Asymptote A

Percentage Accurate: 77.2% → 99.9%

Time: 7.2s

Alternatives: 6

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: 77.2% 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 76.8%

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

   1. sub-neg 76.8%

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

   2. +-commutative 76.8%

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

   3. distribute-neg-frac2 76.8%

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

   4. neg-sub0 76.8%

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

   5. associate-+l- 76.8%

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

   6. neg-sub0 76.8%

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

   7. remove-double-neg 76.8%

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

   8. distribute-neg-in 76.8%

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

   9. sub-neg 76.8%

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

   10. distribute-neg-frac2 76.8%

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

   11. sub-neg 76.8%

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

   12. +-commutative 76.8%

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

   13. unsub-neg 76.8%

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

   14. sub-neg 76.8%

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

   15. +-commutative 76.8%

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

   16. unsub-neg 76.8%

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

   17. metadata-eval 76.8%

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

3. Simplified 76.8%

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

   1. sub-neg 76.8%

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

   2. distribute-neg-frac 76.8%

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

   3. metadata-eval 76.8%

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

6. Applied egg-rr 76.8%

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

   1. metadata-eval 76.8%

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

   2. distribute-neg-frac 76.8%

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

   3. unsub-neg 76.8%

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

   4. *-rgt-identity 76.8%

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

   5. *-inverses 76.8%

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

   6. associate-/r* 54.1%

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

   7. neg-mul-1 54.1%

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

   8. times-frac 76.8%

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

   9. distribute-frac-neg2 76.8%

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

   10. distribute-neg-frac 76.8%

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

   11. metadata-eval 76.8%

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

   12. distribute-lft-out-- 76.8%

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

   13. *-inverses 76.8%

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

   14. div-sub 77.4%

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

   15. associate--r+ 80.6%

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

   16. associate-*r/ 80.6%

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

   17. associate-*l/ 80.6%

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

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%

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

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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x\_m \cdot \left(1 - x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.75) 2.0 (/ 2.0 (* x_m (- 1.0 x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x_m * (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 <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = 2.0d0 / (x_m * (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 <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x_m * (1.0 - x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.75:
		tmp = 2.0
	else:
		tmp = 2.0 / (x_m * (1.0 - x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(2.0 / Float64(x_m * 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 <= 0.75)
		tmp = 2.0;
	else
		tmp = 2.0 / (x_m * (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, 0.75], 2.0, N[(2.0 / N[(x$95$m * N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac2 86.4%

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

      4. neg-sub0 86.4%

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

      5. associate-+l- 86.4%

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

      6. neg-sub0 86.4%

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

      7. remove-double-neg 86.4%

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

      8. distribute-neg-in 86.4%

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

      9. sub-neg 86.4%

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

      10. distribute-neg-frac2 86.4%

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

      11. sub-neg 86.4%

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

      12. +-commutative 86.4%

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

      13. unsub-neg 86.4%

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

      14. sub-neg 86.4%

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

      15. +-commutative 86.4%

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

      16. unsub-neg 86.4%

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

      17. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 69.8%

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

   if 0.75 < x

   1. Initial program 52.3%

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

      1. sub-neg 52.3%

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

      2. +-commutative 52.3%

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

      3. distribute-neg-frac2 52.3%

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

      4. neg-sub0 52.3%

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

      5. associate-+l- 52.3%

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

      6. neg-sub0 52.3%

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

      7. remove-double-neg 52.3%

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

      8. distribute-neg-in 52.3%

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

      9. sub-neg 52.3%

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

      10. distribute-neg-frac2 52.3%

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

      11. sub-neg 52.3%

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

      12. +-commutative 52.3%

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

      13. unsub-neg 52.3%

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

      14. sub-neg 52.3%

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

      15. +-commutative 52.3%

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

      16. unsub-neg 52.3%

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

      17. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

      1. flip-- 7.8%

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

      2. associate-/r/ 7.7%

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

      3. metadata-eval 7.7%

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

      4. pow2 7.7%

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

      5. +-commutative 7.7%

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

   6. Applied egg-rr 7.7%

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

      1. associate-*l/ 7.6%

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

      2. *-lft-identity 7.6%

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

   8. Simplified 7.6%

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

      1. clear-num 7.8%

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

      2. metadata-eval 7.8%

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

      3. unpow2 7.8%

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

      4. +-commutative 7.8%

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

      5. flip-- 52.3%

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

      6. frac-sub 53.0%

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

      7. *-un-lft-identity 53.0%

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

      8. *-commutative 53.0%

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

      9. *-un-lft-identity 53.0%

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

      10. associate--r+ 59.4%

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

      11. *-commutative 59.4%

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

      12. associate-/r* 59.6%

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

      13. +-commutative 59.6%

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

      14. associate--r+ 53.0%

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

   10. Applied egg-rr 53.0%

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

   11. Taylor expanded in x around inf 98.3%

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

       1. *-un-lft-identity 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. *-lft-identity 98.3%

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

       2. associate-/r* 96.1%

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

   15. Simplified 96.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{1 - x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.75) 2.0 (/ (/ 2.0 x_m) (- 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x_m) / (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 <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = (2.0d0 / x_m) / (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 <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x_m) / (1.0 - x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.75:
		tmp = 2.0
	else:
		tmp = (2.0 / x_m) / (1.0 - x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(Float64(2.0 / x_m) / 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 <= 0.75)
		tmp = 2.0;
	else
		tmp = (2.0 / x_m) / (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, 0.75], 2.0, N[(N[(2.0 / x$95$m), $MachinePrecision] / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac2 86.4%

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

      4. neg-sub0 86.4%

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

      5. associate-+l- 86.4%

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

      6. neg-sub0 86.4%

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

      7. remove-double-neg 86.4%

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

      8. distribute-neg-in 86.4%

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

      9. sub-neg 86.4%

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

      10. distribute-neg-frac2 86.4%

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

      11. sub-neg 86.4%

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

      12. +-commutative 86.4%

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

      13. unsub-neg 86.4%

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

      14. sub-neg 86.4%

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

      15. +-commutative 86.4%

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

      16. unsub-neg 86.4%

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

      17. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 69.8%

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

   if 0.75 < x

   1. Initial program 52.3%

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

      1. sub-neg 52.3%

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

      2. +-commutative 52.3%

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

      3. distribute-neg-frac2 52.3%

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

      4. neg-sub0 52.3%

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

      5. associate-+l- 52.3%

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

      6. neg-sub0 52.3%

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

      7. remove-double-neg 52.3%

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

      8. distribute-neg-in 52.3%

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

      9. sub-neg 52.3%

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

      10. distribute-neg-frac2 52.3%

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

      11. sub-neg 52.3%

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

      12. +-commutative 52.3%

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

      13. unsub-neg 52.3%

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

      14. sub-neg 52.3%

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

      15. +-commutative 52.3%

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

      16. unsub-neg 52.3%

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

      17. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

      1. flip-- 7.8%

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

      2. associate-/r/ 7.7%

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

      3. metadata-eval 7.7%

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

      4. pow2 7.7%

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

      5. +-commutative 7.7%

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

   6. Applied egg-rr 7.7%

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

      1. associate-*l/ 7.6%

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

      2. *-lft-identity 7.6%

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

   8. Simplified 7.6%

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

      1. clear-num 7.8%

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

      2. metadata-eval 7.8%

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

      3. unpow2 7.8%

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

      4. +-commutative 7.8%

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

      5. flip-- 52.3%

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

      6. frac-sub 53.0%

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

      7. *-un-lft-identity 53.0%

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

      8. *-commutative 53.0%

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

      9. *-un-lft-identity 53.0%

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

      10. associate--r+ 59.4%

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

      11. *-commutative 59.4%

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

      12. associate-/r* 59.6%

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

      13. +-commutative 59.6%

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

      14. associate--r+ 53.0%

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

   10. Applied egg-rr 53.0%

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

   11. Taylor expanded in x around inf 98.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \]
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 76.8%

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

   1. sub-neg 76.8%

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

   2. +-commutative 76.8%

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

   3. distribute-neg-frac2 76.8%

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

   4. neg-sub0 76.8%

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

   5. associate-+l- 76.8%

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

   6. neg-sub0 76.8%

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

   7. remove-double-neg 76.8%

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

   8. distribute-neg-in 76.8%

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

   9. sub-neg 76.8%

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

   10. distribute-neg-frac2 76.8%

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

   11. sub-neg 76.8%

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

   12. +-commutative 76.8%

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

   13. unsub-neg 76.8%

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

   14. sub-neg 76.8%

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

   15. +-commutative 76.8%

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

   16. unsub-neg 76.8%

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

   17. metadata-eval 76.8%

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

3. Simplified 76.8%

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

   1. sub-neg 76.8%

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

   2. distribute-neg-frac 76.8%

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

   3. metadata-eval 76.8%

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

6. Applied egg-rr 76.8%

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

   1. metadata-eval 76.8%

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

   2. distribute-neg-frac 76.8%

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

   3. unsub-neg 76.8%

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

   4. *-rgt-identity 76.8%

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

   5. *-inverses 76.8%

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

   6. associate-/r* 54.1%

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

   7. neg-mul-1 54.1%

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

   8. times-frac 76.8%

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

   9. distribute-frac-neg2 76.8%

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

   10. distribute-neg-frac 76.8%

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

   11. metadata-eval 76.8%

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

   12. distribute-lft-out-- 76.8%

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

   13. *-inverses 76.8%

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

   14. div-sub 77.4%

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

   15. associate--r+ 80.6%

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

   16. associate-*r/ 80.6%

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

   17. associate-*l/ 80.6%

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

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: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.0) 2.0 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	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 = 0.0;
	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, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac2 86.4%

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

      4. neg-sub0 86.4%

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

      5. associate-+l- 86.4%

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

      6. neg-sub0 86.4%

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

      7. remove-double-neg 86.4%

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

      8. distribute-neg-in 86.4%

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

      9. sub-neg 86.4%

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

      10. distribute-neg-frac2 86.4%

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

      11. sub-neg 86.4%

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

      12. +-commutative 86.4%

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

      13. unsub-neg 86.4%

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

      14. sub-neg 86.4%

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

      15. +-commutative 86.4%

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

      16. unsub-neg 86.4%

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

      17. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 69.8%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. sub-neg 52.3%

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

      2. +-commutative 52.3%

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

      3. distribute-neg-frac2 52.3%

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

      4. neg-sub0 52.3%

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

      5. associate-+l- 52.3%

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

      6. neg-sub0 52.3%

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

      7. remove-double-neg 52.3%

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

      8. distribute-neg-in 52.3%

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

      9. sub-neg 52.3%

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

      10. distribute-neg-frac2 52.3%

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

      11. sub-neg 52.3%

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

      12. +-commutative 52.3%

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

      13. unsub-neg 52.3%

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

      14. sub-neg 52.3%

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

      15. +-commutative 52.3%

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

      16. unsub-neg 52.3%

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

      17. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

      1. flip-- 7.8%

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

      2. associate-/r/ 7.7%

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

      3. metadata-eval 7.7%

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

      4. pow2 7.7%

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

      5. +-commutative 7.7%

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

   6. Applied egg-rr 7.7%

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

      1. associate-*l/ 7.6%

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

      2. *-lft-identity 7.6%

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

   8. Simplified 7.6%

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

      1. clear-num 7.8%

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

      2. metadata-eval 7.8%

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

      3. unpow2 7.8%

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

      4. +-commutative 7.8%

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

      5. flip-- 52.3%

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

      6. frac-sub 53.0%

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

      7. *-un-lft-identity 53.0%

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

      8. *-commutative 53.0%

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

      9. *-un-lft-identity 53.0%

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

      10. associate--r+ 59.4%

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

      11. flip-- 56.7%

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

      12. associate-/l/ 56.7%

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

      13. metadata-eval 56.7%

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

      14. pow2 56.7%

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

   10. Applied egg-rr 56.7%

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

       1. div-sub 56.7%

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

       2. unpow2 56.7%

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

       3. +-commutative 56.7%

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

       4. associate--r- 56.7%

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

       5. +-inverses 56.7%

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

       6. metadata-eval 56.7%

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

       7. +-commutative 56.7%

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

       8. associate--r- 50.2%

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

       9. +-inverses 50.2%

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

       10. metadata-eval 50.2%

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

       11. metadata-eval 50.2%

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

       12. +-inverses 50.3%

           \[\leadsto \color{blue}{0} \]

   12. Simplified 50.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 76.8%

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

   1. sub-neg 76.8%

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

   2. +-commutative 76.8%

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

   3. distribute-neg-frac2 76.8%

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

   4. neg-sub0 76.8%

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

   5. associate-+l- 76.8%

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

   6. neg-sub0 76.8%

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

   7. remove-double-neg 76.8%

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

   8. distribute-neg-in 76.8%

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

   9. sub-neg 76.8%

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

   10. distribute-neg-frac2 76.8%

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

   11. sub-neg 76.8%

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

   12. +-commutative 76.8%

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

   13. unsub-neg 76.8%

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

   14. sub-neg 76.8%

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

   15. +-commutative 76.8%

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

   16. unsub-neg 76.8%

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

   17. metadata-eval 76.8%

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

3. Simplified 76.8%

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

   1. flip-- 54.2%

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

   2. associate-/r/ 54.1%

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

   3. metadata-eval 54.1%

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

   4. pow2 54.1%

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

   5. +-commutative 54.1%

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

6. Applied egg-rr 54.1%

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

   1. associate-*l/ 54.1%

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

   2. *-lft-identity 54.1%

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

8. Simplified 54.1%

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

   1. clear-num 54.2%

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

   2. metadata-eval 54.2%

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

   3. unpow2 54.2%

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

   4. +-commutative 54.2%

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

   5. flip-- 76.8%

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

   6. frac-sub 77.4%

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

   7. *-un-lft-identity 77.4%

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

   8. *-commutative 77.4%

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

   9. *-un-lft-identity 77.4%

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

   10. associate--r+ 80.6%

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

   11. flip-- 28.6%

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

   12. associate-/l/ 28.6%

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

   13. metadata-eval 28.6%

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

   14. pow2 28.6%

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

10. Applied egg-rr 28.6%

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

    1. div-sub 28.6%

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

    2. unpow2 28.6%

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

    3. +-commutative 28.6%

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

    4. associate--r- 28.6%

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

    5. +-inverses 28.6%

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

    6. metadata-eval 28.6%

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

    7. +-commutative 28.6%

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

    8. associate--r- 25.4%

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

    9. +-inverses 25.4%

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

    10. metadata-eval 25.4%

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

    11. metadata-eval 25.4%

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

    12. +-inverses 27.1%

        \[\leadsto \color{blue}{0} \]

12. Simplified 27.1%

    \[\leadsto \color{blue}{0} \]

13. Final simplification 27.1%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))