Asymptote A

Percentage Accurate: 77.8% → 99.9%

Time: 6.5s

Alternatives: 10

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / N[(N[(x$95$m + -1.0), $MachinePrecision] / -2.0), $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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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. clear-num 99.6%

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

   2. inv-pow 99.6%

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

   3. sub-neg 99.6%

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

   4. add-sqr-sqrt 54.9%

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

   5. sqrt-unprod 84.8%

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

   6. sqr-neg 84.8%

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

   7. sqrt-unprod 29.8%

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

   8. add-sqr-sqrt 67.0%

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

   9. +-inverses 67.0%

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

   10. metadata-eval 67.0%

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

   11. metadata-eval 67.0%

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

   12. sub-neg 67.0%

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

   13. add-sqr-sqrt 37.2%

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

   14. sqrt-unprod 81.5%

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

   15. sqr-neg 81.5%

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

   16. sqrt-unprod 44.6%

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

   17. add-sqr-sqrt 99.6%

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

10. Applied egg-rr 99.6%

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

    1. unpow-1 99.6%

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

    2. associate-/r/ 99.6%

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

    3. associate-/r* 99.9%

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

    4. +-commutative 99.9%

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

    5. +-commutative 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;\frac{-2}{x_m + \left(-1 - x_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0) (/ -2.0 (+ x_m (- -1.0 x_m))) (/ -0.5 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 = -0.5 / 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 = (-0.5d0) / 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 = -0.5 / 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 = -0.5 / 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(-0.5 / 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 = -0.5 / 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[(-0.5 / 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. metadata-eval 78.7%

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

      14. metadata-eval 78.7%

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

      15. metadata-eval 78.7%

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

      16. associate-/r* 78.7%

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

      17. metadata-eval 78.7%

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

      18. neg-mul-1 78.7%

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

      19. distribute-neg-in 78.7%

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

      20. sub-neg 78.7%

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

      21. distribute-neg-frac 78.7%

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

      22. neg-mul-1 78.7%

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

   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. metadata-eval 49.1%

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

      14. metadata-eval 49.1%

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

      15. metadata-eval 49.1%

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

      16. associate-/r* 49.1%

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

      17. metadata-eval 49.1%

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

      18. neg-mul-1 49.1%

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

      19. distribute-neg-in 49.1%

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

      20. sub-neg 49.1%

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

      21. distribute-neg-frac 49.1%

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

      22. neg-mul-1 49.1%

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

   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. Taylor expanded in x around inf 96.3%

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

       1. metadata-eval 96.3%

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

       2. distribute-neg-frac 96.3%

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

       3. metadata-eval 96.3%

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

       4. associate-*r/ 96.3%

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

       5. distribute-lft-neg-in 96.3%

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

       6. metadata-eval 96.3%

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

       7. unpow2 96.3%

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

       8. associate-/r* 98.3%

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

       9. metadata-eval 98.3%

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

       10. associate-*r/ 98.3%

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

       11. associate-*l/ 97.9%

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

       12. sqr-neg 97.9%

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

       13. distribute-neg-frac 97.9%

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

       14. metadata-eval 97.9%

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

       15. unpow-1 97.9%

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

       16. distribute-neg-frac 97.9%

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

       17. metadata-eval 97.9%

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

       18. unpow-1 97.9%

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

       19. pow-sqr 98.4%

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

       20. metadata-eval 98.4%

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

   11. Simplified 98.4%

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

   13. Applied egg-rr 7.1%

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

       1. unpow-1 7.1%

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

       2. *-commutative 7.1%

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

       3. associate-/r* 7.1%

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

       4. metadata-eval 7.1%

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

   15. Simplified 7.1%

       \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x + \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.9× speedup

Math

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

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) (+ x_m -1.0))))

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) / (x_m + -1.0);
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.76], 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[(x$95$m + -1.0), $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. metadata-eval 78.7%

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

      14. metadata-eval 78.7%

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

      15. metadata-eval 78.7%

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

      16. associate-/r* 78.7%

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

      17. metadata-eval 78.7%

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

      18. neg-mul-1 78.7%

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

      19. distribute-neg-in 78.7%

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

      20. sub-neg 78.7%

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

      21. distribute-neg-frac 78.7%

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

      22. neg-mul-1 78.7%

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

   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. metadata-eval 49.1%

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

      14. metadata-eval 49.1%

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

      15. metadata-eval 49.1%

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

      16. associate-/r* 49.1%

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

      17. metadata-eval 49.1%

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

      18. neg-mul-1 49.1%

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

      19. distribute-neg-in 49.1%

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

      20. sub-neg 49.1%

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

      21. distribute-neg-frac 49.1%

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

      22. neg-mul-1 49.1%

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

   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. expm1-log1p-u 99.9%

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

      2. expm1-udef 46.7%

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

   10. Applied egg-rr 46.7%

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

       1. expm1-def 99.9%

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

       2. expm1-log1p 99.9%

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

       3. +-commutative 99.9%

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

       4. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 96.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;\frac{-2}{x + \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \]
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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (-2.0 / (1.0 + x_m)) / (x_m + -1.0);
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[(x$95$m + -1.0), $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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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. expm1-log1p-u 98.8%

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

   2. expm1-udef 68.9%

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

10. Applied egg-rr 68.9%

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

    1. expm1-def 98.8%

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

    2. expm1-log1p 99.9%

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

    3. +-commutative 99.9%

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

    4. +-commutative 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 6: 53.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 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. metadata-eval 78.7%

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

      14. metadata-eval 78.7%

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

      15. metadata-eval 78.7%

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

      16. associate-/r* 78.7%

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

      17. metadata-eval 78.7%

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

      18. neg-mul-1 78.7%

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

      19. distribute-neg-in 78.7%

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

      20. sub-neg 78.7%

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

      21. distribute-neg-frac 78.7%

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

      22. neg-mul-1 78.7%

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

   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. metadata-eval 49.1%

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

      14. metadata-eval 49.1%

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

      15. metadata-eval 49.1%

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

      16. associate-/r* 49.1%

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

      17. metadata-eval 49.1%

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

      18. neg-mul-1 49.1%

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

      19. distribute-neg-in 49.1%

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

      20. sub-neg 49.1%

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

      21. distribute-neg-frac 49.1%

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

      22. neg-mul-1 49.1%

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

   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: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.0) 2.0 (/ -0.5 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -0.5 / 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 = (-0.5d0) / 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 = -0.5 / 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 = -0.5 / 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(-0.5 / 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 = -0.5 / 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[(-0.5 / 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. metadata-eval 78.7%

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

      14. metadata-eval 78.7%

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

      15. metadata-eval 78.7%

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

      16. associate-/r* 78.7%

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

      17. metadata-eval 78.7%

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

      18. neg-mul-1 78.7%

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

      19. distribute-neg-in 78.7%

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

      20. sub-neg 78.7%

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

      21. distribute-neg-frac 78.7%

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

      22. neg-mul-1 78.7%

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

   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. metadata-eval 49.1%

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

      14. metadata-eval 49.1%

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

      15. metadata-eval 49.1%

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

      16. associate-/r* 49.1%

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

      17. metadata-eval 49.1%

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

      18. neg-mul-1 49.1%

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

      19. distribute-neg-in 49.1%

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

      20. sub-neg 49.1%

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

      21. distribute-neg-frac 49.1%

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

      22. neg-mul-1 49.1%

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

   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. Taylor expanded in x around inf 96.3%

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

       1. metadata-eval 96.3%

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

       2. distribute-neg-frac 96.3%

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

       3. metadata-eval 96.3%

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

       4. associate-*r/ 96.3%

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

       5. distribute-lft-neg-in 96.3%

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

       6. metadata-eval 96.3%

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

       7. unpow2 96.3%

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

       8. associate-/r* 98.3%

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

       9. metadata-eval 98.3%

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

       10. associate-*r/ 98.3%

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

       11. associate-*l/ 97.9%

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

       12. sqr-neg 97.9%

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

       13. distribute-neg-frac 97.9%

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

       14. metadata-eval 97.9%

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

       15. unpow-1 97.9%

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

       16. distribute-neg-frac 97.9%

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

       17. metadata-eval 97.9%

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

       18. unpow-1 97.9%

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

       19. pow-sqr 98.4%

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

       20. metadata-eval 98.4%

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

   11. Simplified 98.4%

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

   13. Applied egg-rr 7.1%

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

       1. unpow-1 7.1%

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

       2. *-commutative 7.1%

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

       3. associate-/r* 7.1%

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

       4. metadata-eval 7.1%

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

   15. Simplified 7.1%

       \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 2.9% 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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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. clear-num 99.6%

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

   2. inv-pow 99.6%

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

   3. sub-neg 99.6%

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

   4. add-sqr-sqrt 54.9%

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

   5. sqrt-unprod 84.8%

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

   6. sqr-neg 84.8%

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

   7. sqrt-unprod 29.8%

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

   8. add-sqr-sqrt 67.0%

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

   9. +-inverses 67.0%

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

   10. metadata-eval 67.0%

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

   11. metadata-eval 67.0%

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

   12. sub-neg 67.0%

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

   13. add-sqr-sqrt 37.2%

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

   14. sqrt-unprod 81.5%

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

   15. sqr-neg 81.5%

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

   16. sqrt-unprod 44.6%

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

   17. add-sqr-sqrt 99.6%

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

10. Applied egg-rr 99.6%

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

    1. unpow-1 99.6%

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

    2. associate-/r/ 99.6%

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

    3. associate-/r* 99.9%

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

    4. +-commutative 99.9%

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

    5. +-commutative 99.9%

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

12. Simplified 99.9%

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

14. Applied egg-rr 0.0%

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

    1. +-commutative 0.0%

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

    2. associate--r+ 0.0%

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

    3. exp-diff 0.0%

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

    4. +-inverses 0.0%

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

    5. 1-exp 0.0%

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

    6. rem-exp-log 3.2%

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

    7. metadata-eval 3.2%

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

16. Simplified 3.2%

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

17. Final simplification 3.2%

    \[\leadsto -0.5 \]
18. Add Preprocessing

Alternative 9: 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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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 10: 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. metadata-eval 70.9%

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

   14. metadata-eval 70.9%

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

   15. metadata-eval 70.9%

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

   16. associate-/r* 70.9%

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

   17. metadata-eval 70.9%

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

   18. neg-mul-1 70.9%

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

   19. distribute-neg-in 70.9%

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

   20. sub-neg 70.9%

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

   21. distribute-neg-frac 70.9%

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

   22. neg-mul-1 70.9%

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

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))))