Asymptote A

Percentage Accurate: 77.8% → 99.9%

Time: 7.5s

Alternatives: 9

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: 77.8% 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, 0.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return ((-1.0 + (-1.0 + (x_m - x_m))) / (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 = (((-1.0d0) + ((-1.0d0) + (x_m - x_m))) / (1.0d0 - x_m)) / ((-1.0d0) - x_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

   1. frac-sub 71.6%

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

   2. *-rgt-identity 71.6%

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

   3. metadata-eval 71.6%

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

   4. div-inv 71.6%

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

   5. associate-/r* 71.6%

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

   6. *-un-lft-identity 71.6%

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

   7. metadata-eval 71.6%

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

   8. div-inv 71.6%

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

   9. associate--l- 75.4%

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

   10. div-inv 75.4%

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

   11. metadata-eval 75.4%

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

   12. *-rgt-identity 75.4%

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

   13. div-inv 75.4%

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

   14. metadata-eval 75.4%

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

   15. *-rgt-identity 75.4%

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

6. Applied egg-rr 75.4%

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

   1. +-commutative 75.4%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 53.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = -2.0 / (x_m + (-1.0 - x_m));
	} 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) / (x_m + ((-1.0d0) - x_m))
    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 / (x_m + (-1.0 - x_m));
	} 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 / (x_m + (-1.0 - x_m))
	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 = Float64(-2.0 / Float64(x_m + Float64(-1.0 - x_m)));
	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 / (x_m + (-1.0 - x_m));
	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], N[(-2.0 / N[(x$95$m + N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 78.7%

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

      1. sub-neg 78.7%

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

      2. +-commutative 78.7%

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

      3. distribute-neg-frac 78.7%

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

      4. metadata-eval 78.7%

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

      5. metadata-eval 78.7%

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

      6. metadata-eval 78.7%

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

      7. associate-/r* 78.7%

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

      8. metadata-eval 78.7%

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

      9. neg-mul-1 78.7%

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

      10. sub0-neg 78.7%

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

      11. associate-+l- 78.7%

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

      12. neg-sub0 78.7%

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

      13. remove-double-neg 78.7%

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

      14. distribute-neg-in 78.7%

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

      15. sub-neg 78.7%

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

      16. mul-1-neg 78.7%

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

      17. metadata-eval 78.7%

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

      18. associate-/r* 78.7%

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

      19. metadata-eval 78.7%

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

      20. metadata-eval 78.7%

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

      21. metadata-eval 78.7%

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

      22. metadata-eval 78.7%

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

      23. associate-*l/ 78.7%

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

   3. Simplified 78.7%

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

      1. sub-neg 78.7%

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

      2. distribute-neg-frac 78.7%

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

      3. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-un-lft-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around 0 59.9%

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

   if 1 < x

   1. Initial program 49.1%

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

      1. sub-neg 49.1%

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

      2. +-commutative 49.1%

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

      3. distribute-neg-frac 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

      6. metadata-eval 49.1%

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

      7. associate-/r* 49.1%

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

      8. metadata-eval 49.1%

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

      9. neg-mul-1 49.1%

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

      10. sub0-neg 49.1%

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

      11. associate-+l- 49.1%

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

      12. neg-sub0 49.1%

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

      13. remove-double-neg 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. sub-neg 49.1%

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

      16. mul-1-neg 49.1%

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

      17. metadata-eval 49.1%

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

      18. associate-/r* 49.1%

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

      19. metadata-eval 49.1%

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

      20. metadata-eval 49.1%

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

      21. metadata-eval 49.1%

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

      22. metadata-eval 49.1%

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

      23. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in x around 0 2.7%

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

   6. Taylor expanded in x around inf 2.7%

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

   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 46.1%

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

Alternative 3: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.76) {
		tmp = -2.0 / (x_m + (-1.0 - x_m));
	} 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 <= 0.76d0) then
        tmp = (-2.0d0) / (x_m + ((-1.0d0) - x_m))
    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 <= 0.76) {
		tmp = -2.0 / (x_m + (-1.0 - x_m));
	} 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 <= 0.76:
		tmp = -2.0 / (x_m + (-1.0 - x_m))
	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 <= 0.76)
		tmp = Float64(-2.0 / Float64(x_m + Float64(-1.0 - x_m)));
	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 <= 0.76)
		tmp = -2.0 / (x_m + (-1.0 - x_m));
	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, 0.76], N[(-2.0 / N[(x$95$m + N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 0.76000000000000001

   1. Initial program 78.7%

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

      1. sub-neg 78.7%

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

      2. +-commutative 78.7%

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

      3. distribute-neg-frac 78.7%

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

      4. metadata-eval 78.7%

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

      5. metadata-eval 78.7%

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

      6. metadata-eval 78.7%

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

      7. associate-/r* 78.7%

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

      8. metadata-eval 78.7%

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

      9. neg-mul-1 78.7%

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

      10. sub0-neg 78.7%

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

      11. associate-+l- 78.7%

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

      12. neg-sub0 78.7%

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

      13. remove-double-neg 78.7%

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

      14. distribute-neg-in 78.7%

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

      15. sub-neg 78.7%

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

      16. mul-1-neg 78.7%

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

      17. metadata-eval 78.7%

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

      18. associate-/r* 78.7%

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

      19. metadata-eval 78.7%

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

      20. metadata-eval 78.7%

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

      21. metadata-eval 78.7%

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

      22. metadata-eval 78.7%

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

      23. associate-*l/ 78.7%

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

   3. Simplified 78.7%

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

      1. sub-neg 78.7%

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

      2. distribute-neg-frac 78.7%

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

      3. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-un-lft-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around 0 59.9%

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

   if 0.76000000000000001 < x

   1. Initial program 49.1%

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

      1. sub-neg 49.1%

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

      2. +-commutative 49.1%

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

      3. distribute-neg-frac 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

      6. metadata-eval 49.1%

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

      7. associate-/r* 49.1%

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

      8. metadata-eval 49.1%

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

      9. neg-mul-1 49.1%

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

      10. sub0-neg 49.1%

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

      11. associate-+l- 49.1%

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

      12. neg-sub0 49.1%

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

      13. remove-double-neg 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. sub-neg 49.1%

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

      16. mul-1-neg 49.1%

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

      17. metadata-eval 49.1%

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

      18. associate-/r* 49.1%

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

      19. metadata-eval 49.1%

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

      20. metadata-eval 49.1%

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

      21. metadata-eval 49.1%

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

      22. metadata-eval 49.1%

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

      23. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

      1. frac-sub 50.6%

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

      2. *-rgt-identity 50.6%

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

      3. metadata-eval 50.6%

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

      4. div-inv 50.6%

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

      5. associate-/r* 50.6%

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

      6. *-un-lft-identity 50.6%

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

      7. metadata-eval 50.6%

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

      8. div-inv 50.6%

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

      9. associate--l- 56.9%

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

      10. div-inv 56.9%

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

      11. metadata-eval 56.9%

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

      12. *-rgt-identity 56.9%

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

      13. div-inv 56.9%

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

      14. metadata-eval 56.9%

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

      15. *-rgt-identity 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. +-commutative 56.9%

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

      2. associate-+l- 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. div-inv 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. metadata-eval 99.6%

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

      6. sub-neg 99.6%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 43.7%

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

      9. sqr-neg 43.7%

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

      10. sqrt-unprod 43.7%

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

      11. add-sqr-sqrt 43.7%

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

      12. sub-neg 43.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 97.6%

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

      15. sqr-neg 97.6%

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

      16. sqrt-unprod 99.5%

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

      17. add-sqr-sqrt 99.6%

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

      18. +-commutative 99.6%

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

      19. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. distribute-neg-frac 99.9%

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

       4. metadata-eval 99.9%

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

       5. +-commutative 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. metadata-eval 99.9%

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

       8. unsub-neg 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 96.8%

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

4. Final simplification 69.6%

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

Alternative 4: 99.4% 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 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

   1. sub-neg 70.9%

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

   2. distribute-neg-frac 70.9%

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

   3. metadata-eval 70.9%

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

6. Applied egg-rr 70.9%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 5: 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 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

   1. frac-sub 71.6%

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

   2. *-rgt-identity 71.6%

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

   3. metadata-eval 71.6%

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

   4. div-inv 71.6%

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

   5. associate-/r* 71.6%

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

   6. *-un-lft-identity 71.6%

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

   7. metadata-eval 71.6%

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

   8. div-inv 71.6%

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

   9. associate--l- 75.4%

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

   10. div-inv 75.4%

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

   11. metadata-eval 75.4%

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

   12. *-rgt-identity 75.4%

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

   13. div-inv 75.4%

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

   14. metadata-eval 75.4%

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

   15. *-rgt-identity 75.4%

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

6. Applied egg-rr 75.4%

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

   1. +-commutative 75.4%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

   1. frac-2neg 99.9%

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

   2. div-inv 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. metadata-eval 99.7%

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

   6. sub-neg 99.7%

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

   7. add-sqr-sqrt 54.9%

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

   8. sqrt-unprod 84.7%

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

   9. sqr-neg 84.7%

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

   10. sqrt-unprod 29.8%

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

   11. add-sqr-sqrt 67.0%

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

   12. sub-neg 67.0%

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

   13. add-sqr-sqrt 37.2%

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

   14. sqrt-unprod 81.5%

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

   15. sqr-neg 81.5%

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

   16. sqrt-unprod 44.8%

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

   17. add-sqr-sqrt 99.7%

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

   18. +-commutative 99.7%

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

   19. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.9%

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

    2. *-rgt-identity 99.9%

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

    3. distribute-neg-frac 99.9%

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

    4. metadata-eval 99.9%

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

    5. +-commutative 99.9%

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

    6. distribute-neg-in 99.9%

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

    7. metadata-eval 99.9%

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

    8. unsub-neg 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 6: 53.0% accurate, 2.2× 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 78.7%

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

      1. sub-neg 78.7%

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

      2. +-commutative 78.7%

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

      3. distribute-neg-frac 78.7%

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

      4. metadata-eval 78.7%

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

      5. metadata-eval 78.7%

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

      6. metadata-eval 78.7%

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

      7. associate-/r* 78.7%

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

      8. metadata-eval 78.7%

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

      9. neg-mul-1 78.7%

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

      10. sub0-neg 78.7%

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

      11. associate-+l- 78.7%

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

      12. neg-sub0 78.7%

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

      13. remove-double-neg 78.7%

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

      14. distribute-neg-in 78.7%

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

      15. sub-neg 78.7%

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

      16. mul-1-neg 78.7%

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

      17. metadata-eval 78.7%

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

      18. associate-/r* 78.7%

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

      19. metadata-eval 78.7%

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

      20. metadata-eval 78.7%

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

      21. metadata-eval 78.7%

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

      22. metadata-eval 78.7%

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

      23. associate-*l/ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 60.2%

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

   if 1 < x

   1. Initial program 49.1%

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

      1. sub-neg 49.1%

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

      2. +-commutative 49.1%

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

      3. distribute-neg-frac 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

      6. metadata-eval 49.1%

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

      7. associate-/r* 49.1%

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

      8. metadata-eval 49.1%

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

      9. neg-mul-1 49.1%

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

      10. sub0-neg 49.1%

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

      11. associate-+l- 49.1%

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

      12. neg-sub0 49.1%

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

      13. remove-double-neg 49.1%

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

      14. distribute-neg-in 49.1%

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

      15. sub-neg 49.1%

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

      16. mul-1-neg 49.1%

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

      17. metadata-eval 49.1%

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

      18. associate-/r* 49.1%

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

      19. metadata-eval 49.1%

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

      20. metadata-eval 49.1%

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

      21. metadata-eval 49.1%

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

      22. metadata-eval 49.1%

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

      23. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in x around 0 2.7%

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

   6. Taylor expanded in x around inf 2.7%

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

   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 46.3%

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

Alternative 7: 10.0% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.5d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

   1. frac-sub 71.6%

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

   2. *-rgt-identity 71.6%

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

   3. metadata-eval 71.6%

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

   4. div-inv 71.6%

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

   5. associate-/r* 71.6%

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

   6. *-un-lft-identity 71.6%

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

   7. metadata-eval 71.6%

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

   8. div-inv 71.6%

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

   9. associate--l- 75.4%

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

   10. div-inv 75.4%

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

   11. metadata-eval 75.4%

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

   12. *-rgt-identity 75.4%

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

   13. div-inv 75.4%

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

   14. metadata-eval 75.4%

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

   15. *-rgt-identity 75.4%

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

6. Applied egg-rr 75.4%

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

   1. +-commutative 75.4%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

   1. frac-2neg 99.9%

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

   2. div-inv 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. metadata-eval 99.7%

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

   6. sub-neg 99.7%

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

   7. add-sqr-sqrt 54.9%

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

   8. sqrt-unprod 84.7%

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

   9. sqr-neg 84.7%

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

   10. sqrt-unprod 29.8%

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

   11. add-sqr-sqrt 67.0%

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

   12. sub-neg 67.0%

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

   13. add-sqr-sqrt 37.2%

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

   14. sqrt-unprod 81.5%

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

   15. sqr-neg 81.5%

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

   16. sqrt-unprod 44.8%

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

   17. add-sqr-sqrt 99.7%

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

   18. +-commutative 99.7%

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

   19. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.9%

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

    2. *-rgt-identity 99.9%

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

    3. distribute-neg-frac 99.9%

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

    4. metadata-eval 99.9%

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

    5. +-commutative 99.9%

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

    6. distribute-neg-in 99.9%

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

    7. metadata-eval 99.9%

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

    8. unsub-neg 99.9%

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

12. Simplified 99.9%

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

14. Applied egg-rr 8.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) - \left(\log 2 + \mathsf{log1p}\left(x\right)\right)}} \]
15. Step-by-step derivation

    1. +-commutative 8.3%

       \[\leadsto e^{\mathsf{log1p}\left(x\right) - \color{blue}{\left(\mathsf{log1p}\left(x\right) + \log 2\right)}} \]

    2. associate--r+ 8.3%

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

    3. exp-diff 8.3%

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

    4. +-inverses 9.1%

       \[\leadsto \frac{e^{\color{blue}{0}}}{e^{\log 2}} \]

    5. 1-exp 9.1%

       \[\leadsto \frac{\color{blue}{1}}{e^{\log 2}} \]

    6. rem-exp-log 9.1%

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

    7. metadata-eval 9.1%

       \[\leadsto \color{blue}{0.5} \]

16. Simplified 9.1%

    \[\leadsto \color{blue}{0.5} \]

17. Final simplification 9.1%

    \[\leadsto 0.5 \]
18. Add Preprocessing

Alternative 8: 10.7% 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 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

5. Taylor expanded in x around 0 44.7%

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

6. Taylor expanded in x around inf 3.0%

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

7. Taylor expanded in x around inf 9.8%

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

8. Final simplification 9.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Alternative 9: 50.9% 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 70.9%

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

   1. sub-neg 70.9%

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

   2. +-commutative 70.9%

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

   3. distribute-neg-frac 70.9%

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

   4. metadata-eval 70.9%

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

   5. metadata-eval 70.9%

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

   6. metadata-eval 70.9%

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

   7. associate-/r* 70.9%

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

   8. metadata-eval 70.9%

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

   9. neg-mul-1 70.9%

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

   10. sub0-neg 70.9%

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

   11. associate-+l- 70.9%

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

   12. neg-sub0 70.9%

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

   13. remove-double-neg 70.9%

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

   14. distribute-neg-in 70.9%

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

   15. sub-neg 70.9%

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

   16. mul-1-neg 70.9%

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

   17. metadata-eval 70.9%

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

   18. associate-/r* 70.9%

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

   19. metadata-eval 70.9%

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

   20. metadata-eval 70.9%

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

   21. metadata-eval 70.9%

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

   22. metadata-eval 70.9%

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

   23. associate-*l/ 70.9%

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

3. Simplified 70.9%

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

5. Taylor expanded in x around 0 45.2%

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

6. Final simplification 45.2%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))