3frac (problem 3.3.3)

Percentage Accurate: 84.3% → 99.7%

Time: 10.2s

Alternatives: 11

Speedup: 1.0×

Specification

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: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{-2}{x + -1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x))))
   (if (<= x -7e+89)
     (/ (/ -2.0 (+ x -1.0)) t_0)
     (/ 2.0 (- (* x (+ x -1.0)) (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -7e+89) {
		tmp = (-2.0 / (x + -1.0)) / t_0;
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (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)
    if (x <= (-7d+89)) then
        tmp = ((-2.0d0) / (x + (-1.0d0))) / t_0
    else
        tmp = 2.0d0 / ((x * (x + (-1.0d0))) - (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -7e+89) {
		tmp = (-2.0 / (x + -1.0)) / t_0;
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (1.0 - x)
	tmp = 0
	if x <= -7e+89:
		tmp = (-2.0 / (x + -1.0)) / t_0
	else:
		tmp = 2.0 / ((x * (x + -1.0)) - (x * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 - x))
	tmp = 0.0
	if (x <= -7e+89)
		tmp = Float64(Float64(-2.0 / Float64(x + -1.0)) / t_0);
	else
		tmp = Float64(2.0 / Float64(Float64(x * Float64(x + -1.0)) - Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (1.0 - x);
	tmp = 0.0;
	if (x <= -7e+89)
		tmp = (-2.0 / (x + -1.0)) / t_0;
	else
		tmp = 2.0 / ((x * (x + -1.0)) - (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+89], N[(N[(-2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(2.0 / N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000001e89

   1. Initial program 93.2%

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

      1. remove-double-neg 93.2%

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

      2. sub-neg 93.2%

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

      3. sub-neg 93.2%

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

      4. distribute-neg-frac 93.2%

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

      5. metadata-eval 93.2%

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

      6. metadata-eval 93.2%

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

      7. metadata-eval 93.2%

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

      8. associate-/r* 93.2%

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

      9. metadata-eval 93.2%

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

      10. neg-mul-1 93.2%

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

      11. associate--l+ 93.2%

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

      12. +-commutative 93.2%

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

      13. distribute-neg-frac 93.2%

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

      14. metadata-eval 93.2%

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

      15. metadata-eval 93.2%

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

      16. metadata-eval 93.2%

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

      17. associate-/r* 93.2%

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

      18. metadata-eval 93.2%

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

      19. neg-mul-1 93.2%

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

      20. sub0-neg 93.2%

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

      21. associate-+l- 93.2%

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

      22. neg-sub0 93.2%

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

   3. Simplified 93.2%

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

      1. frac-2neg 93.2%

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

      2. metadata-eval 93.2%

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

      3. frac-sub 17.2%

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

      4. frac-add 20.1%

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

      5. fma-def 20.1%

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

      6. +-commutative 20.1%

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

      7. distribute-neg-in 20.1%

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

      8. neg-mul-1 20.1%

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

      9. metadata-eval 20.1%

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

      10. fma-def 20.1%

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

      11. *-rgt-identity 20.1%

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

      12. +-commutative 20.1%

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

      13. distribute-neg-in 20.1%

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

      14. neg-mul-1 20.1%

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

      15. metadata-eval 20.1%

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

      16. fma-def 20.1%

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

   5. Applied egg-rr 20.1%

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

   6. Taylor expanded in x around 0 98.3%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-udef 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. expm1-def 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-log1p 98.3%

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

      3. associate-/r* 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. neg-sub0 100.0%

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

       2. distribute-neg-frac 100.0%

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

       3. metadata-eval 100.0%

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

       4. +-commutative 100.0%

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

   14. Simplified 100.0%

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

   if -7.0000000000000001e89 < x

   1. Initial program 82.0%

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

      1. remove-double-neg 82.0%

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

      2. sub-neg 82.0%

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

      3. sub-neg 82.0%

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

      4. distribute-neg-frac 82.0%

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

      5. metadata-eval 82.0%

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

      6. metadata-eval 82.0%

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

      7. metadata-eval 82.0%

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

      8. associate-/r* 82.0%

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

      9. metadata-eval 82.0%

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

      10. neg-mul-1 82.0%

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

      11. associate--l+ 82.0%

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

      12. +-commutative 82.0%

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

      13. distribute-neg-frac 82.0%

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

      14. metadata-eval 82.0%

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

      15. metadata-eval 82.0%

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

      16. metadata-eval 82.0%

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

      17. associate-/r* 82.0%

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

      18. metadata-eval 82.0%

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

      19. neg-mul-1 82.0%

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

      20. sub0-neg 82.0%

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

      21. associate-+l- 82.0%

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

      22. neg-sub0 82.0%

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

   3. Simplified 82.0%

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

      1. frac-2neg 82.0%

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

      2. metadata-eval 82.0%

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

      3. frac-sub 68.7%

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

      4. frac-add 68.6%

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

      5. fma-def 68.6%

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

      6. +-commutative 68.6%

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

      7. distribute-neg-in 68.6%

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

      8. neg-mul-1 68.6%

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

      9. metadata-eval 68.6%

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

      10. fma-def 68.6%

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

      11. *-rgt-identity 68.6%

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

      12. +-commutative 68.6%

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

      13. distribute-neg-in 68.6%

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

      14. neg-mul-1 68.6%

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

      15. metadata-eval 68.6%

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

      16. fma-def 68.6%

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

   5. Applied egg-rr 68.6%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. neg-mul-1 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{-2}{x + -1}}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg 84.3%

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

   2. sub-neg 84.3%

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

   3. sub-neg 84.3%

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

   4. distribute-neg-frac 84.3%

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

   5. metadata-eval 84.3%

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

   6. metadata-eval 84.3%

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

   7. metadata-eval 84.3%

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

   8. associate-/r* 84.3%

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

   9. metadata-eval 84.3%

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

   10. neg-mul-1 84.3%

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

   11. associate--l+ 84.3%

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

   12. +-commutative 84.3%

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

   13. distribute-neg-frac 84.3%

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

   14. metadata-eval 84.3%

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

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

   17. associate-/r* 84.3%

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

   18. metadata-eval 84.3%

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

   19. neg-mul-1 84.3%

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

   20. sub0-neg 84.3%

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

   21. associate-+l- 84.3%

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

   22. neg-sub0 84.3%

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

3. Simplified 84.3%

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

   1. frac-2neg 84.3%

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

   2. metadata-eval 84.3%

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

   3. frac-sub 58.2%

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

   4. frac-add 58.7%

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

   5. fma-def 58.7%

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

   6. +-commutative 58.7%

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

   7. distribute-neg-in 58.7%

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

   8. neg-mul-1 58.7%

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

   9. metadata-eval 58.7%

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

   10. fma-def 58.7%

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

   11. *-rgt-identity 58.7%

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

   12. +-commutative 58.7%

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

   13. distribute-neg-in 58.7%

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

   14. neg-mul-1 58.7%

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

   15. metadata-eval 58.7%

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

   16. fma-def 58.7%

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

5. Applied egg-rr 58.7%

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

6. Taylor expanded in x around 0 99.6%

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

   1. expm1-log1p-u 70.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

   2. expm1-udef 55.5%

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

8. Applied egg-rr 55.5%

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

   1. expm1-def 70.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

   2. expm1-log1p 99.6%

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

   3. associate-/r* 99.9%

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

10. Simplified 99.9%

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

12. Applied egg-rr 99.9%

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

    1. associate-*r* 99.9%

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

    2. associate-*r/ 99.9%

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

    3. metadata-eval 99.9%

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

    4. associate-*l/ 99.9%

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

    5. associate-*r/ 99.9%

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

    6. metadata-eval 99.9%

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

    7. +-commutative 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

    \[\leadsto \frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x + -1} \]

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -340000 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{\frac{-2}{x + -1}}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -340000.0) (not (<= x 450000.0)))
   (/ (/ -2.0 (+ x -1.0)) (* x (- 1.0 x)))
   (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -340000.0) || !(x <= 450000.0)) {
		tmp = (-2.0 / (x + -1.0)) / (x * (1.0 - x));
	} else {
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-340000.0d0)) .or. (.not. (x <= 450000.0d0))) then
        tmp = ((-2.0d0) / (x + (-1.0d0))) / (x * (1.0d0 - x))
    else
        tmp = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -340000.0) || !(x <= 450000.0)) {
		tmp = (-2.0 / (x + -1.0)) / (x * (1.0 - x));
	} else {
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -340000.0) or not (x <= 450000.0):
		tmp = (-2.0 / (x + -1.0)) / (x * (1.0 - x))
	else:
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -340000.0) || !(x <= 450000.0))
		tmp = Float64(Float64(-2.0 / Float64(x + -1.0)) / Float64(x * Float64(1.0 - x)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -340000.0) || ~((x <= 450000.0)))
		tmp = (-2.0 / (x + -1.0)) / (x * (1.0 - x));
	else
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -340000.0], N[Not[LessEqual[x, 450000.0]], $MachinePrecision]], N[(N[(-2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if x < -3.4e5 or 4.5e5 < x

   1. Initial program 68.1%

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

      1. remove-double-neg 68.1%

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

      2. sub-neg 68.1%

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

      3. sub-neg 68.1%

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

      4. distribute-neg-frac 68.1%

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

      5. metadata-eval 68.1%

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

      6. metadata-eval 68.1%

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

      7. metadata-eval 68.1%

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

      8. associate-/r* 68.1%

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

      9. metadata-eval 68.1%

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

      10. neg-mul-1 68.1%

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

      11. associate--l+ 68.1%

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

      12. +-commutative 68.1%

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

      13. distribute-neg-frac 68.1%

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

      14. metadata-eval 68.1%

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

      15. metadata-eval 68.1%

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

      16. metadata-eval 68.1%

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

      17. associate-/r* 68.1%

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

      18. metadata-eval 68.1%

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

      19. neg-mul-1 68.1%

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

      20. sub0-neg 68.1%

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

      21. associate-+l- 68.1%

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

      22. neg-sub0 68.1%

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

   3. Simplified 68.1%

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

      1. frac-2neg 68.1%

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

      2. metadata-eval 68.1%

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

      3. frac-sub 14.2%

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

      4. frac-add 14.8%

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

      5. fma-def 14.8%

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

      6. +-commutative 14.8%

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

      7. distribute-neg-in 14.8%

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

      8. neg-mul-1 14.8%

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

      9. metadata-eval 14.8%

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

      10. fma-def 14.8%

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

      11. *-rgt-identity 14.8%

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

      12. +-commutative 14.8%

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

      13. distribute-neg-in 14.8%

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

      14. neg-mul-1 14.8%

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

      15. metadata-eval 14.8%

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

      16. fma-def 14.8%

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

   5. Applied egg-rr 14.8%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-udef 68.1%

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

   8. Applied egg-rr 68.1%

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

      1. expm1-def 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-log1p 99.1%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   12. Applied egg-rr 99.2%

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

       1. neg-sub0 99.2%

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

       2. distribute-neg-frac 99.2%

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

       3. metadata-eval 99.2%

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

       4. +-commutative 99.2%

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

   14. Simplified 99.2%

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

   if -3.4e5 < x < 4.5e5

   1. Initial program 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -340000 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{\frac{-2}{x + -1}}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (/ -2.0 (+ x -1.0)) (* x (- 1.0 x)))
   (- (* x -2.0) (/ 2.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-2.0d0) / (x + (-1.0d0))) / (x * (1.0d0 - x))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-2.0 / (x + -1.0)) / (x * (1.0 - x))
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.1%

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

      1. remove-double-neg 68.1%

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

      2. sub-neg 68.1%

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

      3. sub-neg 68.1%

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

      4. distribute-neg-frac 68.1%

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

      5. metadata-eval 68.1%

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

      6. metadata-eval 68.1%

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

      7. metadata-eval 68.1%

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

      8. associate-/r* 68.1%

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

      9. metadata-eval 68.1%

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

      10. neg-mul-1 68.1%

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

      11. associate--l+ 68.1%

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

      12. +-commutative 68.1%

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

      13. distribute-neg-frac 68.1%

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

      14. metadata-eval 68.1%

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

      15. metadata-eval 68.1%

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

      16. metadata-eval 68.1%

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

      17. associate-/r* 68.1%

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

      18. metadata-eval 68.1%

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

      19. neg-mul-1 68.1%

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

      20. sub0-neg 68.1%

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

      21. associate-+l- 68.1%

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

      22. neg-sub0 68.1%

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

   3. Simplified 68.1%

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

      1. frac-2neg 68.1%

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

      2. metadata-eval 68.1%

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

      3. frac-sub 15.1%

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

      4. frac-add 16.2%

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

      5. fma-def 16.2%

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

      6. +-commutative 16.2%

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

      7. distribute-neg-in 16.2%

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

      8. neg-mul-1 16.2%

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

      9. metadata-eval 16.2%

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

      10. fma-def 16.2%

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

      11. *-rgt-identity 16.2%

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

      12. +-commutative 16.2%

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

      13. distribute-neg-in 16.2%

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

      14. neg-mul-1 16.2%

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

      15. metadata-eval 16.2%

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

      16. fma-def 16.2%

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

   5. Applied egg-rr 16.2%

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

   6. Taylor expanded in x around 0 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-udef 67.9%

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

   8. Applied egg-rr 67.9%

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

      1. expm1-def 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(-1, x, -1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\right)\right)} \]

      2. expm1-log1p 99.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   12. Applied egg-rr 98.2%

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

       1. neg-sub0 98.2%

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

       2. distribute-neg-frac 98.2%

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

       3. metadata-eval 98.2%

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

       4. +-commutative 98.2%

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

   14. Simplified 98.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.1%

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

Alternative 5: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (+ (/ -1.0 x) (/ 1.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ -1.0 x) (/ 1.0 (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-1.0 / x) + (1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-1.0d0) / x) + (1.0d0 / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = ((-1.0d0) / x) + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-1.0 / x) + (1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-1.0 / x) + (1.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (-1.0 / x) + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-1.0 / x) + (1.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. sub-neg 72.5%

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

      3. sub-neg 72.5%

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

      4. distribute-neg-frac 72.5%

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

      5. metadata-eval 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. associate-/r* 72.5%

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

      9. metadata-eval 72.5%

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

      10. neg-mul-1 72.5%

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

      11. associate--l+ 72.5%

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

      12. +-commutative 72.5%

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

      13. distribute-neg-frac 72.5%

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

      14. metadata-eval 72.5%

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

      15. metadata-eval 72.5%

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

      16. metadata-eval 72.5%

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

      17. associate-/r* 72.5%

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

      18. metadata-eval 72.5%

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

      19. neg-mul-1 72.5%

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

      20. sub0-neg 72.5%

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

      21. associate-+l- 72.5%

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

      22. neg-sub0 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around inf 70.4%

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

   5. Taylor expanded in x around inf 70.5%

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

   if -1 < x < 0.650000000000000022

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 0.650000000000000022 < x

   1. Initial program 62.7%

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

      1. remove-double-neg 62.7%

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

      2. sub-neg 62.7%

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

      3. sub-neg 62.7%

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

      4. distribute-neg-frac 62.7%

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

      5. metadata-eval 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

      8. associate-/r* 62.7%

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

      9. metadata-eval 62.7%

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

      10. neg-mul-1 62.7%

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

      11. associate--l+ 62.7%

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

      12. +-commutative 62.7%

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

      13. distribute-neg-frac 62.7%

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

      14. metadata-eval 62.7%

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

      15. metadata-eval 62.7%

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

      16. metadata-eval 62.7%

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

      17. associate-/r* 62.7%

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

      18. metadata-eval 62.7%

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

      19. neg-mul-1 62.7%

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

      20. sub0-neg 62.7%

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

      21. associate-+l- 62.7%

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

      22. neg-sub0 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   6. Applied egg-rr 7.7%

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

      1. fma-udef 36.2%

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

      2. pow-sqr 63.1%

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

      3. metadata-eval 63.1%

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

      4. metadata-eval 63.1%

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

      5. +-commutative 63.1%

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

      6. metadata-eval 63.1%

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

      7. unpow-1 63.1%

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

      8. +-commutative 63.1%

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

   8. Simplified 63.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \]

Alternative 6: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (+ (/ -1.0 x) (/ 1.0 (+ x 1.0)))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ -1.0 x) (/ 1.0 (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (-1.0 / x) + (1.0 / (x + 1.0));
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = ((-1.0d0) / x) + (1.0d0 / (x + 1.0d0))
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = ((-1.0d0) / x) + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (-1.0 / x) + (1.0 / (x + 1.0));
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = (-1.0 / x) + (1.0 / (x + 1.0))
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (-1.0 / x) + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = (-1.0 / x) + (1.0 / (x + 1.0));
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. sub-neg 72.5%

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

      3. sub-neg 72.5%

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

      4. distribute-neg-frac 72.5%

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

      5. metadata-eval 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. associate-/r* 72.5%

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

      9. metadata-eval 72.5%

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

      10. neg-mul-1 72.5%

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

      11. associate--l+ 72.5%

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

      12. +-commutative 72.5%

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

      13. distribute-neg-frac 72.5%

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

      14. metadata-eval 72.5%

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

      15. metadata-eval 72.5%

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

      16. metadata-eval 72.5%

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

      17. associate-/r* 72.5%

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

      18. metadata-eval 72.5%

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

      19. neg-mul-1 72.5%

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

      20. sub0-neg 72.5%

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

      21. associate-+l- 72.5%

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

      22. neg-sub0 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around inf 70.8%

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

   if -0.650000000000000022 < x < 0.650000000000000022

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 0.650000000000000022 < x

   1. Initial program 62.7%

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

      1. remove-double-neg 62.7%

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

      2. sub-neg 62.7%

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

      3. sub-neg 62.7%

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

      4. distribute-neg-frac 62.7%

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

      5. metadata-eval 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

      8. associate-/r* 62.7%

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

      9. metadata-eval 62.7%

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

      10. neg-mul-1 62.7%

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

      11. associate--l+ 62.7%

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

      12. +-commutative 62.7%

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

      13. distribute-neg-frac 62.7%

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

      14. metadata-eval 62.7%

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

      15. metadata-eval 62.7%

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

      16. metadata-eval 62.7%

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

      17. associate-/r* 62.7%

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

      18. metadata-eval 62.7%

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

      19. neg-mul-1 62.7%

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

      20. sub0-neg 62.7%

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

      21. associate-+l- 62.7%

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

      22. neg-sub0 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   6. Applied egg-rr 7.7%

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

      1. fma-udef 36.2%

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

      2. pow-sqr 63.1%

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

      3. metadata-eval 63.1%

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

      4. metadata-eval 63.1%

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

      5. +-commutative 63.1%

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

      6. metadata-eval 63.1%

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

      7. unpow-1 63.1%

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

      8. +-commutative 63.1%

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

   8. Simplified 63.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \]

Alternative 7: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x + -1} - \frac{-1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ -1.0 (+ x -1.0)) (/ -1.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ -1.0 x) (/ 1.0 (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-1.0 / (x + -1.0)) - (-1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-1.0d0) / (x + (-1.0d0))) - ((-1.0d0) / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = ((-1.0d0) / x) + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-1.0 / (x + -1.0)) - (-1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-1.0 / (x + -1.0)) - (-1.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (-1.0 / x) + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) - Float64(-1.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-1.0 / (x + -1.0)) - (-1.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. sub-neg 72.5%

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

      3. sub-neg 72.5%

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

      4. distribute-neg-frac 72.5%

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

      5. metadata-eval 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. associate-/r* 72.5%

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

      9. metadata-eval 72.5%

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

      10. neg-mul-1 72.5%

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

      11. associate--l+ 72.5%

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

      12. +-commutative 72.5%

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

      13. distribute-neg-frac 72.5%

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

      14. metadata-eval 72.5%

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

      15. metadata-eval 72.5%

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

      16. metadata-eval 72.5%

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

      17. associate-/r* 72.5%

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

      18. metadata-eval 72.5%

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

      19. neg-mul-1 72.5%

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

      20. sub0-neg 72.5%

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

      21. associate-+l- 72.5%

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

      22. neg-sub0 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in x around inf 70.8%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 20.1%

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

      3. frac-times 15.8%

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

      4. metadata-eval 15.8%

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

      5. metadata-eval 15.8%

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

      6. frac-times 20.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 6.0%

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

      9. *-un-lft-identity 6.0%

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

      10. metadata-eval 6.0%

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

      11. cancel-sign-sub-inv 6.0%

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

   6. Applied egg-rr 70.8%

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

      1. +-commutative 70.8%

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

   8. Simplified 70.8%

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

   if -1 < x < 0.650000000000000022

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 0.650000000000000022 < x

   1. Initial program 62.7%

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

      1. remove-double-neg 62.7%

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

      2. sub-neg 62.7%

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

      3. sub-neg 62.7%

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

      4. distribute-neg-frac 62.7%

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

      5. metadata-eval 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

      8. associate-/r* 62.7%

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

      9. metadata-eval 62.7%

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

      10. neg-mul-1 62.7%

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

      11. associate--l+ 62.7%

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

      12. +-commutative 62.7%

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

      13. distribute-neg-frac 62.7%

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

      14. metadata-eval 62.7%

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

      15. metadata-eval 62.7%

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

      16. metadata-eval 62.7%

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

      17. associate-/r* 62.7%

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

      18. metadata-eval 62.7%

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

      19. neg-mul-1 62.7%

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

      20. sub0-neg 62.7%

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

      21. associate-+l- 62.7%

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

      22. neg-sub0 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   6. Applied egg-rr 7.7%

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

      1. fma-udef 36.2%

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

      2. pow-sqr 63.1%

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

      3. metadata-eval 63.1%

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

      4. metadata-eval 63.1%

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

      5. +-commutative 63.1%

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

      6. metadata-eval 63.1%

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

      7. unpow-1 63.1%

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

      8. +-commutative 63.1%

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

   8. Simplified 63.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x + -1} - \frac{-1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \]

Alternative 8: 82.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (+ (/ -1.0 x) (/ 1.0 x))
   (- (- (/ 2.0 x)) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-1.0 / x) + (1.0 / x);
	} else {
		tmp = -(2.0 / x) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-1.0d0) / x) + (1.0d0 / x)
    else
        tmp = -(2.0d0 / x) - x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-1.0 / x) + (1.0 / x)
	else:
		tmp = -(2.0 / x) - x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-1.0 / x) + (1.0 / x);
	else
		tmp = -(2.0 / x) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[((-N[(2.0 / x), $MachinePrecision]) - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.1%

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

      1. remove-double-neg 68.1%

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

      2. sub-neg 68.1%

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

      3. sub-neg 68.1%

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

      4. distribute-neg-frac 68.1%

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

      5. metadata-eval 68.1%

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

      6. metadata-eval 68.1%

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

      7. metadata-eval 68.1%

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

      8. associate-/r* 68.1%

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

      9. metadata-eval 68.1%

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

      10. neg-mul-1 68.1%

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

      11. associate--l+ 68.1%

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

      12. +-commutative 68.1%

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

      13. distribute-neg-frac 68.1%

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

      14. metadata-eval 68.1%

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

      15. metadata-eval 68.1%

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

      16. metadata-eval 68.1%

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

      17. associate-/r* 68.1%

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

      18. metadata-eval 68.1%

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

      19. neg-mul-1 68.1%

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

      20. sub0-neg 68.1%

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

      21. associate-+l- 68.1%

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

      22. neg-sub0 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around inf 67.2%

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

   5. Taylor expanded in x around inf 67.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 83.6%

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

Alternative 9: 83.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (+ (/ -1.0 x) (/ 1.0 x))
   (- (* x -2.0) (/ 2.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-1.0d0) / x) + (1.0d0 / x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-1.0 / x) + (1.0 / x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.1%

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

      1. remove-double-neg 68.1%

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

      2. sub-neg 68.1%

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

      3. sub-neg 68.1%

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

      4. distribute-neg-frac 68.1%

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

      5. metadata-eval 68.1%

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

      6. metadata-eval 68.1%

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

      7. metadata-eval 68.1%

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

      8. associate-/r* 68.1%

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

      9. metadata-eval 68.1%

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

      10. neg-mul-1 68.1%

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

      11. associate--l+ 68.1%

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

      12. +-commutative 68.1%

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

      13. distribute-neg-frac 68.1%

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

      14. metadata-eval 68.1%

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

      15. metadata-eval 68.1%

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

      16. metadata-eval 68.1%

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

      17. associate-/r* 68.1%

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

      18. metadata-eval 68.1%

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

      19. neg-mul-1 68.1%

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

      20. sub0-neg 68.1%

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

      21. associate-+l- 68.1%

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

      22. neg-sub0 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around inf 67.2%

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

   5. Taylor expanded in x around inf 67.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 83.8%

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

Alternative 10: 51.8% 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 84.3%

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

   1. remove-double-neg 84.3%

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

   2. sub-neg 84.3%

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

   3. sub-neg 84.3%

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

   4. distribute-neg-frac 84.3%

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

   5. metadata-eval 84.3%

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

   6. metadata-eval 84.3%

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

   7. metadata-eval 84.3%

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

   8. associate-/r* 84.3%

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

   9. metadata-eval 84.3%

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

   10. neg-mul-1 84.3%

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

   11. associate--l+ 84.3%

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

   12. +-commutative 84.3%

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

   13. distribute-neg-frac 84.3%

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

   14. metadata-eval 84.3%

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

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

   17. associate-/r* 84.3%

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

   18. metadata-eval 84.3%

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

   19. neg-mul-1 84.3%

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

   20. sub0-neg 84.3%

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

   21. associate-+l- 84.3%

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

   22. neg-sub0 84.3%

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

3. Simplified 84.3%

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

4. Taylor expanded in x around 0 53.0%

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

5. Final simplification 53.0%

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

Alternative 11: 3.3% 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 84.3%

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

   1. remove-double-neg 84.3%

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

   2. sub-neg 84.3%

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

   3. sub-neg 84.3%

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

   4. distribute-neg-frac 84.3%

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

   5. metadata-eval 84.3%

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

   6. metadata-eval 84.3%

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

   7. metadata-eval 84.3%

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

   8. associate-/r* 84.3%

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

   9. metadata-eval 84.3%

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

   10. neg-mul-1 84.3%

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

   11. associate--l+ 84.3%

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

   12. +-commutative 84.3%

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

   13. distribute-neg-frac 84.3%

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

   14. metadata-eval 84.3%

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

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

   17. associate-/r* 84.3%

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

   18. metadata-eval 84.3%

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

   19. neg-mul-1 84.3%

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

   20. sub0-neg 84.3%

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

   21. associate-+l- 84.3%

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

   22. neg-sub0 84.3%

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

3. Simplified 84.3%

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

4. Taylor expanded in x around 0 52.2%

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

5. Taylor expanded in x around inf 3.2%

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

6. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023305 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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