Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-*l/ 100.0%

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

   15. /-rgt-identity 100.0%

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

   16. distribute-rgt1-in 100.0%

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

   17. *-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. neg-mul-1 100.0%

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

   20. unsub-neg 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. frac-sub 75.8%

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

   2. associate-/r* 76.5%

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

   3. *-un-lft-identity 76.5%

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

   4. *-commutative 76.5%

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

5. Applied egg-rr 76.5%

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

   1. div-sub 76.5%

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

   2. sub-neg 76.5%

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

   3. +-commutative 76.5%

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

   4. *-un-lft-identity 76.5%

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

   5. times-frac 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

   8. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

   1. *-inverses 100.0%

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

   2. distribute-lft-neg-in 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. unsub-neg 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.78 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - \frac{x}{-1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.78) (not (<= x 1.0)))
   (/ (- (/ -1.0 x) x) (- x))
   (- (- -1.0 x) (/ x (- -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.78) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.78d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (((-1.0d0) / x) - x) / -x
    else
        tmp = ((-1.0d0) - x) - (x / ((-1.0d0) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.78) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.78) or not (x <= 1.0):
		tmp = ((-1.0 / x) - x) / -x
	else:
		tmp = (-1.0 - x) - (x / (-1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.78) || !(x <= 1.0))
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-x));
	else
		tmp = Float64(Float64(-1.0 - x) - Float64(x / Float64(-1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.78) || ~((x <= 1.0)))
		tmp = ((-1.0 / x) - x) / -x;
	else
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.78], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(-1.0 - x), $MachinePrecision] - N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.78000000000000003 or 1 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate--r+ 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

      7. associate--l- 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. +-commutative 100.0%

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

       4. associate-+l- 100.0%

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

       5. sub-neg 100.0%

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

       6. +-commutative 100.0%

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

       7. metadata-eval 100.0%

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

       8. associate-+r+ 100.0%

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

       9. +-commutative 100.0%

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

       10. associate-+r+ 100.0%

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

       11. metadata-eval 100.0%

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

       12. +-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 99.5%

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

       1. neg-mul-1 99.5%

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

   15. Simplified 99.5%

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

   if -1.78000000000000003 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. neg-mul-1 97.5%

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

      3. metadata-eval 97.5%

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

      4. +-commutative 97.5%

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

      5. sub-neg 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.78 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - \frac{x}{-1 - x}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-1 - x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} - t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-1 - x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (- -1.0 x))))
   (if (<= x -1.0)
     (- (/ 1.0 x) t_0)
     (if (<= x 1.0) (- (- -1.0 x) t_0) (/ (- (/ -1.0 x) x) (- x))))))

C

double code(double x) {
	double t_0 = x / (-1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) - t_0;
	} else if (x <= 1.0) {
		tmp = (-1.0 - x) - t_0;
	} else {
		tmp = ((-1.0 / x) - x) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((-1.0d0) - x)
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) - t_0
    else if (x <= 1.0d0) then
        tmp = ((-1.0d0) - x) - t_0
    else
        tmp = (((-1.0d0) / x) - x) / -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (-1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) - t_0;
	} else if (x <= 1.0) {
		tmp = (-1.0 - x) - t_0;
	} else {
		tmp = ((-1.0 / x) - x) / -x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (-1.0 - x)
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) - t_0
	elif x <= 1.0:
		tmp = (-1.0 - x) - t_0
	else:
		tmp = ((-1.0 / x) - x) / -x
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(-1.0 - x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) - t_0);
	elseif (x <= 1.0)
		tmp = Float64(Float64(-1.0 - x) - t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (-1.0 - x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) - t_0;
	elseif (x <= 1.0)
		tmp = (-1.0 - x) - t_0;
	else
		tmp = ((-1.0 / x) - x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(-1.0 - x), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. sub-neg 97.5%

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

      2. neg-mul-1 97.5%

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

      3. metadata-eval 97.5%

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

      4. +-commutative 97.5%

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

      5. sub-neg 97.5%

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

   6. Simplified 97.5%

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

   if 1 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate--r+ 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

      7. associate--l- 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. +-commutative 100.0%

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

       4. associate-+l- 100.0%

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

       5. sub-neg 100.0%

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

       6. +-commutative 100.0%

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

       7. metadata-eval 100.0%

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

       8. associate-+r+ 100.0%

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

       9. +-commutative 100.0%

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

       10. associate-+r+ 100.0%

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

       11. metadata-eval 100.0%

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

       12. +-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 99.7%

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

       1. neg-mul-1 99.7%

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

   15. Simplified 99.7%

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

4. Final simplification 98.5%

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

Alternative 4: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.75)))
   (/ (- (/ -1.0 x) x) (- x))
   (+ x (/ 1.0 (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.75d0))) then
        tmp = (((-1.0d0) / x) - x) / -x
    else
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.75):
		tmp = ((-1.0 / x) - x) / -x
	else:
		tmp = x + (1.0 / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.75))
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-x));
	else
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.75)))
		tmp = ((-1.0 / x) - x) / -x;
	else
		tmp = x + (1.0 / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.75]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.75 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate--r+ 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

      7. associate--l- 100.0%

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

      8. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. +-commutative 100.0%

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

       4. associate-+l- 100.0%

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

       5. sub-neg 100.0%

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

       6. +-commutative 100.0%

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

       7. metadata-eval 100.0%

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

       8. associate-+r+ 100.0%

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

       9. +-commutative 100.0%

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

       10. associate-+r+ 100.0%

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

       11. metadata-eval 100.0%

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

       12. +-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 99.5%

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

       1. neg-mul-1 99.5%

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

   15. Simplified 99.5%

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

   if -1 < x < 1.75

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. fma-neg 0.0%

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

      3. inv-pow 0.0%

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

      4. sqrt-pow1 0.0%

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

      5. +-commutative 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x + -1}}, -\left(-x\right)\right) \]

      7. inv-pow 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{\left(x + -1\right)}^{-1}}}, -\left(-x\right)\right) \]

      8. sqrt-pow1 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{\left(x + -1\right)}^{\left(\frac{-1}{2}\right)}}, -\left(-x\right)\right) \]

      9. +-commutative 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\color{blue}{\left(-1 + x\right)}}^{\left(\frac{-1}{2}\right)}, -\left(-x\right)\right) \]

      10. metadata-eval 0.0%

          \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{\color{blue}{-0.5}}, -\left(-x\right)\right) \]

      11. remove-double-neg 0.0%

          \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{-0.5}, \color{blue}{x}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{-0.5}, x\right)} \]
   9. Step-by-step derivation

      1. fma-udef 0.0%

         \[\leadsto \color{blue}{{\left(-1 + x\right)}^{-0.5} \cdot {\left(-1 + x\right)}^{-0.5} + x} \]

      2. pow-sqr 97.5%

         \[\leadsto \color{blue}{{\left(-1 + x\right)}^{\left(2 \cdot -0.5\right)}} + x \]

      3. metadata-eval 97.5%

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

      4. unpow-1 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.95) (+ x (/ 1.0 (+ -1.0 x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.95) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.95d0) then
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.95) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.95:
		tmp = x + (1.0 / (-1.0 + x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.95)
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.95)
		tmp = x + (1.0 / (-1.0 + x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.95], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.94999999999999996 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.5%

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

   if -1 < x < 1.94999999999999996

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. fma-neg 0.0%

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

      3. inv-pow 0.0%

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

      4. sqrt-pow1 0.0%

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

      5. +-commutative 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x + -1}}, -\left(-x\right)\right) \]

      7. inv-pow 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{\left(x + -1\right)}^{-1}}}, -\left(-x\right)\right) \]

      8. sqrt-pow1 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{\left(x + -1\right)}^{\left(\frac{-1}{2}\right)}}, -\left(-x\right)\right) \]

      9. +-commutative 0.0%

         \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\color{blue}{\left(-1 + x\right)}}^{\left(\frac{-1}{2}\right)}, -\left(-x\right)\right) \]

      10. metadata-eval 0.0%

          \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{\color{blue}{-0.5}}, -\left(-x\right)\right) \]

      11. remove-double-neg 0.0%

          \[\leadsto \mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{-0.5}, \color{blue}{x}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-1 + x\right)}^{-0.5}, {\left(-1 + x\right)}^{-0.5}, x\right)} \]
   9. Step-by-step derivation

      1. fma-udef 0.0%

         \[\leadsto \color{blue}{{\left(-1 + x\right)}^{-0.5} \cdot {\left(-1 + x\right)}^{-0.5} + x} \]

      2. pow-sqr 97.5%

         \[\leadsto \color{blue}{{\left(-1 + x\right)}^{\left(2 \cdot -0.5\right)}} + x \]

      3. metadata-eval 97.5%

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

      4. unpow-1 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 7: 99.0% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) -1.0 1.0)))

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. *-commutative 99.2%

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

      5. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 98.4%

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

Alternative 8: 50.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-*l/ 100.0%

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

   15. /-rgt-identity 100.0%

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

   16. distribute-rgt1-in 100.0%

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

   17. *-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. neg-mul-1 100.0%

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

   20. unsub-neg 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. frac-sub 99.6%

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

   3. *-un-lft-identity 99.6%

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

   4. *-commutative 99.6%

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

   5. *-un-lft-identity 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in x around 0 50.2%

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

7. Final simplification 50.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))