3frac (problem 3.3.3)

Percentage Accurate: 84.5% → 99.6%

Time: 15.6s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - x\\ t_1 := \frac{2 - x}{1 - x}\\ \mathbf{if}\;x \leq -65000000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot t_1\\ \mathbf{elif}\;x \leq 1050000:\\ \;\;\;\;\frac{t_0 + \left(x + 1\right) \cdot \left(x - \left(1 - x\right) \cdot -2\right)}{t_0 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x x) x)) (t_1 (/ (- 2.0 x) (- 1.0 x))))
   (if (<= x -65000000.0)
     (* (/ (/ 2.0 (* x x)) x) t_1)
     (if (<= x 1050000.0)
       (/ (+ t_0 (* (+ x 1.0) (- x (* (- 1.0 x) -2.0)))) (* t_0 (+ x 1.0)))
       (* t_1 (/ (/ (+ (/ 2.0 x) (/ 4.0 (* x x))) (+ x 1.0)) x))))))

C

double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -65000000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_1;
	} else if (x <= 1050000.0) {
		tmp = (t_0 + ((x + 1.0) * (x - ((1.0 - x) * -2.0)))) / (t_0 * (x + 1.0));
	} else {
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * x) - x
    t_1 = (2.0d0 - x) / (1.0d0 - x)
    if (x <= (-65000000.0d0)) then
        tmp = ((2.0d0 / (x * x)) / x) * t_1
    else if (x <= 1050000.0d0) then
        tmp = (t_0 + ((x + 1.0d0) * (x - ((1.0d0 - x) * (-2.0d0))))) / (t_0 * (x + 1.0d0))
    else
        tmp = t_1 * ((((2.0d0 / x) + (4.0d0 / (x * x))) / (x + 1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -65000000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_1;
	} else if (x <= 1050000.0) {
		tmp = (t_0 + ((x + 1.0) * (x - ((1.0 - x) * -2.0)))) / (t_0 * (x + 1.0));
	} else {
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) - x
	t_1 = (2.0 - x) / (1.0 - x)
	tmp = 0
	if x <= -65000000.0:
		tmp = ((2.0 / (x * x)) / x) * t_1
	elif x <= 1050000.0:
		tmp = (t_0 + ((x + 1.0) * (x - ((1.0 - x) * -2.0)))) / (t_0 * (x + 1.0))
	else:
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) - x)
	t_1 = Float64(Float64(2.0 - x) / Float64(1.0 - x))
	tmp = 0.0
	if (x <= -65000000.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * t_1);
	elseif (x <= 1050000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(x + 1.0) * Float64(x - Float64(Float64(1.0 - x) * -2.0)))) / Float64(t_0 * Float64(x + 1.0)));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x))) / Float64(x + 1.0)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) - x;
	t_1 = (2.0 - x) / (1.0 - x);
	tmp = 0.0;
	if (x <= -65000000.0)
		tmp = ((2.0 / (x * x)) / x) * t_1;
	elseif (x <= 1050000.0)
		tmp = (t_0 + ((x + 1.0) * (x - ((1.0 - x) * -2.0)))) / (t_0 * (x + 1.0));
	else
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -65000000.0], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 1050000.0], N[(N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(x - N[(N[(1.0 - x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e7

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. metadata-eval 75.8%

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

      3. frac-sub 19.3%

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

      4. +-commutative 19.3%

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

      5. distribute-neg-in 19.3%

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

      6. metadata-eval 19.3%

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

      7. sub-neg 19.3%

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

      8. *-commutative 19.3%

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

      9. neg-mul-1 19.3%

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

      10. +-commutative 19.3%

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

      11. distribute-neg-in 19.3%

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

      12. metadata-eval 19.3%

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

      13. sub-neg 19.3%

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

   5. Applied egg-rr 19.3%

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

   6. Taylor expanded in x around 0 19.3%

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

      1. neg-mul-1 19.3%

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

      2. unsub-neg 19.3%

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

   8. Simplified 19.3%

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

      1. clear-num 23.9%

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

      2. frac-sub 18.5%

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

      3. *-un-lft-identity 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. /-rgt-identity 18.5%

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

       2. associate-/r* 18.5%

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

       3. associate-/l* 19.4%

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

       4. associate-/r/ 19.4%

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

   12. Simplified 75.8%

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

   13. Taylor expanded in x around inf 99.9%

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

       1. unpow2 99.9%

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

   15. Simplified 99.9%

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

   if -6.5e7 < x < 1.05e6

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. frac-2neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. frac-sub 99.7%

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

      5. metadata-eval 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. neg-mul-1 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-lft-in 99.7%

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

      8. sqr-neg 99.7%

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

      9. unpow2 99.7%

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

      10. sqr-pow 99.7%

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

      11. *-rgt-identity 99.7%

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

      12. sqr-pow 99.7%

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

      13. sub-neg 99.7%

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

      14. unpow2 99.7%

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

   7. Simplified 99.7%

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

      1. frac-sub 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.05e6 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. frac-2neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. frac-sub 13.7%

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

      4. +-commutative 13.7%

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

      5. distribute-neg-in 13.7%

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

      6. metadata-eval 13.7%

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

      7. sub-neg 13.7%

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

      8. *-commutative 13.7%

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

      9. neg-mul-1 13.7%

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

      10. +-commutative 13.7%

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

      11. distribute-neg-in 13.7%

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

      12. metadata-eval 13.7%

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

      13. sub-neg 13.7%

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

   5. Applied egg-rr 13.7%

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

   6. Taylor expanded in x around 0 13.7%

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

      1. neg-mul-1 13.7%

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

      2. unsub-neg 13.7%

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

   8. Simplified 13.7%

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

      1. clear-num 14.9%

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

      2. frac-sub 9.6%

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

      3. *-un-lft-identity 9.6%

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

   10. Applied egg-rr 9.6%

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

       1. /-rgt-identity 9.6%

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

       2. associate-/r* 9.6%

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

       3. associate-/l* 10.5%

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

       4. associate-/r/ 10.5%

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

   12. Simplified 69.6%

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

   13. Taylor expanded in x around inf 99.9%

       \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
   14. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot 1}{x}} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       2. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{2}}{x} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       3. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{\color{blue}{4}}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       5. unpow2 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{4}{\color{blue}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

   15. Simplified 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{2}{x} + \frac{4}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65000000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{elif}\;x \leq 1050000:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \left(x - \left(1 - x\right) \cdot -2\right)}{\left(x \cdot x - x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - x}{1 - x} \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -0.0002 \lor \neg \left(t_0 \leq 10^{-24}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (or (<= t_0 -0.0002) (not (<= t_0 1e-24)))
     t_0
     (* (/ (/ 2.0 (* x x)) x) (/ (- 2.0 x) (- 1.0 x))))))

C

double code(double x) {
	double t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-24)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if ((t_0 <= (-0.0002d0)) .or. (.not. (t_0 <= 1d-24))) then
        tmp = t_0
    else
        tmp = ((2.0d0 / (x * x)) / x) * ((2.0d0 - x) / (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-24)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if (t_0 <= -0.0002) or not (t_0 <= 1e-24):
		tmp = t_0
	else:
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if ((t_0 <= -0.0002) || !(t_0 <= 1e-24))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * Float64(Float64(2.0 - x) / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if ((t_0 <= -0.0002) || ~((t_0 <= 1e-24)))
		tmp = t_0;
	else
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0002], N[Not[LessEqual[t$95$0, 1e-24]], $MachinePrecision]], t$95$0, N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2.0000000000000001e-4 or 9.99999999999999924e-25 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

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

   if -2.0000000000000001e-4 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 9.99999999999999924e-25

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. frac-2neg 72.6%

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

      2. metadata-eval 72.6%

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

      3. frac-sub 16.3%

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

      4. +-commutative 16.3%

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

      5. distribute-neg-in 16.3%

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

      6. metadata-eval 16.3%

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

      7. sub-neg 16.3%

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

      8. *-commutative 16.3%

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

      9. neg-mul-1 16.3%

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

      10. +-commutative 16.3%

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

      11. distribute-neg-in 16.3%

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

      12. metadata-eval 16.3%

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

      13. sub-neg 16.3%

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

   5. Applied egg-rr 16.3%

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

   6. Taylor expanded in x around 0 16.3%

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

      1. neg-mul-1 16.3%

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

      2. unsub-neg 16.3%

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

   8. Simplified 16.3%

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

      1. clear-num 19.1%

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

      2. frac-sub 13.8%

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

      3. *-un-lft-identity 13.8%

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

   10. Applied egg-rr 13.8%

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

       1. /-rgt-identity 13.8%

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

       2. associate-/r* 13.8%

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

       3. associate-/l* 14.7%

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

       4. associate-/r/ 14.7%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in x around inf 99.3%

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

       1. unpow2 99.3%

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

   15. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -0.0002 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-24}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -0.0002 \lor \neg \left(t_1 \leq 10^{-24}\right):\\ \;\;\;\;t_0 + \frac{x - 2}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x 1.0))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (or (<= t_1 -0.0002) (not (<= t_1 1e-24)))
     (+ t_0 (/ (- x 2.0) (* x (- 1.0 x))))
     (* (/ (/ 2.0 (* x x)) x) (/ (- 2.0 x) (- 1.0 x))))))

C

double code(double x) {
	double t_0 = 1.0 / (x + 1.0);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_1 <= -0.0002) || !(t_1 <= 1e-24)) {
		tmp = t_0 + ((x - 2.0) / (x * (1.0 - x)));
	} else {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (x + 1.0d0)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if ((t_1 <= (-0.0002d0)) .or. (.not. (t_1 <= 1d-24))) then
        tmp = t_0 + ((x - 2.0d0) / (x * (1.0d0 - x)))
    else
        tmp = ((2.0d0 / (x * x)) / x) * ((2.0d0 - x) / (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (x + 1.0);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_1 <= -0.0002) || !(t_1 <= 1e-24)) {
		tmp = t_0 + ((x - 2.0) / (x * (1.0 - x)));
	} else {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (x + 1.0)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if (t_1 <= -0.0002) or not (t_1 <= 1e-24):
		tmp = t_0 + ((x - 2.0) / (x * (1.0 - x)))
	else:
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(x + 1.0))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if ((t_1 <= -0.0002) || !(t_1 <= 1e-24))
		tmp = Float64(t_0 + Float64(Float64(x - 2.0) / Float64(x * Float64(1.0 - x))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * Float64(Float64(2.0 - x) / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (x + 1.0);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if ((t_1 <= -0.0002) || ~((t_1 <= 1e-24)))
		tmp = t_0 + ((x - 2.0) / (x * (1.0 - x)));
	else
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.0002], N[Not[LessEqual[t$95$1, 1e-24]], $MachinePrecision]], N[(t$95$0 + N[(N[(x - 2.0), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2.0000000000000001e-4 or 9.99999999999999924e-25 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. frac-sub 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

      8. *-commutative 99.7%

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

      9. neg-mul-1 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

   if -2.0000000000000001e-4 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 9.99999999999999924e-25

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. frac-2neg 72.6%

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

      2. metadata-eval 72.6%

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

      3. frac-sub 16.3%

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

      4. +-commutative 16.3%

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

      5. distribute-neg-in 16.3%

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

      6. metadata-eval 16.3%

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

      7. sub-neg 16.3%

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

      8. *-commutative 16.3%

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

      9. neg-mul-1 16.3%

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

      10. +-commutative 16.3%

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

      11. distribute-neg-in 16.3%

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

      12. metadata-eval 16.3%

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

      13. sub-neg 16.3%

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

   5. Applied egg-rr 16.3%

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

   6. Taylor expanded in x around 0 16.3%

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

      1. neg-mul-1 16.3%

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

      2. unsub-neg 16.3%

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

   8. Simplified 16.3%

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

      1. clear-num 19.1%

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

      2. frac-sub 13.8%

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

      3. *-un-lft-identity 13.8%

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

   10. Applied egg-rr 13.8%

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

       1. /-rgt-identity 13.8%

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

       2. associate-/r* 13.8%

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

       3. associate-/l* 14.7%

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

       4. associate-/r/ 14.7%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in x around inf 99.3%

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

       1. unpow2 99.3%

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

   15. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -0.0002 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-24}\right):\\ \;\;\;\;\frac{1}{x + 1} + \frac{x - 2}{x \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 - x}{1 - x}\\ \mathbf{if}\;x \leq -350000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 17000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{x - \left(1 - x\right) \cdot -2}{x \cdot x - x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 2.0 x) (- 1.0 x))))
   (if (<= x -350000.0)
     (* (/ (/ 2.0 (* x x)) x) t_0)
     (if (<= x 17000.0)
       (+ (/ 1.0 (+ x 1.0)) (/ (- x (* (- 1.0 x) -2.0)) (- (* x x) x)))
       (* t_0 (/ (/ (+ (/ 2.0 x) (/ 4.0 (* x x))) (+ x 1.0)) x))))))

C

double code(double x) {
	double t_0 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -350000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_0;
	} else if (x <= 17000.0) {
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	} else {
		tmp = t_0 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 - x) / (1.0d0 - x)
    if (x <= (-350000.0d0)) then
        tmp = ((2.0d0 / (x * x)) / x) * t_0
    else if (x <= 17000.0d0) then
        tmp = (1.0d0 / (x + 1.0d0)) + ((x - ((1.0d0 - x) * (-2.0d0))) / ((x * x) - x))
    else
        tmp = t_0 * ((((2.0d0 / x) + (4.0d0 / (x * x))) / (x + 1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -350000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_0;
	} else if (x <= 17000.0) {
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	} else {
		tmp = t_0 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (2.0 - x) / (1.0 - x)
	tmp = 0
	if x <= -350000.0:
		tmp = ((2.0 / (x * x)) / x) * t_0
	elif x <= 17000.0:
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x))
	else:
		tmp = t_0 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(2.0 - x) / Float64(1.0 - x))
	tmp = 0.0
	if (x <= -350000.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * t_0);
	elseif (x <= 17000.0)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(Float64(x - Float64(Float64(1.0 - x) * -2.0)) / Float64(Float64(x * x) - x)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x))) / Float64(x + 1.0)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (2.0 - x) / (1.0 - x);
	tmp = 0.0;
	if (x <= -350000.0)
		tmp = ((2.0 / (x * x)) / x) * t_0;
	elseif (x <= 17000.0)
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	else
		tmp = t_0 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -350000.0], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 17000.0], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - N[(N[(1.0 - x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e5

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. metadata-eval 75.8%

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

      3. frac-sub 19.3%

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

      4. +-commutative 19.3%

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

      5. distribute-neg-in 19.3%

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

      6. metadata-eval 19.3%

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

      7. sub-neg 19.3%

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

      8. *-commutative 19.3%

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

      9. neg-mul-1 19.3%

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

      10. +-commutative 19.3%

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

      11. distribute-neg-in 19.3%

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

      12. metadata-eval 19.3%

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

      13. sub-neg 19.3%

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

   5. Applied egg-rr 19.3%

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

   6. Taylor expanded in x around 0 19.3%

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

      1. neg-mul-1 19.3%

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

      2. unsub-neg 19.3%

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

   8. Simplified 19.3%

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

      1. clear-num 23.9%

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

      2. frac-sub 18.5%

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

      3. *-un-lft-identity 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. /-rgt-identity 18.5%

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

       2. associate-/r* 18.5%

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

       3. associate-/l* 19.4%

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

       4. associate-/r/ 19.4%

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

   12. Simplified 75.8%

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

   13. Taylor expanded in x around inf 99.9%

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

       1. unpow2 99.9%

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

   15. Simplified 99.9%

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

   if -3.5e5 < x < 17000

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. frac-2neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. frac-sub 99.7%

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

      5. metadata-eval 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. neg-mul-1 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-lft-in 99.7%

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

      8. sqr-neg 99.7%

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

      9. unpow2 99.7%

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

      10. sqr-pow 99.7%

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

      11. *-rgt-identity 99.7%

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

      12. sqr-pow 99.7%

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

      13. sub-neg 99.7%

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

      14. unpow2 99.7%

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

   7. Simplified 99.7%

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

   if 17000 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. frac-2neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. frac-sub 13.7%

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

      4. +-commutative 13.7%

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

      5. distribute-neg-in 13.7%

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

      6. metadata-eval 13.7%

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

      7. sub-neg 13.7%

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

      8. *-commutative 13.7%

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

      9. neg-mul-1 13.7%

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

      10. +-commutative 13.7%

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

      11. distribute-neg-in 13.7%

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

      12. metadata-eval 13.7%

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

      13. sub-neg 13.7%

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

   5. Applied egg-rr 13.7%

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

   6. Taylor expanded in x around 0 13.7%

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

      1. neg-mul-1 13.7%

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

      2. unsub-neg 13.7%

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

   8. Simplified 13.7%

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

      1. clear-num 14.9%

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

      2. frac-sub 9.6%

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

      3. *-un-lft-identity 9.6%

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

   10. Applied egg-rr 9.6%

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

       1. /-rgt-identity 9.6%

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

       2. associate-/r* 9.6%

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

       3. associate-/l* 10.5%

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

       4. associate-/r/ 10.5%

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

   12. Simplified 69.6%

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

   13. Taylor expanded in x around inf 99.9%

       \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
   14. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot 1}{x}} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       2. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{2}}{x} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       3. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{\color{blue}{4}}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       5. unpow2 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{4}{\color{blue}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

   15. Simplified 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{2}{x} + \frac{4}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{elif}\;x \leq 17000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{x - \left(1 - x\right) \cdot -2}{x \cdot x - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - x}{1 - x} \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \]

Alternative 5: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ t_1 := \frac{2 - x}{1 - x}\\ \mathbf{if}\;x \leq -135000000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot t_1\\ \mathbf{elif}\;x \leq 140000000:\\ \;\;\;\;\frac{t_0 + \left(2 - x\right) \cdot \left(-1 - x\right)}{\left(x + 1\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x))) (t_1 (/ (- 2.0 x) (- 1.0 x))))
   (if (<= x -135000000.0)
     (* (/ (/ 2.0 (* x x)) x) t_1)
     (if (<= x 140000000.0)
       (/ (+ t_0 (* (- 2.0 x) (- -1.0 x))) (* (+ x 1.0) t_0))
       (* t_1 (/ (/ (+ (/ 2.0 x) (/ 4.0 (* x x))) (+ x 1.0)) x))))))

C

double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -135000000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_1;
	} else if (x <= 140000000.0) {
		tmp = (t_0 + ((2.0 - x) * (-1.0 - x))) / ((x + 1.0) * t_0);
	} else {
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 - x)
    t_1 = (2.0d0 - x) / (1.0d0 - x)
    if (x <= (-135000000.0d0)) then
        tmp = ((2.0d0 / (x * x)) / x) * t_1
    else if (x <= 140000000.0d0) then
        tmp = (t_0 + ((2.0d0 - x) * ((-1.0d0) - x))) / ((x + 1.0d0) * t_0)
    else
        tmp = t_1 * ((((2.0d0 / x) + (4.0d0 / (x * x))) / (x + 1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = (2.0 - x) / (1.0 - x);
	double tmp;
	if (x <= -135000000.0) {
		tmp = ((2.0 / (x * x)) / x) * t_1;
	} else if (x <= 140000000.0) {
		tmp = (t_0 + ((2.0 - x) * (-1.0 - x))) / ((x + 1.0) * t_0);
	} else {
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (1.0 - x)
	t_1 = (2.0 - x) / (1.0 - x)
	tmp = 0
	if x <= -135000000.0:
		tmp = ((2.0 / (x * x)) / x) * t_1
	elif x <= 140000000.0:
		tmp = (t_0 + ((2.0 - x) * (-1.0 - x))) / ((x + 1.0) * t_0)
	else:
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 - x))
	t_1 = Float64(Float64(2.0 - x) / Float64(1.0 - x))
	tmp = 0.0
	if (x <= -135000000.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * t_1);
	elseif (x <= 140000000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(2.0 - x) * Float64(-1.0 - x))) / Float64(Float64(x + 1.0) * t_0));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x))) / Float64(x + 1.0)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (1.0 - x);
	t_1 = (2.0 - x) / (1.0 - x);
	tmp = 0.0;
	if (x <= -135000000.0)
		tmp = ((2.0 / (x * x)) / x) * t_1;
	elseif (x <= 140000000.0)
		tmp = (t_0 + ((2.0 - x) * (-1.0 - x))) / ((x + 1.0) * t_0);
	else
		tmp = t_1 * ((((2.0 / x) + (4.0 / (x * x))) / (x + 1.0)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -135000000.0], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 140000000.0], N[(N[(t$95$0 + N[(N[(2.0 - x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e8

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. metadata-eval 75.8%

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

      3. frac-sub 19.3%

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

      4. +-commutative 19.3%

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

      5. distribute-neg-in 19.3%

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

      6. metadata-eval 19.3%

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

      7. sub-neg 19.3%

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

      8. *-commutative 19.3%

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

      9. neg-mul-1 19.3%

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

      10. +-commutative 19.3%

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

      11. distribute-neg-in 19.3%

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

      12. metadata-eval 19.3%

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

      13. sub-neg 19.3%

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

   5. Applied egg-rr 19.3%

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

   6. Taylor expanded in x around 0 19.3%

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

      1. neg-mul-1 19.3%

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

      2. unsub-neg 19.3%

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

   8. Simplified 19.3%

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

      1. clear-num 23.9%

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

      2. frac-sub 18.5%

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

      3. *-un-lft-identity 18.5%

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

   10. Applied egg-rr 18.5%

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

       1. /-rgt-identity 18.5%

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

       2. associate-/r* 18.5%

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

       3. associate-/l* 19.4%

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

       4. associate-/r/ 19.4%

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

   12. Simplified 75.8%

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

   13. Taylor expanded in x around inf 99.9%

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

       1. unpow2 99.9%

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

   15. Simplified 99.9%

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

   if -1.35e8 < x < 1.4e8

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. frac-sub 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

      8. *-commutative 99.7%

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

      9. neg-mul-1 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

      1. frac-sub 100.0%

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

      2. *-un-lft-identity 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 1.4e8 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. frac-2neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. frac-sub 13.7%

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

      4. +-commutative 13.7%

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

      5. distribute-neg-in 13.7%

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

      6. metadata-eval 13.7%

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

      7. sub-neg 13.7%

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

      8. *-commutative 13.7%

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

      9. neg-mul-1 13.7%

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

      10. +-commutative 13.7%

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

      11. distribute-neg-in 13.7%

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

      12. metadata-eval 13.7%

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

      13. sub-neg 13.7%

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

   5. Applied egg-rr 13.7%

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

   6. Taylor expanded in x around 0 13.7%

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

      1. neg-mul-1 13.7%

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

      2. unsub-neg 13.7%

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

   8. Simplified 13.7%

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

      1. clear-num 14.9%

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

      2. frac-sub 9.6%

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

      3. *-un-lft-identity 9.6%

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

   10. Applied egg-rr 9.6%

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

       1. /-rgt-identity 9.6%

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

       2. associate-/r* 9.6%

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

       3. associate-/l* 10.5%

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

       4. associate-/r/ 10.5%

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

   12. Simplified 69.6%

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

   13. Taylor expanded in x around inf 99.9%

       \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
   14. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot 1}{x}} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       2. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{\color{blue}{2}}{x} + 4 \cdot \frac{1}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       3. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{\color{blue}{4}}{{x}^{2}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

       5. unpow2 99.9%

          \[\leadsto \frac{\frac{\frac{2}{x} + \frac{4}{\color{blue}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]

   15. Simplified 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{2}{x} + \frac{4}{x \cdot x}}}{x + 1}}{x} \cdot \frac{2 - x}{1 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -135000000:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{elif}\;x \leq 140000000:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(2 - x\right) \cdot \left(-1 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - x}{1 - x} \cdot \frac{\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{x + 1}}{x}\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350000 \lor \neg \left(x \leq 250000\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \frac{x - \left(1 - x\right) \cdot -2}{x \cdot x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -350000.0) (not (<= x 250000.0)))
   (* (/ (/ 2.0 (* x x)) x) (/ (- 2.0 x) (- 1.0 x)))
   (+ (/ 1.0 (+ x 1.0)) (/ (- x (* (- 1.0 x) -2.0)) (- (* x x) x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -350000.0) || !(x <= 250000.0)) {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	} else {
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-350000.0d0)) .or. (.not. (x <= 250000.0d0))) then
        tmp = ((2.0d0 / (x * x)) / x) * ((2.0d0 - x) / (1.0d0 - x))
    else
        tmp = (1.0d0 / (x + 1.0d0)) + ((x - ((1.0d0 - x) * (-2.0d0))) / ((x * x) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -350000.0) || !(x <= 250000.0)) {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	} else {
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -350000.0) or not (x <= 250000.0):
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x))
	else:
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -350000.0) || !(x <= 250000.0))
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * Float64(Float64(2.0 - x) / Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(Float64(x - Float64(Float64(1.0 - x) * -2.0)) / Float64(Float64(x * x) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -350000.0) || ~((x <= 250000.0)))
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	else
		tmp = (1.0 / (x + 1.0)) + ((x - ((1.0 - x) * -2.0)) / ((x * x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -350000.0], N[Not[LessEqual[x, 250000.0]], $MachinePrecision]], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - N[(N[(1.0 - x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e5 or 2.5e5 < x

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. frac-2neg 72.6%

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

      2. metadata-eval 72.6%

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

      3. frac-sub 16.3%

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

      4. +-commutative 16.3%

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

      5. distribute-neg-in 16.3%

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

      6. metadata-eval 16.3%

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

      7. sub-neg 16.3%

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

      8. *-commutative 16.3%

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

      9. neg-mul-1 16.3%

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

      10. +-commutative 16.3%

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

      11. distribute-neg-in 16.3%

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

      12. metadata-eval 16.3%

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

      13. sub-neg 16.3%

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

   5. Applied egg-rr 16.3%

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

   6. Taylor expanded in x around 0 16.3%

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

      1. neg-mul-1 16.3%

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

      2. unsub-neg 16.3%

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

   8. Simplified 16.3%

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

      1. clear-num 19.1%

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

      2. frac-sub 13.8%

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

      3. *-un-lft-identity 13.8%

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

   10. Applied egg-rr 13.8%

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

       1. /-rgt-identity 13.8%

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

       2. associate-/r* 13.8%

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

       3. associate-/l* 14.7%

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

       4. associate-/r/ 14.7%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in x around inf 99.3%

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

       1. unpow2 99.3%

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

   15. Simplified 99.3%

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

   if -3.5e5 < x < 2.5e5

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. frac-2neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. frac-sub 99.7%

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

      5. metadata-eval 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. neg-mul-1 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-lft-in 99.7%

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

      8. sqr-neg 99.7%

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

      9. unpow2 99.7%

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

      10. sqr-pow 99.7%

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

      11. *-rgt-identity 99.7%

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

      12. sqr-pow 99.7%

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

      13. sub-neg 99.7%

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

      14. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350000 \lor \neg \left(x \leq 250000\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \frac{x - \left(1 - x\right) \cdot -2}{x \cdot x - x}\\ \end{array} \]

Alternative 7: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.2)))
   (* (/ (/ 2.0 (* x x)) x) (/ (- 2.0 x) (- 1.0 x)))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2)) {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = ((2.0d0 / (x * x)) / x) * ((2.0d0 - x) / (1.0d0 - x))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2)) {
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.2):
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x))
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.2))
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) / x) * Float64(Float64(2.0 - x) / Float64(1.0 - x)));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.2)))
		tmp = ((2.0 / (x * x)) / x) * ((2.0 - x) / (1.0 - x));
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(2.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.19999999999999996 < x

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. frac-2neg 72.6%

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

      2. metadata-eval 72.6%

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

      3. frac-sub 17.3%

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

      4. +-commutative 17.3%

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

      5. distribute-neg-in 17.3%

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

      6. metadata-eval 17.3%

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

      7. sub-neg 17.3%

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

      8. *-commutative 17.3%

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

      9. neg-mul-1 17.3%

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

      10. +-commutative 17.3%

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

      11. distribute-neg-in 17.3%

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

      12. metadata-eval 17.3%

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

      13. sub-neg 17.3%

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

   5. Applied egg-rr 17.3%

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

   6. Taylor expanded in x around 0 17.3%

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

      1. neg-mul-1 17.3%

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

      2. unsub-neg 17.3%

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

   8. Simplified 17.3%

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

      1. clear-num 20.1%

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

      2. frac-sub 14.8%

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

      3. *-un-lft-identity 14.8%

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

   10. Applied egg-rr 14.8%

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

       1. /-rgt-identity 14.8%

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

       2. associate-/r* 14.8%

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

       3. associate-/l* 15.7%

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

       4. associate-/r/ 15.7%

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

   12. Simplified 72.6%

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

   13. Taylor expanded in x around inf 98.3%

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

       1. unpow2 98.3%

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

   15. Simplified 98.3%

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

   if -1 < x < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x} \cdot \frac{2 - x}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 8: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-0.3333333333333333}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ -0.3333333333333333 (* x x))
   (- (- x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 / (x * x);
	} else {
		tmp = -x - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-0.3333333333333333d0) / (x * x)
    else
        tmp = -x - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 / (x * x);
	} else {
		tmp = -x - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -0.3333333333333333 / (x * x)
	else:
		tmp = -x - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-0.3333333333333333 / Float64(x * x));
	else
		tmp = Float64(Float64(-x) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -0.3333333333333333 / (x * x);
	else
		tmp = -x - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[((-x) - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. sub-neg 72.6%

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

      2. flip-+ 17.2%

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

   5. Applied egg-rr 11.5%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{4}{x \cdot x} - \frac{1}{1 - x} \cdot \frac{1}{1 - x}}{\frac{2}{x} - \frac{1}{1 - x}}} \]

   6. Taylor expanded in x around inf 11.0%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{1}{1 - x} \cdot \frac{1}{1 - x}}{\color{blue}{\frac{3}{x}}} \]

   7. Taylor expanded in x around inf 59.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 59.0%

         \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot x}} \]

   9. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-0.3333333333333333}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) - \frac{2}{x}\\ \end{array} \]

Alternative 9: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-0.3333333333333333}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ -0.3333333333333333 (* x x))
   (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-0.3333333333333333d0) / (x * x)
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -0.3333333333333333 / (x * x)
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-0.3333333333333333 / Float64(x * x));
	else
		tmp = Float64(-2.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -0.3333333333333333 / (x * x);
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. sub-neg 72.6%

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

      2. flip-+ 17.2%

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

   5. Applied egg-rr 11.5%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{4}{x \cdot x} - \frac{1}{1 - x} \cdot \frac{1}{1 - x}}{\frac{2}{x} - \frac{1}{1 - x}}} \]

   6. Taylor expanded in x around inf 11.0%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{1}{1 - x} \cdot \frac{1}{1 - x}}{\color{blue}{\frac{3}{x}}} \]

   7. Taylor expanded in x around inf 59.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 59.0%

         \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot x}} \]

   9. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-0.3333333333333333}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 10: 83.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- -1.0 (/ 2.0 x))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Python

def code(x):
	return 1.0 + (-1.0 - (2.0 / x))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 + N[(-1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

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

3. Simplified 85.8%

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

4. Taylor expanded in x around 0 49.6%

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

5. Taylor expanded in x around 0 84.9%

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

6. Final simplification 84.9%

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

Alternative 11: 51.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

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

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

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

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 85.8%

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

3. Simplified 85.8%

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

4. Taylor expanded in x around 0 50.6%

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

5. Final simplification 50.6%

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

Alternative 12: 3.3% accurate, 15.0× 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 85.8%

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

3. Simplified 85.8%

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

4. Taylor expanded in x around 0 49.6%

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

5. Taylor expanded in x around inf 3.2%

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

6. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023285 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))