3frac (problem 3.3.3)

Percentage Accurate: 85.5% → 99.6%

Time: 7.9s

Alternatives: 12

Speedup: 1.3×

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: 85.5% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - x\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_2 := t_0 \cdot \left(1 + x\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{t_2}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + -2\right) \cdot \left(-1 - x\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x x) x))
        (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_2 (* t_0 (+ 1.0 x))))
   (if (<= t_1 -5e-24)
     (/ (+ t_0 (* (+ 1.0 x) (+ x (* -2.0 (+ x -1.0))))) t_2)
     (if (<= t_1 0.0)
       (* 2.0 (pow x -3.0))
       (/ (+ t_0 (* (+ x -2.0) (- -1.0 x))) t_2)))))

C

double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_2 = t_0 * (1.0 + x);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * x) - x
    t_1 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_2 = t_0 * (1.0d0 + x)
    if (t_1 <= (-5d-24)) then
        tmp = (t_0 + ((1.0d0 + x) * (x + ((-2.0d0) * (x + (-1.0d0)))))) / t_2
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (x ** (-3.0d0))
    else
        tmp = (t_0 + ((x + (-2.0d0)) * ((-1.0d0) - x))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_2 = t_0 * (1.0 + x);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * Math.pow(x, -3.0);
	} else {
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) - x
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_2 = t_0 * (1.0 + x)
	tmp = 0
	if t_1 <= -5e-24:
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2
	elif t_1 <= 0.0:
		tmp = 2.0 * math.pow(x, -3.0)
	else:
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) - x)
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_2 = Float64(t_0 * Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-24)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x + Float64(-2.0 * Float64(x + -1.0))))) / t_2);
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(x + -2.0) * Float64(-1.0 - x))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) - x;
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_2 = t_0 * (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e-24)
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	elseif (t_1 <= 0.0)
		tmp = 2.0 * (x ^ -3.0);
	else
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	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[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-24], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(x + -2.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.9999999999999998e-24

   1. Initial program 98.6%

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

      1. associate-+l- 98.6%

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

      2. sub-neg 98.6%

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

      3. neg-mul-1 98.6%

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

      4. metadata-eval 98.6%

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

      5. cancel-sign-sub-inv 98.6%

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

      6. +-commutative 98.6%

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

      7. *-lft-identity 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

   3. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. frac-2neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. frac-sub 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. distribute-neg-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. +-commutative 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. metadata-eval 98.7%

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

      13. sub-neg 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. cancel-sign-sub 98.7%

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

      2. *-commutative 98.7%

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

      3. neg-mul-1 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. sqr-neg 98.7%

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

      9. *-rgt-identity 98.7%

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

      10. fma-def 98.7%

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

      11. fma-neg 98.7%

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

   7. Simplified 98.7%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -4.9999999999999998e-24 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 74.3%

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

      1. associate-+l- 74.3%

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

      2. sub-neg 74.3%

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

      3. neg-mul-1 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. +-commutative 74.3%

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

      7. *-lft-identity 74.3%

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

      8. sub-neg 74.3%

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

      9. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 98.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} \]

      2. expm1-udef 74.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1} \]

      3. div-inv 74.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1 \]

      4. pow-flip 74.3%

         \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1 \]

      5. metadata-eval 74.3%

         \[\leadsto e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1 \]

   6. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} \]

      2. expm1-log1p 99.9%

         \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. sub-neg 98.9%

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

      3. neg-mul-1 98.9%

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

      4. metadata-eval 98.9%

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

      5. cancel-sign-sub-inv 98.9%

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

      6. +-commutative 98.9%

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

      7. *-lft-identity 98.9%

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

      8. sub-neg 98.9%

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

      9. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. frac-2neg 98.9%

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

      2. frac-2neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. frac-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-neg-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. sub-neg 98.9%

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

      10. +-commutative 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. metadata-eval 98.9%

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

      13. sub-neg 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. cancel-sign-sub 98.9%

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

      2. *-commutative 98.9%

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

      3. neg-mul-1 98.9%

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

      4. unsub-neg 98.9%

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

      5. sub-neg 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-lft-in 98.9%

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

      8. sqr-neg 98.9%

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

      9. *-rgt-identity 98.9%

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

      10. fma-def 98.9%

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

      11. fma-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around 0 98.9%

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

      1. frac-sub 99.9%

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

      2. /-rgt-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. /-rgt-identity 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(x + -2\right) \cdot \left(-1 - x\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-24} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{t_1 + \left(x + -2\right) \cdot \left(-1 - x\right)}{t_1 \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1 (- (* x x) x)))
   (if (or (<= t_0 -5e-24) (not (<= t_0 0.0)))
     (/ (+ t_1 (* (+ x -2.0) (- -1.0 x))) (* t_1 (+ 1.0 x)))
     (/ 2.0 (* x (* x x))))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if ((t_0 <= -5e-24) || !(t_0 <= 0.0)) {
		tmp = (t_1 + ((x + -2.0) * (-1.0 - x))) / (t_1 * (1.0 + x));
	} else {
		tmp = 2.0 / (x * (x * 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 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = (x * x) - x
    if ((t_0 <= (-5d-24)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (t_1 + ((x + (-2.0d0)) * ((-1.0d0) - x))) / (t_1 * (1.0d0 + x))
    else
        tmp = 2.0d0 / (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if ((t_0 <= -5e-24) || !(t_0 <= 0.0)) {
		tmp = (t_1 + ((x + -2.0) * (-1.0 - x))) / (t_1 * (1.0 + x));
	} else {
		tmp = 2.0 / (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_1 = (x * x) - x
	tmp = 0
	if (t_0 <= -5e-24) or not (t_0 <= 0.0):
		tmp = (t_1 + ((x + -2.0) * (-1.0 - x))) / (t_1 * (1.0 + x))
	else:
		tmp = 2.0 / (x * (x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(Float64(x * x) - x)
	tmp = 0.0
	if ((t_0 <= -5e-24) || !(t_0 <= 0.0))
		tmp = Float64(Float64(t_1 + Float64(Float64(x + -2.0) * Float64(-1.0 - x))) / Float64(t_1 * Float64(1.0 + x)));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_1 = (x * x) - x;
	tmp = 0.0;
	if ((t_0 <= -5e-24) || ~((t_0 <= 0.0)))
		tmp = (t_1 + ((x + -2.0) * (-1.0 - x))) / (t_1 * (1.0 + x));
	else
		tmp = 2.0 / (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-24], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(t$95$1 + N[(N[(x + -2.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * 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))) < -4.9999999999999998e-24 or 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 98.8%

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

      1. associate-+l- 98.8%

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

      2. sub-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. metadata-eval 98.8%

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

      5. cancel-sign-sub-inv 98.8%

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

      6. +-commutative 98.8%

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

      7. *-lft-identity 98.8%

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

      8. sub-neg 98.8%

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

      9. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. frac-2neg 98.8%

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

      3. metadata-eval 98.8%

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

      4. frac-sub 98.8%

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

      5. metadata-eval 98.8%

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

      6. +-commutative 98.8%

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

      7. distribute-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. sub-neg 98.8%

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

      10. +-commutative 98.8%

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

      11. distribute-neg-in 98.8%

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

      12. metadata-eval 98.8%

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

      13. sub-neg 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. cancel-sign-sub 98.8%

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

      2. *-commutative 98.8%

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

      3. neg-mul-1 98.8%

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

      4. unsub-neg 98.8%

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

      5. sub-neg 98.8%

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

      6. +-commutative 98.8%

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

      7. distribute-lft-in 98.8%

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

      8. sqr-neg 98.8%

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

      9. *-rgt-identity 98.8%

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

      10. fma-def 98.8%

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

      11. fma-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. frac-sub 99.9%

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

      2. /-rgt-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. /-rgt-identity 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -4.9999999999999998e-24 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 74.3%

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

      1. associate-+l- 74.3%

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

      2. sub-neg 74.3%

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

      3. neg-mul-1 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. +-commutative 74.3%

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

      7. *-lft-identity 74.3%

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

      8. sub-neg 74.3%

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

      9. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.9%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-24} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0\right):\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(x + -2\right) \cdot \left(-1 - x\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - x\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_2 := t_0 \cdot \left(1 + x\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{t_2}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + -2\right) \cdot \left(-1 - x\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x x) x))
        (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_2 (* t_0 (+ 1.0 x))))
   (if (<= t_1 -5e-24)
     (/ (+ t_0 (* (+ 1.0 x) (+ x (* -2.0 (+ x -1.0))))) t_2)
     (if (<= t_1 0.0)
       (/ 2.0 (* x (* x x)))
       (/ (+ t_0 (* (+ x -2.0) (- -1.0 x))) t_2)))))

C

double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_2 = t_0 * (1.0 + x);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * x) - x
    t_1 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_2 = t_0 * (1.0d0 + x)
    if (t_1 <= (-5d-24)) then
        tmp = (t_0 + ((1.0d0 + x) * (x + ((-2.0d0) * (x + (-1.0d0)))))) / t_2
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = (t_0 + ((x + (-2.0d0)) * ((-1.0d0) - x))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_2 = t_0 * (1.0 + x);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) - x
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_2 = t_0 * (1.0 + x)
	tmp = 0
	if t_1 <= -5e-24:
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2
	elif t_1 <= 0.0:
		tmp = 2.0 / (x * (x * x))
	else:
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) - x)
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_2 = Float64(t_0 * Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -5e-24)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x + Float64(-2.0 * Float64(x + -1.0))))) / t_2);
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(x + -2.0) * Float64(-1.0 - x))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) - x;
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_2 = t_0 * (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -5e-24)
		tmp = (t_0 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / t_2;
	elseif (t_1 <= 0.0)
		tmp = 2.0 / (x * (x * x));
	else
		tmp = (t_0 + ((x + -2.0) * (-1.0 - x))) / t_2;
	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[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-24], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(x + -2.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.9999999999999998e-24

   1. Initial program 98.6%

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

      1. associate-+l- 98.6%

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

      2. sub-neg 98.6%

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

      3. neg-mul-1 98.6%

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

      4. metadata-eval 98.6%

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

      5. cancel-sign-sub-inv 98.6%

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

      6. +-commutative 98.6%

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

      7. *-lft-identity 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

   3. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. frac-2neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. frac-sub 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. distribute-neg-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. +-commutative 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. metadata-eval 98.7%

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

      13. sub-neg 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. cancel-sign-sub 98.7%

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

      2. *-commutative 98.7%

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

      3. neg-mul-1 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. sqr-neg 98.7%

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

      9. *-rgt-identity 98.7%

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

      10. fma-def 98.7%

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

      11. fma-neg 98.7%

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

   7. Simplified 98.7%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -4.9999999999999998e-24 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 74.3%

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

      1. associate-+l- 74.3%

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

      2. sub-neg 74.3%

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

      3. neg-mul-1 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. +-commutative 74.3%

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

      7. *-lft-identity 74.3%

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

      8. sub-neg 74.3%

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

      9. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.9%

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

   6. Applied egg-rr 98.9%

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

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 98.9%

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

      1. associate-+l- 98.9%

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

      2. sub-neg 98.9%

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

      3. neg-mul-1 98.9%

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

      4. metadata-eval 98.9%

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

      5. cancel-sign-sub-inv 98.9%

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

      6. +-commutative 98.9%

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

      7. *-lft-identity 98.9%

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

      8. sub-neg 98.9%

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

      9. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. frac-2neg 98.9%

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

      2. frac-2neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. frac-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-neg-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. sub-neg 98.9%

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

      10. +-commutative 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. metadata-eval 98.9%

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

      13. sub-neg 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. cancel-sign-sub 98.9%

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

      2. *-commutative 98.9%

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

      3. neg-mul-1 98.9%

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

      4. unsub-neg 98.9%

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

      5. sub-neg 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-lft-in 98.9%

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

      8. sqr-neg 98.9%

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

      9. *-rgt-identity 98.9%

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

      10. fma-def 98.9%

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

      11. fma-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around 0 98.9%

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

      1. frac-sub 99.9%

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

      2. /-rgt-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. /-rgt-identity 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(x + -2\right) \cdot \left(-1 - x\right)}{\left(x \cdot x - x\right) \cdot \left(1 + x\right)}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0 + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{\frac{-1 - x \cdot -0.5}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-6)
     (+ t_0 (/ (+ x (* -2.0 (+ x -1.0))) (- (* x x) x)))
     (if (<= t_1 2e-18)
       (/ 2.0 (* x (* x x)))
       (+ t_0 (/ (/ (- -1.0 (* x -0.5)) x) (+ 0.5 (* x -0.5))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + (((-1.0 - (x * -0.5)) / x) / (0.5 + (x * -0.5)));
	}
	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 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-1d-6)) then
        tmp = t_0 + ((x + ((-2.0d0) * (x + (-1.0d0)))) / ((x * x) - x))
    else if (t_1 <= 2d-18) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = t_0 + ((((-1.0d0) - (x * (-0.5d0))) / x) / (0.5d0 + (x * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + (((-1.0 - (x * -0.5)) / x) / (0.5 + (x * -0.5)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -1e-6:
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x))
	elif t_1 <= 2e-18:
		tmp = 2.0 / (x * (x * x))
	else:
		tmp = t_0 + (((-1.0 - (x * -0.5)) / x) / (0.5 + (x * -0.5)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(t_0 + Float64(Float64(x + Float64(-2.0 * Float64(x + -1.0))) / Float64(Float64(x * x) - x)));
	elseif (t_1 <= 2e-18)
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(-1.0 - Float64(x * -0.5)) / x) / Float64(0.5 + Float64(x * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	elseif (t_1 <= 2e-18)
		tmp = 2.0 / (x * (x * x));
	else
		tmp = t_0 + (((-1.0 - (x * -0.5)) / x) / (0.5 + (x * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $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[LessEqual[t$95$1, -1e-6], N[(t$95$0 + N[(N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. +-commutative 99.7%

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

      7. *-lft-identity 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      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. *-rgt-identity 99.7%

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

      10. fma-def 99.7%

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

      11. fma-neg 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 -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000001e-18

   1. Initial program 73.6%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. neg-mul-1 73.6%

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

      4. metadata-eval 73.6%

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

      5. cancel-sign-sub-inv 73.6%

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

      6. +-commutative 73.6%

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

      7. *-lft-identity 73.6%

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

      8. sub-neg 73.6%

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

      9. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.0000000000000001e-18 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. metadata-eval 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. *-lft-identity 99.8%

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

      8. sub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. frac-sub 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. div-inv 99.8%

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

      11. metadata-eval 99.8%

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

      12. div-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

      15. distribute-neg-in 99.8%

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

      16. metadata-eval 99.8%

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

      17. sub-neg 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. associate-/r* 99.9%

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

      3. *-commutative 99.9%

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

      4. cancel-sign-sub-inv 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-lft-identity 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-rgt-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{-0.5 \cdot x + 1}}{x}}{0.5 + x \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{-1 - x \cdot -0.5}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot x - x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{0.5 - \frac{1}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-6)
     (+ t_0 (/ (- 2.0 x) (- (* x x) x)))
     (if (<= t_1 2e-18)
       (/ 2.0 (* x (* x x)))
       (+ t_0 (/ (- 0.5 (/ 1.0 x)) (+ 0.5 (* x -0.5))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	}
	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 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-1d-6)) then
        tmp = t_0 + ((2.0d0 - x) / ((x * x) - x))
    else if (t_1 <= 2d-18) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = t_0 + ((0.5d0 - (1.0d0 / x)) / (0.5d0 + (x * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -1e-6:
		tmp = t_0 + ((2.0 - x) / ((x * x) - x))
	elif t_1 <= 2e-18:
		tmp = 2.0 / (x * (x * x))
	else:
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(t_0 + Float64(Float64(2.0 - x) / Float64(Float64(x * x) - x)));
	elseif (t_1 <= 2e-18)
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(t_0 + Float64(Float64(0.5 - Float64(1.0 / x)) / Float64(0.5 + Float64(x * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	elseif (t_1 <= 2e-18)
		tmp = 2.0 / (x * (x * x));
	else
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $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[LessEqual[t$95$1, -1e-6], N[(t$95$0 + N[(N[(2.0 - x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(0.5 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. +-commutative 99.7%

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

      7. *-lft-identity 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      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. *-rgt-identity 99.7%

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

      10. fma-def 99.7%

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

      11. fma-neg 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. Taylor expanded in x around 0 99.7%

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

   if -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000001e-18

   1. Initial program 73.6%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. neg-mul-1 73.6%

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

      4. metadata-eval 73.6%

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

      5. cancel-sign-sub-inv 73.6%

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

      6. +-commutative 73.6%

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

      7. *-lft-identity 73.6%

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

      8. sub-neg 73.6%

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

      9. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.0000000000000001e-18 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. metadata-eval 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. *-lft-identity 99.8%

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

      8. sub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. frac-sub 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. div-inv 99.8%

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

      11. metadata-eval 99.8%

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

      12. div-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

      15. distribute-neg-in 99.8%

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

      16. metadata-eval 99.8%

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

      17. sub-neg 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. associate-/r* 99.9%

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

      3. *-commutative 99.9%

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

      4. cancel-sign-sub-inv 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-lft-identity 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-rgt-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{-0.5 \cdot x + 1}}{x}}{0.5 + x \cdot -0.5} \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{\frac{1}{x} - 0.5}}{0.5 + x \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{2 - x}{x \cdot x - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{0.5 - \frac{1}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0 + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{0.5 - \frac{1}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-6)
     (+ t_0 (/ (+ x (* -2.0 (+ x -1.0))) (- (* x x) x)))
     (if (<= t_1 2e-18)
       (/ 2.0 (* x (* x x)))
       (+ t_0 (/ (- 0.5 (/ 1.0 x)) (+ 0.5 (* x -0.5))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	}
	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 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-1d-6)) then
        tmp = t_0 + ((x + ((-2.0d0) * (x + (-1.0d0)))) / ((x * x) - x))
    else if (t_1 <= 2d-18) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = t_0 + ((0.5d0 - (1.0d0 / x)) / (0.5d0 + (x * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -1e-6:
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x))
	elif t_1 <= 2e-18:
		tmp = 2.0 / (x * (x * x))
	else:
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(t_0 + Float64(Float64(x + Float64(-2.0 * Float64(x + -1.0))) / Float64(Float64(x * x) - x)));
	elseif (t_1 <= 2e-18)
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(t_0 + Float64(Float64(0.5 - Float64(1.0 / x)) / Float64(0.5 + Float64(x * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	elseif (t_1 <= 2e-18)
		tmp = 2.0 / (x * (x * x));
	else
		tmp = t_0 + ((0.5 - (1.0 / x)) / (0.5 + (x * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $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[LessEqual[t$95$1, -1e-6], N[(t$95$0 + N[(N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(0.5 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. +-commutative 99.7%

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

      7. *-lft-identity 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      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. *-rgt-identity 99.7%

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

      10. fma-def 99.7%

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

      11. fma-neg 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 -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000001e-18

   1. Initial program 73.6%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. neg-mul-1 73.6%

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

      4. metadata-eval 73.6%

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

      5. cancel-sign-sub-inv 73.6%

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

      6. +-commutative 73.6%

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

      7. *-lft-identity 73.6%

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

      8. sub-neg 73.6%

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

      9. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.0000000000000001e-18 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. metadata-eval 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. *-lft-identity 99.8%

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

      8. sub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-2neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. frac-sub 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. div-inv 99.8%

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

      11. metadata-eval 99.8%

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

      12. div-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

      15. distribute-neg-in 99.8%

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

      16. metadata-eval 99.8%

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

      17. sub-neg 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. associate-/r* 99.9%

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

      3. *-commutative 99.9%

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

      4. cancel-sign-sub-inv 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-lft-identity 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-rgt-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{-0.5 \cdot x + 1}}{x}}{0.5 + x \cdot -0.5} \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{\frac{1}{x} - 0.5}}{0.5 + x \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{0.5 - \frac{1}{x}}{0.5 + x \cdot -0.5}\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-6} \lor \neg \left(t_0 \leq 2 \cdot 10^{-18}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (or (<= t_0 -1e-6) (not (<= t_0 2e-18))) t_0 (/ 2.0 (* x (* x x))))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-18)) {
		tmp = t_0;
	} else {
		tmp = 2.0 / (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if ((t_0 <= (-1d-6)) .or. (.not. (t_0 <= 2d-18))) then
        tmp = t_0
    else
        tmp = 2.0d0 / (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-18)) {
		tmp = t_0;
	} else {
		tmp = 2.0 / (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if (t_0 <= -1e-6) or not (t_0 <= 2e-18):
		tmp = t_0
	else:
		tmp = 2.0 / (x * (x * x))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-18))
		tmp = t_0;
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if ((t_0 <= -1e-6) || ~((t_0 <= 2e-18)))
		tmp = t_0;
	else
		tmp = 2.0 / (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-6], N[Not[LessEqual[t$95$0, 2e-18]], $MachinePrecision]], t$95$0, N[(2.0 / N[(x * N[(x * 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))) < -9.99999999999999955e-7 or 2.0000000000000001e-18 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

   if -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000001e-18

   1. Initial program 73.6%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. neg-mul-1 73.6%

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

      4. metadata-eval 73.6%

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

      5. cancel-sign-sub-inv 73.6%

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

      6. +-commutative 73.6%

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

      7. *-lft-identity 73.6%

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

      8. sub-neg 73.6%

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

      9. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 8: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot x - x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-6)
     (+ t_0 (/ (- 2.0 x) (- (* x x) x)))
     (if (<= t_1 2e-18) (/ 2.0 (* x (* x x))) t_1))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_1;
	}
	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 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-1d-6)) then
        tmp = t_0 + ((2.0d0 - x) / ((x * x) - x))
    else if (t_1 <= 2d-18) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	} else if (t_1 <= 2e-18) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -1e-6:
		tmp = t_0 + ((2.0 - x) / ((x * x) - x))
	elif t_1 <= 2e-18:
		tmp = 2.0 / (x * (x * x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(t_0 + Float64(Float64(2.0 - x) / Float64(Float64(x * x) - x)));
	elseif (t_1 <= 2e-18)
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	elseif (t_1 <= 2e-18)
		tmp = 2.0 / (x * (x * x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $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[LessEqual[t$95$1, -1e-6], N[(t$95$0 + N[(N[(2.0 - x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-18], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. neg-mul-1 99.7%

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

      4. metadata-eval 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. +-commutative 99.7%

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

      7. *-lft-identity 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      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. *-rgt-identity 99.7%

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

      10. fma-def 99.7%

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

      11. fma-neg 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. Taylor expanded in x around 0 99.7%

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

   if -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000001e-18

   1. Initial program 73.6%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. neg-mul-1 73.6%

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

      4. metadata-eval 73.6%

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

      5. cancel-sign-sub-inv 73.6%

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

      6. +-commutative 73.6%

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

      7. *-lft-identity 73.6%

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

      8. sub-neg 73.6%

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

      9. metadata-eval 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.0000000000000001e-18 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{2 - x}{x \cdot x - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \end{array} \]

Alternative 9: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (1.0 - x) + ((-1.0 - 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 = 2.0d0 / (x * (x * x))
    else
        tmp = (1.0d0 - x) + (((-1.0d0) - 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 = 2.0 / (x * (x * x));
	} else {
		tmp = (1.0 - x) + ((-1.0 - x) - (2.0 / x));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 - x) + Float64(Float64(-1.0 - 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 = 2.0 / (x * (x * x));
	else
		tmp = (1.0 - x) + ((-1.0 - 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[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.7%

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

      1. associate-+l- 73.7%

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

      2. sub-neg 73.7%

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

      3. neg-mul-1 73.7%

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

      4. metadata-eval 73.7%

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

      5. cancel-sign-sub-inv 73.7%

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

      6. +-commutative 73.7%

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

      7. *-lft-identity 73.7%

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

      8. sub-neg 73.7%

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

      9. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 97.6%

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

   6. Applied egg-rr 97.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-sub 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. div-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. div-inv 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. associate-/r* 100.0%

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

      3. *-commutative 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+l+ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around 0 99.9%

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

       1. neg-mul-1 99.9%

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

       2. sub-neg 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 98.8%

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

Alternative 10: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 2.0 / (x * (x * 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 <= 1.0d0))) then
        tmp = 2.0d0 / (x * (x * 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 <= 1.0)) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(2.0 / Float64(x * Float64(x * 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 <= 1.0)))
		tmp = 2.0 / (x * (x * 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, 1.0]], $MachinePrecision]], N[(2.0 / N[(x * N[(x * 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 < x

   1. Initial program 73.7%

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

      1. associate-+l- 73.7%

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

      2. sub-neg 73.7%

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

      3. neg-mul-1 73.7%

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

      4. metadata-eval 73.7%

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

      5. cancel-sign-sub-inv 73.7%

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

      6. +-commutative 73.7%

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

      7. *-lft-identity 73.7%

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

      8. sub-neg 73.7%

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

      9. metadata-eval 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 97.6%

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

   6. Applied egg-rr 97.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 98.8%

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

Alternative 11: 52.0% 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 87.5%

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

   1. associate-+l- 87.5%

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

   2. sub-neg 87.5%

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

   3. neg-mul-1 87.5%

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

   4. metadata-eval 87.5%

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

   5. cancel-sign-sub-inv 87.5%

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

   6. +-commutative 87.5%

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

   7. *-lft-identity 87.5%

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

   8. sub-neg 87.5%

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

   9. metadata-eval 87.5%

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

3. Simplified 87.5%

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

4. Taylor expanded in x around 0 54.5%

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

5. Final simplification 54.5%

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

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 87.5%

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

   1. associate-+l- 87.5%

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

   2. sub-neg 87.5%

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

   3. neg-mul-1 87.5%

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

   4. metadata-eval 87.5%

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

   5. cancel-sign-sub-inv 87.5%

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

   6. +-commutative 87.5%

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

   7. *-lft-identity 87.5%

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

   8. sub-neg 87.5%

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

   9. metadata-eval 87.5%

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

3. Simplified 87.5%

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

4. Taylor expanded in x around 0 53.4%

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

5. Taylor expanded in x around inf 3.4%

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

6. Final simplification 3.4%

   \[\leadsto -1 \]

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 2023258 
(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))))