3frac (problem 3.3.3)

Percentage Accurate: 84.4% → 99.5%

Time: 8.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.4% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - x\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \mathsf{fma}\left(2, 1 - x, x\right) \cdot \left(-1 - x\right)}{\left(1 - x\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array} \end{array} \]

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)))))
   (if (<= t_1 -1e-7)
     (/ (+ t_0 (* (+ 1.0 x) (- x (fma x 2.0 -2.0)))) (* (+ 1.0 x) t_0))
     (if (<= t_1 0.0)
       (* 2.0 (pow x -3.0))
       (/
        (+ (* x (- 1.0 x)) (* (fma 2.0 (- 1.0 x) x) (- -1.0 x)))
        (* (- 1.0 x) (* x (+ 1.0 x))))))))

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 tmp;
	if (t_1 <= -1e-7) {
		tmp = (t_0 + ((1.0 + x) * (x - fma(x, 2.0, -2.0)))) / ((1.0 + x) * t_0);
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = ((x * (1.0 - x)) + (fma(2.0, (1.0 - x), x) * (-1.0 - x))) / ((1.0 - x) * (x * (1.0 + x)));
	}
	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)))
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x - fma(x, 2.0, -2.0)))) / Float64(Float64(1.0 + x) * t_0));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 - x)) + Float64(fma(2.0, Float64(1.0 - x), x) * Float64(-1.0 - x))) / Float64(Float64(1.0 - x) * Float64(x * Float64(1.0 + x))));
	end
	return 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]}, If[LessEqual[t$95$1, -1e-7], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - N[(x * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(x * N[(1.0 + x), $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))) < -9.9999999999999995e-8

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. neg-mul-1 99.5%

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

      4. metadata-eval 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. *-lft-identity 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. frac-sub 99.5%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. fma-def 100.0%

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

      12. metadata-eval 100.0%

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

      13. distribute-rgt-in 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-udef 100.0%

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

      8. distribute-lft-neg-in 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. fma-udef 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. remove-double-neg 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -9.9999999999999995e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 67.0%

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

      1. associate-+l- 67.0%

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

      2. sub-neg 67.0%

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

      3. neg-mul-1 67.0%

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

      4. metadata-eval 67.0%

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

      5. cancel-sign-sub-inv 67.0%

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

      6. +-commutative 67.0%

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

      7. *-lft-identity 67.0%

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

      8. sub-neg 67.0%

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

      9. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. div-inv 99.4%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

      2. pow-flip 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 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%

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

      2. metadata-eval 99.7%

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

      3. frac-sub 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. neg-mul-1 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-neg-in 99.8%

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

      12. metadata-eval 99.8%

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

      13. sub-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. fma-def 99.9%

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

      6. metadata-eval 99.9%

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

      7. fma-neg 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. fma-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. *-un-lft-identity 99.9%

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

      12. fma-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. fma-neg 99.9%

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

      15. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. remove-double-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r* 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \mathsf{fma}\left(2, 1 - x, x\right) \cdot \left(-1 - x\right)}{\left(1 - x\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \mathsf{fma}\left(2, 1 - x, x\right) \cdot \left(-1 - x\right)}{\left(1 - x\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x)))
        (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-7)
     (/ (+ t_0 (* (+ 1.0 x) (- x 2.0))) (* (- -1.0 x) (* x (+ x -1.0))))
     (if (<= t_1 0.0)
       (* 2.0 (pow x -3.0))
       (/
        (+ t_0 (* (fma 2.0 (- 1.0 x) x) (- -1.0 x)))
        (* (- 1.0 x) (* x (+ 1.0 x))))))))

C

double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((-1.0 - x) * (x * (x + -1.0)));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (t_0 + (fma(2.0, (1.0 - x), x) * (-1.0 - x))) / ((1.0 - x) * (x * (1.0 + x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 - x))
	t_1 = 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_1 <= -1e-7)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x - 2.0))) / Float64(Float64(-1.0 - x) * Float64(x * Float64(x + -1.0))));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(t_0 + Float64(fma(2.0, Float64(1.0 - x), x) * Float64(-1.0 - x))) / Float64(Float64(1.0 - x) * Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $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]}, If[LessEqual[t$95$1, -1e-7], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - x), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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[(2.0 * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(x * N[(1.0 + x), $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))) < -9.9999999999999995e-8

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. neg-mul-1 99.5%

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

      4. metadata-eval 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. *-lft-identity 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. frac-sub 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. sub-neg 99.5%

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

      8. *-commutative 99.5%

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

      9. neg-mul-1 99.5%

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

      10. +-commutative 99.5%

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

      11. distribute-neg-in 99.5%

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

      12. metadata-eval 99.5%

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

      13. sub-neg 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. neg-mul-1 99.4%

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

      2. unsub-neg 99.4%

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

   8. Simplified 99.4%

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

      1. frac-sub 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -9.9999999999999995e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 67.0%

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

      1. associate-+l- 67.0%

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

      2. sub-neg 67.0%

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

      3. neg-mul-1 67.0%

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

      4. metadata-eval 67.0%

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

      5. cancel-sign-sub-inv 67.0%

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

      6. +-commutative 67.0%

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

      7. *-lft-identity 67.0%

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

      8. sub-neg 67.0%

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

      9. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. div-inv 99.4%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

      2. pow-flip 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 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%

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

      2. metadata-eval 99.7%

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

      3. frac-sub 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. neg-mul-1 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-neg-in 99.8%

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

      12. metadata-eval 99.8%

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

      13. sub-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. fma-def 99.9%

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

      6. metadata-eval 99.9%

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

      7. fma-neg 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. fma-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. *-un-lft-identity 99.9%

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

      12. fma-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. fma-neg 99.9%

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

      15. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. remove-double-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r* 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \mathsf{fma}\left(2, 1 - x, x\right) \cdot \left(-1 - x\right)}{\left(1 - x\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-7} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \end{array} \]

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-7) (not (<= t_0 0.0)))
     (/
      (+ (* x (- 1.0 x)) (* (+ 1.0 x) (- x 2.0)))
      (* (- -1.0 x) (* x (+ x -1.0))))
     (* 2.0 (pow x -3.0)))))

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-7) || !(t_0 <= 0.0)) {
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x - 2.0))) / ((-1.0 - x) * (x * (x + -1.0)));
	} else {
		tmp = 2.0 * pow(x, -3.0);
	}
	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-7)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = ((x * (1.0d0 - x)) + ((1.0d0 + x) * (x - 2.0d0))) / (((-1.0d0) - x) * (x * (x + (-1.0d0))))
    else
        tmp = 2.0d0 * (x ** (-3.0d0))
    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-7) || !(t_0 <= 0.0)) {
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x - 2.0))) / ((-1.0 - x) * (x * (x + -1.0)));
	} else {
		tmp = 2.0 * Math.pow(x, -3.0);
	}
	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-7) or not (t_0 <= 0.0):
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x - 2.0))) / ((-1.0 - x) * (x * (x + -1.0)))
	else:
		tmp = 2.0 * math.pow(x, -3.0)
	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-7) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(x * Float64(1.0 - x)) + Float64(Float64(1.0 + x) * Float64(x - 2.0))) / Float64(Float64(-1.0 - x) * Float64(x * Float64(x + -1.0))));
	else
		tmp = Float64(2.0 * (x ^ -3.0));
	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-7) || ~((t_0 <= 0.0)))
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x - 2.0))) / ((-1.0 - x) * (x * (x + -1.0)));
	else
		tmp = 2.0 * (x ^ -3.0);
	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-7], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - x), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[x, -3.0], $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))) < -9.9999999999999995e-8 or 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. sub-neg 99.6%

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

      3. neg-mul-1 99.6%

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

      4. metadata-eval 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. +-commutative 99.6%

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

      7. *-lft-identity 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. frac-sub 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. sub-neg 99.6%

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

      8. *-commutative 99.6%

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

      9. neg-mul-1 99.6%

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

      10. +-commutative 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. metadata-eval 99.6%

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

      13. sub-neg 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. unsub-neg 99.6%

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

   8. Simplified 99.6%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -9.9999999999999995e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 67.0%

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

      1. associate-+l- 67.0%

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

      2. sub-neg 67.0%

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

      3. neg-mul-1 67.0%

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

      4. metadata-eval 67.0%

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

      5. cancel-sign-sub-inv 67.0%

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

      6. +-commutative 67.0%

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

      7. *-lft-identity 67.0%

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

      8. sub-neg 67.0%

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

      9. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. div-inv 99.4%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

      2. pow-flip 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-7} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0\right):\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternative 4: 84.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-+l- 83.3%

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

   2. sub-neg 83.3%

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

   3. neg-mul-1 83.3%

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

   4. metadata-eval 83.3%

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

   5. cancel-sign-sub-inv 83.3%

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

   6. +-commutative 83.3%

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

   7. *-lft-identity 83.3%

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

   8. sub-neg 83.3%

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

   9. metadata-eval 83.3%

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

3. Simplified 83.3%

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

   1. frac-2neg 83.3%

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

   2. metadata-eval 83.3%

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

   3. frac-sub 56.0%

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

   4. +-commutative 56.0%

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

   5. distribute-neg-in 56.0%

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

   6. metadata-eval 56.0%

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

   7. sub-neg 56.0%

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

   8. *-commutative 56.0%

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

   9. neg-mul-1 56.0%

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

   10. +-commutative 56.0%

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

   11. distribute-neg-in 56.0%

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

   12. metadata-eval 56.0%

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

   13. sub-neg 56.0%

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

5. Applied egg-rr 56.0%

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

6. Taylor expanded in x around 0 55.9%

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

   1. neg-mul-1 55.9%

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

   2. unsub-neg 55.9%

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

8. Simplified 55.9%

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

   1. associate-/r* 83.3%

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

   2. frac-sub 83.3%

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

   3. *-un-lft-identity 83.3%

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

   4. +-commutative 83.3%

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

   5. +-commutative 83.3%

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

10. Applied egg-rr 83.3%

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

    1. *-commutative 83.3%

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

    2. *-commutative 83.3%

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

12. Simplified 83.3%

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

13. Final simplification 83.3%

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

Alternative 5: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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]

Derivation

1. Initial program 83.3%

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

2. Final simplification 83.3%

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

Alternative 6: 76.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 67.4%

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

      1. associate-+l- 67.4%

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

      2. sub-neg 67.4%

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

      3. neg-mul-1 67.4%

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

      4. metadata-eval 67.4%

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

      5. cancel-sign-sub-inv 67.4%

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

      6. +-commutative 67.4%

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

      7. *-lft-identity 67.4%

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

      8. sub-neg 67.4%

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

      9. metadata-eval 67.4%

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

   3. Simplified 67.4%

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

      1. flip-+ 14.6%

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

      2. sub-neg 14.6%

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

      3. metadata-eval 14.6%

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

      4. distribute-neg-in 14.6%

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

      5. +-commutative 14.6%

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

      6. associate-/r/ 12.3%

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

      7. metadata-eval 12.3%

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

      8. +-commutative 12.3%

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

      9. distribute-neg-in 12.3%

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

      10. metadata-eval 12.3%

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

      11. sub-neg 12.3%

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

   5. Applied egg-rr 12.3%

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

      1. associate-*l/ 13.9%

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

      2. *-lft-identity 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 12.2%

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

   9. Taylor expanded in x around inf 58.0%

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

       1. unpow2 58.0%

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

   11. Simplified 58.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 78.3%

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

Alternative 7: 76.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 67.4%

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

      1. associate-+l- 67.4%

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

      2. sub-neg 67.4%

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

      3. neg-mul-1 67.4%

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

      4. metadata-eval 67.4%

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

      5. cancel-sign-sub-inv 67.4%

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

      6. +-commutative 67.4%

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

      7. *-lft-identity 67.4%

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

      8. sub-neg 67.4%

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

      9. metadata-eval 67.4%

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

   3. Simplified 67.4%

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

      1. flip-+ 14.6%

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

      2. sub-neg 14.6%

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

      3. metadata-eval 14.6%

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

      4. distribute-neg-in 14.6%

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

      5. +-commutative 14.6%

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

      6. associate-/r/ 12.3%

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

      7. metadata-eval 12.3%

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

      8. +-commutative 12.3%

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

      9. distribute-neg-in 12.3%

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

      10. metadata-eval 12.3%

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

      11. sub-neg 12.3%

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

   5. Applied egg-rr 12.3%

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

      1. associate-*l/ 13.9%

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

      2. *-lft-identity 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 12.2%

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

   9. Taylor expanded in x around inf 58.0%

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

       1. unpow2 58.0%

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

   11. Simplified 58.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   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 99.6%

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

4. Final simplification 78.3%

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

Alternative 8: 83.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-+l- 83.3%

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

   2. sub-neg 83.3%

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

   3. neg-mul-1 83.3%

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

   4. metadata-eval 83.3%

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

   5. cancel-sign-sub-inv 83.3%

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

   6. +-commutative 83.3%

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

   7. *-lft-identity 83.3%

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

   8. sub-neg 83.3%

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

   9. metadata-eval 83.3%

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

3. Simplified 83.3%

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

4. Taylor expanded in x around 0 50.3%

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

5. Taylor expanded in x around 0 82.2%

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

6. Final simplification 82.2%

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

Alternative 9: 52.4% 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 83.3%

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

   1. associate-+l- 83.3%

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

   2. sub-neg 83.3%

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

   3. neg-mul-1 83.3%

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

   4. metadata-eval 83.3%

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

   5. cancel-sign-sub-inv 83.3%

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

   6. +-commutative 83.3%

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

   7. *-lft-identity 83.3%

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

   8. sub-neg 83.3%

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

   9. metadata-eval 83.3%

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

3. Simplified 83.3%

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

4. Taylor expanded in x around 0 51.3%

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

5. Final simplification 51.3%

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

Alternative 10: 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 83.3%

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

   1. associate-+l- 83.3%

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

   2. sub-neg 83.3%

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

   3. neg-mul-1 83.3%

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

   4. metadata-eval 83.3%

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

   5. cancel-sign-sub-inv 83.3%

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

   6. +-commutative 83.3%

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

   7. *-lft-identity 83.3%

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

   8. sub-neg 83.3%

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

   9. metadata-eval 83.3%

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

3. Simplified 83.3%

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

4. Taylor expanded in x around 0 50.3%

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

5. Taylor expanded in x around inf 3.2%

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

6. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023256 
(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))))