Asymptote A

Percentage Accurate: 77.1% → 99.9%

Time: 7.1s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{1 - x\_m}}{-1 - x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ -2.0 (- 1.0 x_m)) (- -1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(-2.0 / Float64(1.0 - x_m)) / Float64(-1.0 - x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. sub-neg 80.0%

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

   2. +-commutative 80.0%

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

   3. distribute-neg-frac2 80.0%

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

   4. neg-sub0 80.0%

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

   5. associate-+l- 80.0%

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

   6. neg-sub0 80.0%

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

   7. remove-double-neg 80.0%

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

   8. distribute-neg-in 80.0%

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

   9. sub-neg 80.0%

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

   10. distribute-neg-frac2 80.0%

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

   11. sub-neg 80.0%

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

   12. +-commutative 80.0%

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

   13. unsub-neg 80.0%

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

   14. sub-neg 80.0%

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

   15. +-commutative 80.0%

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

   16. unsub-neg 80.0%

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

   17. metadata-eval 80.0%

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

3. Simplified 80.0%

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

   1. sub-neg 80.0%

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

   2. distribute-neg-frac 80.0%

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

   3. metadata-eval 80.0%

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

6. Applied egg-rr 80.0%

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

   1. metadata-eval 80.0%

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

   2. distribute-neg-frac 80.0%

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

   3. unsub-neg 80.0%

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

   4. *-rgt-identity 80.0%

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

   5. *-inverses 80.0%

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

   6. associate-/r* 54.4%

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

   7. *-commutative 54.4%

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

   8. *-lft-identity 54.4%

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

   9. associate-/r* 80.0%

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

   10. associate-*r/ 80.0%

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

   11. associate-*l/ 80.0%

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

   12. distribute-lft-out-- 80.1%

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

   13. *-inverses 80.1%

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

   14. div-sub 80.5%

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

   15. associate--r+ 83.1%

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

   16. *-commutative 83.1%

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

   17. times-frac 83.1%

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

8. Simplified 99.2%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

10. Applied egg-rr 99.8%

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

    1. un-div-inv 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

    \[\leadsto \frac{\frac{-2}{1 - x}}{-1 - x} \]
14. Add Preprocessing

Alternative 2: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x\_m \cdot \left(x\_m + -1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.76) 2.0 (/ -2.0 (* x_m (+ x_m -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x_m * (x_m + -1.0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.76d0) then
        tmp = 2.0d0
    else
        tmp = (-2.0d0) / (x_m * (x_m + (-1.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x_m * (x_m + -1.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.76:
		tmp = 2.0
	else:
		tmp = -2.0 / (x_m * (x_m + -1.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.76)
		tmp = 2.0;
	else
		tmp = Float64(-2.0 / Float64(x_m * Float64(x_m + -1.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.76)
		tmp = 2.0;
	else
		tmp = -2.0 / (x_m * (x_m + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.76], 2.0, N[(-2.0 / N[(x$95$m * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 88.5%

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

      1. sub-neg 88.5%

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

      2. +-commutative 88.5%

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

      3. distribute-neg-frac2 88.5%

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

      4. neg-sub0 88.5%

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

      5. associate-+l- 88.5%

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

      6. neg-sub0 88.5%

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

      7. remove-double-neg 88.5%

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

      8. distribute-neg-in 88.5%

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

      9. sub-neg 88.5%

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

      10. distribute-neg-frac2 88.5%

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

      11. sub-neg 88.5%

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

      12. +-commutative 88.5%

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

      13. unsub-neg 88.5%

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

      14. sub-neg 88.5%

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

      15. +-commutative 88.5%

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

      16. unsub-neg 88.5%

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

      17. metadata-eval 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around 0 66.7%

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

   if 0.76000000000000001 < x

   1. Initial program 56.2%

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

      1. sub-neg 56.2%

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

      2. +-commutative 56.2%

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

      3. distribute-neg-frac2 56.2%

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

      4. neg-sub0 56.2%

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

      5. associate-+l- 56.2%

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

      6. neg-sub0 56.2%

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

      7. remove-double-neg 56.2%

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

      8. distribute-neg-in 56.2%

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

      9. sub-neg 56.2%

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

      10. distribute-neg-frac2 56.2%

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

      11. sub-neg 56.2%

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

      12. +-commutative 56.2%

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

      13. unsub-neg 56.2%

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

      14. sub-neg 56.2%

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

      15. +-commutative 56.2%

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

      16. unsub-neg 56.2%

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

      17. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

      1. sub-neg 56.2%

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

      2. distribute-neg-frac 56.2%

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

      3. metadata-eval 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. metadata-eval 56.2%

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

      2. distribute-neg-frac 56.2%

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

      3. unsub-neg 56.2%

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

      4. *-rgt-identity 56.2%

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

      5. *-inverses 56.2%

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

      6. associate-/r* 11.2%

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

      7. *-commutative 11.2%

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

      8. *-lft-identity 11.2%

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

      9. associate-/r* 56.1%

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

      10. associate-*r/ 56.1%

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

      11. associate-*l/ 56.2%

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

      12. distribute-lft-out-- 56.3%

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

      13. *-inverses 56.3%

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

      14. div-sub 57.8%

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

      15. associate--r+ 63.6%

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

      16. *-commutative 63.6%

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

      17. times-frac 63.6%

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

   8. Simplified 99.8%

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

      1. flip3-- 38.4%

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

      2. associate-*r/ 34.2%

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

      3. metadata-eval 34.2%

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

      4. metadata-eval 34.2%

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

      5. distribute-rgt-out 34.2%

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

   10. Applied egg-rr 34.2%

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

       1. associate-/l* 38.4%

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

   12. Simplified 38.4%

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

   13. Taylor expanded in x around inf 95.7%

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

       1. neg-mul-1 95.7%

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

   15. Simplified 95.7%

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

   16. Taylor expanded in x around 0 95.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{1 + x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0) 2.0 (/ (/ -2.0 (+ 1.0 x_m)) x_m)))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / (1.0 + x_m)) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = (-2.0 / (1.0 + x_m)) / x_m
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], 2.0, N[(N[(-2.0 / N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.5%

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

      1. sub-neg 88.5%

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

      2. +-commutative 88.5%

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

      3. distribute-neg-frac2 88.5%

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

      4. neg-sub0 88.5%

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

      5. associate-+l- 88.5%

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

      6. neg-sub0 88.5%

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

      7. remove-double-neg 88.5%

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

      8. distribute-neg-in 88.5%

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

      9. sub-neg 88.5%

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

      10. distribute-neg-frac2 88.5%

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

      11. sub-neg 88.5%

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

      12. +-commutative 88.5%

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

      13. unsub-neg 88.5%

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

      14. sub-neg 88.5%

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

      15. +-commutative 88.5%

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

      16. unsub-neg 88.5%

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

      17. metadata-eval 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around 0 66.7%

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

   if 1 < x

   1. Initial program 56.2%

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

      1. sub-neg 56.2%

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

      2. +-commutative 56.2%

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

      3. distribute-neg-frac2 56.2%

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

      4. neg-sub0 56.2%

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

      5. associate-+l- 56.2%

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

      6. neg-sub0 56.2%

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

      7. remove-double-neg 56.2%

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

      8. distribute-neg-in 56.2%

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

      9. sub-neg 56.2%

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

      10. distribute-neg-frac2 56.2%

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

      11. sub-neg 56.2%

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

      12. +-commutative 56.2%

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

      13. unsub-neg 56.2%

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

      14. sub-neg 56.2%

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

      15. +-commutative 56.2%

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

      16. unsub-neg 56.2%

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

      17. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

      1. sub-neg 56.2%

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

      2. distribute-neg-frac 56.2%

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

      3. metadata-eval 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. metadata-eval 56.2%

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

      2. distribute-neg-frac 56.2%

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

      3. unsub-neg 56.2%

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

      4. *-rgt-identity 56.2%

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

      5. *-inverses 56.2%

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

      6. associate-/r* 11.2%

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

      7. *-commutative 11.2%

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

      8. *-lft-identity 11.2%

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

      9. associate-/r* 56.1%

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

      10. associate-*r/ 56.1%

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

      11. associate-*l/ 56.2%

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

      12. distribute-lft-out-- 56.3%

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

      13. *-inverses 56.3%

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

      14. div-sub 57.8%

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

      15. associate--r+ 63.6%

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

      16. *-commutative 63.6%

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

      17. times-frac 63.6%

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

   8. Simplified 99.8%

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

      1. flip3-- 38.4%

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

      2. associate-*r/ 34.2%

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

      3. metadata-eval 34.2%

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

      4. metadata-eval 34.2%

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

      5. distribute-rgt-out 34.2%

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

   10. Applied egg-rr 34.2%

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

       1. associate-/l* 38.4%

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

   12. Simplified 38.4%

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

   13. Taylor expanded in x around inf 95.7%

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

       1. neg-mul-1 95.7%

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

   15. Simplified 95.7%

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

       1. associate-/r* 95.8%

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

       2. div-inv 95.6%

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

       3. sub-neg 95.6%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 49.7%

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

       6. sqr-neg 49.7%

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

       7. sqrt-unprod 49.7%

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

       8. add-sqr-sqrt 49.7%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 95.6%

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

       11. sqr-neg 95.6%

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

       12. sqrt-unprod 95.5%

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

       13. add-sqr-sqrt 95.6%

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

   17. Applied egg-rr 95.6%

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

       1. associate-*r/ 95.8%

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

       2. *-rgt-identity 95.8%

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

       3. +-commutative 95.8%

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

   19. Simplified 95.8%

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

4. Final simplification 74.3%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ -2.0 (* (- 1.0 x_m) (- -1.0 x_m))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return -2.0 / ((1.0 - x_m) * (-1.0 - x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return -2.0 / ((1.0 - x_m) * (-1.0 - x_m))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-2.0 / N[(N[(1.0 - x$95$m), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. sub-neg 80.0%

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

   2. +-commutative 80.0%

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

   3. distribute-neg-frac2 80.0%

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

   4. neg-sub0 80.0%

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

   5. associate-+l- 80.0%

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

   6. neg-sub0 80.0%

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

   7. remove-double-neg 80.0%

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

   8. distribute-neg-in 80.0%

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

   9. sub-neg 80.0%

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

   10. distribute-neg-frac2 80.0%

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

   11. sub-neg 80.0%

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

   12. +-commutative 80.0%

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

   13. unsub-neg 80.0%

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

   14. sub-neg 80.0%

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

   15. +-commutative 80.0%

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

   16. unsub-neg 80.0%

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

   17. metadata-eval 80.0%

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

3. Simplified 80.0%

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

   1. sub-neg 80.0%

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

   2. distribute-neg-frac 80.0%

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

   3. metadata-eval 80.0%

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

6. Applied egg-rr 80.0%

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

   1. metadata-eval 80.0%

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

   2. distribute-neg-frac 80.0%

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

   3. unsub-neg 80.0%

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

   4. *-rgt-identity 80.0%

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

   5. *-inverses 80.0%

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

   6. associate-/r* 54.4%

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

   7. *-commutative 54.4%

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

   8. *-lft-identity 54.4%

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

   9. associate-/r* 80.0%

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

   10. associate-*r/ 80.0%

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

   11. associate-*l/ 80.0%

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

   12. distribute-lft-out-- 80.1%

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

   13. *-inverses 80.1%

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

   14. div-sub 80.5%

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

   15. associate--r+ 83.1%

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

   16. *-commutative 83.1%

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

   17. times-frac 83.1%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 53.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.0) 2.0 (/ -1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -1.0 / x_m;
	}
	return tmp;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -1.0 / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = -1.0 / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(-1.0 / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 2.0;
	else
		tmp = -1.0 / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], 2.0, N[(-1.0 / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.5%

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

      1. sub-neg 88.5%

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

      2. +-commutative 88.5%

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

      3. distribute-neg-frac2 88.5%

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

      4. neg-sub0 88.5%

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

      5. associate-+l- 88.5%

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

      6. neg-sub0 88.5%

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

      7. remove-double-neg 88.5%

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

      8. distribute-neg-in 88.5%

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

      9. sub-neg 88.5%

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

      10. distribute-neg-frac2 88.5%

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

      11. sub-neg 88.5%

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

      12. +-commutative 88.5%

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

      13. unsub-neg 88.5%

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

      14. sub-neg 88.5%

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

      15. +-commutative 88.5%

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

      16. unsub-neg 88.5%

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

      17. metadata-eval 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around 0 66.7%

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

   if 1 < x

   1. Initial program 56.2%

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

      1. sub-neg 56.2%

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

      2. +-commutative 56.2%

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

      3. distribute-neg-frac2 56.2%

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

      4. neg-sub0 56.2%

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

      5. associate-+l- 56.2%

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

      6. neg-sub0 56.2%

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

      7. remove-double-neg 56.2%

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

      8. distribute-neg-in 56.2%

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

      9. sub-neg 56.2%

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

      10. distribute-neg-frac2 56.2%

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

      11. sub-neg 56.2%

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

      12. +-commutative 56.2%

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

      13. unsub-neg 56.2%

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

      14. sub-neg 56.2%

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

      15. +-commutative 56.2%

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

      16. unsub-neg 56.2%

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

      17. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in x around 0 2.6%

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

   6. Taylor expanded in x around inf 2.6%

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

   7. Taylor expanded in x around 0 7.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 80.0%

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

   1. sub-neg 80.0%

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

   2. +-commutative 80.0%

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

   3. distribute-neg-frac2 80.0%

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

   4. neg-sub0 80.0%

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

   5. associate-+l- 80.0%

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

   6. neg-sub0 80.0%

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

   7. remove-double-neg 80.0%

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

   8. distribute-neg-in 80.0%

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

   9. sub-neg 80.0%

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

   10. distribute-neg-frac2 80.0%

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

   11. sub-neg 80.0%

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

   12. +-commutative 80.0%

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

   13. unsub-neg 80.0%

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

   14. sub-neg 80.0%

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

   15. +-commutative 80.0%

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

   16. unsub-neg 80.0%

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

   17. metadata-eval 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in x around 0 49.4%

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

6. Taylor expanded in x around inf 10.6%

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

7. Final simplification 10.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 7: 51.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

x_m = abs(x)
function code(x_m)
	return 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 80.0%

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

   1. sub-neg 80.0%

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

   2. +-commutative 80.0%

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

   3. distribute-neg-frac2 80.0%

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

   4. neg-sub0 80.0%

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

   5. associate-+l- 80.0%

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

   6. neg-sub0 80.0%

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

   7. remove-double-neg 80.0%

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

   8. distribute-neg-in 80.0%

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

   9. sub-neg 80.0%

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

   10. distribute-neg-frac2 80.0%

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

   11. sub-neg 80.0%

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

   12. +-commutative 80.0%

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

   13. unsub-neg 80.0%

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

   14. sub-neg 80.0%

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

   15. +-commutative 80.0%

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

   16. unsub-neg 80.0%

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

   17. metadata-eval 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in x around 0 49.9%

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

6. Final simplification 49.9%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))