68.4%

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

   6. distribute-neg-frac 68.4%

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

   7. metadata-eval 68.4%

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

   8. remove-double-neg 68.4%

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

   9. sub-neg 68.4%

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

   10. metadata-eval 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around inf 66.7%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Alternative 10: 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.4%

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

   11. +-commutative 68.4%

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

   12. neg-mul-1 68.4%

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

   13. metadata-eval 68.4%

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

   14. associate-/r* 68.4%

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

   15. metadata-eval 68.4%

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

   16. metadata-eval 68.4%

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

   17. +-commutative 68.4%

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

   18. +-commutative 68.4%

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

3. Simplified 68.4%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 99.2% 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 2024010 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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