Asymptote A

Percentage Accurate: 76.9% → 99.9%

Time: 5.3s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. sub-neg 82.9%

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

   2. +-commutative 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. metadata-eval 82.9%

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

   6. metadata-eval 82.9%

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

   7. associate-/r* 82.9%

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

   8. metadata-eval 82.9%

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

   9. neg-mul-1 82.9%

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

   10. sub0-neg 82.9%

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

   11. associate-+l- 82.9%

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

   12. neg-sub0 82.9%

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

   13. remove-double-neg 82.9%

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

   14. distribute-neg-in 82.9%

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

   15. sub-neg 82.9%

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

   16. mul-1-neg 82.9%

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

   17. metadata-eval 82.9%

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

   18. associate-/r* 82.9%

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

   19. metadata-eval 82.9%

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

   20. metadata-eval 82.9%

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

   21. metadata-eval 82.9%

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

   22. metadata-eval 82.9%

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

   23. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

   1. frac-sub 83.4%

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

   2. *-rgt-identity 83.4%

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

   3. metadata-eval 83.4%

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

   4. div-inv 83.4%

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

   5. associate-/r* 83.4%

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

   6. *-un-lft-identity 83.4%

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

   7. metadata-eval 83.4%

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

   8. div-inv 83.4%

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

   9. associate--l- 85.7%

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

   10. div-inv 85.7%

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

   11. metadata-eval 85.7%

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

   12. *-rgt-identity 85.7%

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

   13. div-inv 85.7%

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

   14. metadata-eval 85.7%

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

   15. *-rgt-identity 85.7%

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

5. Applied egg-rr 85.7%

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

   1. frac-2neg 85.7%

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

   2. div-inv 85.6%

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

   3. distribute-neg-frac 85.6%

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

   4. associate--r+ 83.4%

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

   5. associate--r- 83.4%

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

   6. neg-mul-1 83.4%

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

   7. sub-neg 83.4%

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

   8. distribute-rgt-in 83.4%

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

   9. metadata-eval 83.4%

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

   10. cancel-sign-sub-inv 83.4%

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

   11. *-commutative 83.4%

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

   12. neg-mul-1 83.4%

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

7. Applied egg-rr 83.4%

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

   1. associate-*l/ 83.4%

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

   2. sub-neg 83.4%

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

   3. remove-double-neg 83.4%

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

   4. associate-*r/ 83.4%

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

   5. associate--l- 83.4%

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

   6. +-commutative 83.4%

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

   7. associate-+l- 85.6%

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

   8. associate--l+ 99.9%

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

   9. +-inverses 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. distribute-neg-frac 90.4%

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

      4. metadata-eval 90.4%

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

      5. metadata-eval 90.4%

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

      6. metadata-eval 90.4%

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

      7. associate-/r* 90.4%

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

      8. metadata-eval 90.4%

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

      9. neg-mul-1 90.4%

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

      10. sub0-neg 90.4%

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

      11. associate-+l- 90.4%

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

      12. neg-sub0 90.4%

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

      13. remove-double-neg 90.4%

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

      14. distribute-neg-in 90.4%

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

      15. sub-neg 90.4%

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

      16. mul-1-neg 90.4%

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

      17. metadata-eval 90.4%

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

      18. associate-/r* 90.4%

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

      19. metadata-eval 90.4%

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

      20. metadata-eval 90.4%

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

      21. metadata-eval 90.4%

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

      22. metadata-eval 90.4%

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

      23. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 70.0%

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

   if 1 < x

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. +-commutative 53.4%

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

      3. distribute-neg-frac 53.4%

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

      4. metadata-eval 53.4%

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

      5. metadata-eval 53.4%

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

      6. metadata-eval 53.4%

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

      7. associate-/r* 53.4%

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

      8. metadata-eval 53.4%

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

      9. neg-mul-1 53.4%

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

      10. sub0-neg 53.4%

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

      11. associate-+l- 53.4%

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

      12. neg-sub0 53.4%

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

      13. remove-double-neg 53.4%

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

      14. distribute-neg-in 53.4%

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

      15. sub-neg 53.4%

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

      16. mul-1-neg 53.4%

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

      17. metadata-eval 53.4%

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

      18. associate-/r* 53.4%

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

      19. metadata-eval 53.4%

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

      20. metadata-eval 53.4%

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

      21. metadata-eval 53.4%

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

      22. metadata-eval 53.4%

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

      23. associate-*l/ 53.4%

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

   3. Simplified 53.4%

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

      1. frac-sub 54.7%

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

      2. *-rgt-identity 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 54.7%

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

      5. associate-/r* 54.7%

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

      6. *-un-lft-identity 54.7%

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

      7. metadata-eval 54.7%

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

      8. div-inv 54.7%

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

      9. associate--l- 60.9%

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

      10. div-inv 60.9%

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

      11. metadata-eval 60.9%

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

      12. *-rgt-identity 60.9%

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

      13. div-inv 60.9%

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

      14. metadata-eval 60.9%

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

      15. *-rgt-identity 60.9%

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

   5. Applied egg-rr 60.9%

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

   6. Taylor expanded in x around inf 98.1%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 51.7%

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

      3. associate-/l/ 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. expm1-def 98.2%

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

      2. expm1-log1p 98.2%

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

      3. *-commutative 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 75.7%

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

Alternative 3: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = 2.0d0 / (x_m * (1.0d0 - x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x_m * (1.0 - x_m));
	}
	return tmp;
}

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(2.0 / Float64(x_m * Float64(1.0 - x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.75)
		tmp = 2.0;
	else
		tmp = 2.0 / (x_m * (1.0 - x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.75], 2.0, N[(2.0 / N[(x$95$m * N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. distribute-neg-frac 90.4%

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

      4. metadata-eval 90.4%

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

      5. metadata-eval 90.4%

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

      6. metadata-eval 90.4%

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

      7. associate-/r* 90.4%

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

      8. metadata-eval 90.4%

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

      9. neg-mul-1 90.4%

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

      10. sub0-neg 90.4%

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

      11. associate-+l- 90.4%

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

      12. neg-sub0 90.4%

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

      13. remove-double-neg 90.4%

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

      14. distribute-neg-in 90.4%

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

      15. sub-neg 90.4%

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

      16. mul-1-neg 90.4%

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

      17. metadata-eval 90.4%

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

      18. associate-/r* 90.4%

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

      19. metadata-eval 90.4%

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

      20. metadata-eval 90.4%

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

      21. metadata-eval 90.4%

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

      22. metadata-eval 90.4%

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

      23. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 70.0%

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

   if 0.75 < x

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. +-commutative 53.4%

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

      3. distribute-neg-frac 53.4%

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

      4. metadata-eval 53.4%

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

      5. metadata-eval 53.4%

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

      6. metadata-eval 53.4%

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

      7. associate-/r* 53.4%

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

      8. metadata-eval 53.4%

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

      9. neg-mul-1 53.4%

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

      10. sub0-neg 53.4%

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

      11. associate-+l- 53.4%

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

      12. neg-sub0 53.4%

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

      13. remove-double-neg 53.4%

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

      14. distribute-neg-in 53.4%

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

      15. sub-neg 53.4%

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

      16. mul-1-neg 53.4%

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

      17. metadata-eval 53.4%

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

      18. associate-/r* 53.4%

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

      19. metadata-eval 53.4%

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

      20. metadata-eval 53.4%

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

      21. metadata-eval 53.4%

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

      22. metadata-eval 53.4%

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

      23. associate-*l/ 53.4%

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

   3. Simplified 53.4%

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

      1. frac-sub 54.7%

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

      2. *-rgt-identity 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 54.7%

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

      5. associate-/r* 54.7%

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

      6. *-un-lft-identity 54.7%

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

      7. metadata-eval 54.7%

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

      8. div-inv 54.7%

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

      9. associate--l- 60.9%

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

      10. div-inv 60.9%

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

      11. metadata-eval 60.9%

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

      12. *-rgt-identity 60.9%

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

      13. div-inv 60.9%

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

      14. metadata-eval 60.9%

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

      15. *-rgt-identity 60.9%

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

   5. Applied egg-rr 60.9%

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

      1. frac-2neg 60.9%

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

      2. div-inv 60.9%

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

      3. distribute-neg-frac 60.9%

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

      4. associate--r+ 54.6%

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

      5. associate--r- 54.6%

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

      6. neg-mul-1 54.6%

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

      7. sub-neg 54.6%

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

      8. distribute-rgt-in 54.6%

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

      9. metadata-eval 54.6%

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

      10. cancel-sign-sub-inv 54.6%

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

      11. *-commutative 54.6%

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

      12. neg-mul-1 54.6%

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

   7. Applied egg-rr 54.6%

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

      1. associate-*l/ 54.7%

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

      2. sub-neg 54.7%

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

      3. remove-double-neg 54.7%

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

      4. associate-*r/ 54.7%

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

      5. associate--l- 54.7%

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

      6. +-commutative 54.7%

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

      7. associate-+l- 60.9%

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

      8. associate--l+ 99.8%

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

      9. +-inverses 99.8%

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

      10. metadata-eval 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 98.1%

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

       1. expm1-log1p-u 98.1%

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

       2. expm1-udef 51.7%

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

       3. associate-/l/ 51.7%

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

       4. *-commutative 51.7%

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

   12. Applied egg-rr 51.7%

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

       1. expm1-def 98.2%

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

       2. expm1-log1p 98.2%

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

   14. Simplified 98.2%

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

4. Final simplification 75.7%

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

Alternative 4: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. distribute-neg-frac 90.4%

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

      4. metadata-eval 90.4%

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

      5. metadata-eval 90.4%

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

      6. metadata-eval 90.4%

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

      7. associate-/r* 90.4%

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

      8. metadata-eval 90.4%

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

      9. neg-mul-1 90.4%

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

      10. sub0-neg 90.4%

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

      11. associate-+l- 90.4%

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

      12. neg-sub0 90.4%

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

      13. remove-double-neg 90.4%

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

      14. distribute-neg-in 90.4%

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

      15. sub-neg 90.4%

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

      16. mul-1-neg 90.4%

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

      17. metadata-eval 90.4%

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

      18. associate-/r* 90.4%

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

      19. metadata-eval 90.4%

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

      20. metadata-eval 90.4%

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

      21. metadata-eval 90.4%

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

      22. metadata-eval 90.4%

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

      23. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 70.0%

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

   if 1 < x

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. +-commutative 53.4%

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

      3. distribute-neg-frac 53.4%

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

      4. metadata-eval 53.4%

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

      5. metadata-eval 53.4%

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

      6. metadata-eval 53.4%

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

      7. associate-/r* 53.4%

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

      8. metadata-eval 53.4%

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

      9. neg-mul-1 53.4%

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

      10. sub0-neg 53.4%

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

      11. associate-+l- 53.4%

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

      12. neg-sub0 53.4%

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

      13. remove-double-neg 53.4%

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

      14. distribute-neg-in 53.4%

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

      15. sub-neg 53.4%

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

      16. mul-1-neg 53.4%

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

      17. metadata-eval 53.4%

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

      18. associate-/r* 53.4%

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

      19. metadata-eval 53.4%

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

      20. metadata-eval 53.4%

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

      21. metadata-eval 53.4%

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

      22. metadata-eval 53.4%

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

      23. associate-*l/ 53.4%

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

   3. Simplified 53.4%

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

      1. frac-sub 54.7%

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

      2. *-rgt-identity 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 54.7%

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

      5. associate-/r* 54.7%

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

      6. *-un-lft-identity 54.7%

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

      7. metadata-eval 54.7%

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

      8. div-inv 54.7%

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

      9. associate--l- 60.9%

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

      10. div-inv 60.9%

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

      11. metadata-eval 60.9%

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

      12. *-rgt-identity 60.9%

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

      13. div-inv 60.9%

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

      14. metadata-eval 60.9%

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

      15. *-rgt-identity 60.9%

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

   5. Applied egg-rr 60.9%

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

   6. Taylor expanded in x around inf 98.1%

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

4. Final simplification 75.7%

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

Alternative 5: 99.4% accurate, 1.2× speedup

Math

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

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(2.0 / Float64(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[(2.0 / N[(N[(-1.0 - x$95$m), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. sub-neg 82.9%

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

   2. +-commutative 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. metadata-eval 82.9%

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

   6. metadata-eval 82.9%

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

   7. associate-/r* 82.9%

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

   8. metadata-eval 82.9%

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

   9. neg-mul-1 82.9%

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

   10. sub0-neg 82.9%

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

   11. associate-+l- 82.9%

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

   12. neg-sub0 82.9%

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

   13. remove-double-neg 82.9%

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

   14. distribute-neg-in 82.9%

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

   15. sub-neg 82.9%

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

   16. mul-1-neg 82.9%

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

   17. metadata-eval 82.9%

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

   18. associate-/r* 82.9%

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

   19. metadata-eval 82.9%

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

   20. metadata-eval 82.9%

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

   21. metadata-eval 82.9%

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

   22. metadata-eval 82.9%

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

   23. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

   1. sub-neg 82.9%

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

   2. distribute-neg-frac 82.9%

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

   3. metadata-eval 82.9%

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

5. Applied egg-rr 82.9%

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

   1. *-rgt-identity 82.9%

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

   2. cancel-sign-sub 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. *-rgt-identity 82.9%

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

   6. *-inverses 82.9%

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

   7. associate-/r* 60.0%

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

   8. distribute-lft-neg-in 60.0%

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

   9. distribute-rgt-neg-in 60.0%

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

   10. *-commutative 60.0%

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

   11. *-lft-identity 60.0%

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

   12. *-inverses 60.0%

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

   13. times-frac 82.9%

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

   14. div-sub 83.4%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 6: 52.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = -2.0 / 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(-2.0 / 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 = -2.0 / 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[(-2.0 / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. distribute-neg-frac 90.4%

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

      4. metadata-eval 90.4%

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

      5. metadata-eval 90.4%

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

      6. metadata-eval 90.4%

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

      7. associate-/r* 90.4%

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

      8. metadata-eval 90.4%

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

      9. neg-mul-1 90.4%

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

      10. sub0-neg 90.4%

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

      11. associate-+l- 90.4%

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

      12. neg-sub0 90.4%

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

      13. remove-double-neg 90.4%

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

      14. distribute-neg-in 90.4%

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

      15. sub-neg 90.4%

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

      16. mul-1-neg 90.4%

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

      17. metadata-eval 90.4%

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

      18. associate-/r* 90.4%

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

      19. metadata-eval 90.4%

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

      20. metadata-eval 90.4%

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

      21. metadata-eval 90.4%

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

      22. metadata-eval 90.4%

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

      23. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 70.0%

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

   if 1 < x

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. +-commutative 53.4%

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

      3. distribute-neg-frac 53.4%

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

      4. metadata-eval 53.4%

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

      5. metadata-eval 53.4%

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

      6. metadata-eval 53.4%

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

      7. associate-/r* 53.4%

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

      8. metadata-eval 53.4%

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

      9. neg-mul-1 53.4%

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

      10. sub0-neg 53.4%

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

      11. associate-+l- 53.4%

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

      12. neg-sub0 53.4%

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

      13. remove-double-neg 53.4%

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

      14. distribute-neg-in 53.4%

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

      15. sub-neg 53.4%

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

      16. mul-1-neg 53.4%

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

      17. metadata-eval 53.4%

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

      18. associate-/r* 53.4%

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

      19. metadata-eval 53.4%

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

      20. metadata-eval 53.4%

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

      21. metadata-eval 53.4%

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

      22. metadata-eval 53.4%

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

      23. associate-*l/ 53.4%

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

   3. Simplified 53.4%

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

      1. frac-sub 54.7%

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

      2. *-rgt-identity 54.7%

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

      3. metadata-eval 54.7%

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

      4. div-inv 54.7%

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

      5. associate-/r* 54.7%

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

      6. *-un-lft-identity 54.7%

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

      7. metadata-eval 54.7%

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

      8. div-inv 54.7%

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

      9. associate--l- 60.9%

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

      10. div-inv 60.9%

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

      11. metadata-eval 60.9%

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

      12. *-rgt-identity 60.9%

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

      13. div-inv 60.9%

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

      14. metadata-eval 60.9%

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

      15. *-rgt-identity 60.9%

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

   5. Applied egg-rr 60.9%

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

   6. Taylor expanded in x around inf 98.1%

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

   7. Taylor expanded in x around 0 7.1%

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

4. Final simplification 57.2%

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

Alternative 7: 51.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return -2.0 / (-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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. sub-neg 82.9%

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

   2. +-commutative 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. metadata-eval 82.9%

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

   6. metadata-eval 82.9%

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

   7. associate-/r* 82.9%

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

   8. metadata-eval 82.9%

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

   9. neg-mul-1 82.9%

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

   10. sub0-neg 82.9%

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

   11. associate-+l- 82.9%

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

   12. neg-sub0 82.9%

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

   13. remove-double-neg 82.9%

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

   14. distribute-neg-in 82.9%

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

   15. sub-neg 82.9%

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

   16. mul-1-neg 82.9%

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

   17. metadata-eval 82.9%

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

   18. associate-/r* 82.9%

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

   19. metadata-eval 82.9%

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

   20. metadata-eval 82.9%

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

   21. metadata-eval 82.9%

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

   22. metadata-eval 82.9%

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

   23. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

   1. frac-sub 83.4%

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

   2. *-rgt-identity 83.4%

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

   3. metadata-eval 83.4%

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

   4. div-inv 83.4%

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

   5. associate-/r* 83.4%

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

   6. *-un-lft-identity 83.4%

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

   7. metadata-eval 83.4%

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

   8. div-inv 83.4%

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

   9. associate--l- 85.7%

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

   10. div-inv 85.7%

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

   11. metadata-eval 85.7%

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

   12. *-rgt-identity 85.7%

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

   13. div-inv 85.7%

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

   14. metadata-eval 85.7%

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

   15. *-rgt-identity 85.7%

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

5. Applied egg-rr 85.7%

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

6. Taylor expanded in x around 0 57.3%

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

7. Final simplification 57.3%

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

Alternative 8: 10.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. sub-neg 82.9%

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

   2. +-commutative 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. metadata-eval 82.9%

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

   6. metadata-eval 82.9%

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

   7. associate-/r* 82.9%

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

   8. metadata-eval 82.9%

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

   9. neg-mul-1 82.9%

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

   10. sub0-neg 82.9%

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

   11. associate-+l- 82.9%

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

   12. neg-sub0 82.9%

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

   13. remove-double-neg 82.9%

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

   14. distribute-neg-in 82.9%

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

   15. sub-neg 82.9%

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

   16. mul-1-neg 82.9%

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

   17. metadata-eval 82.9%

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

   18. associate-/r* 82.9%

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

   19. metadata-eval 82.9%

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

   20. metadata-eval 82.9%

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

   21. metadata-eval 82.9%

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

   22. metadata-eval 82.9%

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

   23. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in x around 0 55.8%

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

5. Taylor expanded in x around inf 11.6%

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

6. Final simplification 11.6%

   \[\leadsto 1 \]

Alternative 9: 50.6% 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 82.9%

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

   1. sub-neg 82.9%

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

   2. +-commutative 82.9%

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

   3. distribute-neg-frac 82.9%

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

   4. metadata-eval 82.9%

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

   5. metadata-eval 82.9%

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

   6. metadata-eval 82.9%

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

   7. associate-/r* 82.9%

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

   8. metadata-eval 82.9%

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

   9. neg-mul-1 82.9%

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

   10. sub0-neg 82.9%

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

   11. associate-+l- 82.9%

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

   12. neg-sub0 82.9%

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

   13. remove-double-neg 82.9%

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

   14. distribute-neg-in 82.9%

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

   15. sub-neg 82.9%

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

   16. mul-1-neg 82.9%

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

   17. metadata-eval 82.9%

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

   18. associate-/r* 82.9%

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

   19. metadata-eval 82.9%

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

   20. metadata-eval 82.9%

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

   21. metadata-eval 82.9%

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

   22. metadata-eval 82.9%

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

   23. associate-*l/ 82.9%

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

3. Simplified 82.9%

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

4. Taylor expanded in x around 0 56.3%

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

5. Final simplification 56.3%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))