3frac (problem 3.3.3)

Percentage Accurate: 84.0% → 99.7%

Time: 11.2s

Alternatives: 15

Speedup: 0.9×

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: 84.0% 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: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;t_1 \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 + {x}^{3}}, \mathsf{fma}\left(x, x, 1 - x\right), t_0 + \frac{-2}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x -1.0))) (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) t_0)))
   (if (<= t_1 -4e-8)
     (/
      (+ (* (+ x x) (/ x 2.0)) (* (+ x -1.0) (- -1.0 x)))
      (* (+ x -1.0) (* (+ 1.0 x) (/ x 2.0))))
     (if (<= t_1 1e-22)
       (* 2.0 (+ (pow x -5.0) (pow x -3.0)))
       (fma
        (/ 1.0 (+ 1.0 (pow x 3.0)))
        (fma x x (- 1.0 x))
        (+ t_0 (/ -2.0 x)))))))

C

double code(double x) {
	double t_0 = 1.0 / (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0;
	double tmp;
	if (t_1 <= -4e-8) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else if (t_1 <= 1e-22) {
		tmp = 2.0 * (pow(x, -5.0) + pow(x, -3.0));
	} else {
		tmp = fma((1.0 / (1.0 + pow(x, 3.0))), fma(x, x, (1.0 - x)), (t_0 + (-2.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(x + -1.0))
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + t_0)
	tmp = 0.0
	if (t_1 <= -4e-8)
		tmp = Float64(Float64(Float64(Float64(x + x) * Float64(x / 2.0)) + Float64(Float64(x + -1.0) * Float64(-1.0 - x))) / Float64(Float64(x + -1.0) * Float64(Float64(1.0 + x) * Float64(x / 2.0))));
	elseif (t_1 <= 1e-22)
		tmp = Float64(2.0 * Float64((x ^ -5.0) + (x ^ -3.0)));
	else
		tmp = fma(Float64(1.0 / Float64(1.0 + (x ^ 3.0))), fma(x, x, Float64(1.0 - x)), Float64(t_0 + Float64(-2.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-8], N[(N[(N[(N[(x + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-22], N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(-2.0 / x), $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))) < -4.0000000000000001e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-add 99.5%

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

      2. clear-num 99.5%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-+r+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \frac{\left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{x}{2} - \left(x + -1\right) \cdot \left(x + 1\right)}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x}{2}\right)} \]

       4. +-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   if -4.0000000000000001e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-22

   1. Initial program 73.7%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   4. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\right)} \]

      2. expm1-udef 73.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)} - 1} \]

      3. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{5}}} + \frac{2}{{x}^{3}}\right)} - 1 \]

      4. fma-def 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)}\right)} - 1 \]

      5. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      6. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      7. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)\right)} - 1 \]

      8. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)\right)} - 1 \]

      9. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{\color{blue}{-3}}\right)\right)} - 1 \]

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)} \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{2 \cdot {x}^{-5} + 2 \cdot {x}^{-3}} \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   if 1e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-+l+ 99.9%

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

      3. flip3-+ 99.9%

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

      4. associate-/r/ 99.9%

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

      5. fma-def 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 + {x}^{3}}, \mathsf{fma}\left(x, x, 1 - x\right), \frac{1}{x + -1} + \frac{-2}{x}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;t_1 \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}, 1 + x, t_0 + \frac{-2}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -4e-8)
     (/
      (+ (* (+ x x) (/ x 2.0)) (* (+ x -1.0) (- -1.0 x)))
      (* (+ x -1.0) (* (+ 1.0 x) (/ x 2.0))))
     (if (<= t_1 1e-22)
       (* 2.0 (+ (pow x -5.0) (pow x -3.0)))
       (fma (/ 1.0 (fma x x -1.0)) (+ 1.0 x) (+ t_0 (/ -2.0 x)))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -4e-8) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else if (t_1 <= 1e-22) {
		tmp = 2.0 * (pow(x, -5.0) + pow(x, -3.0));
	} else {
		tmp = fma((1.0 / fma(x, x, -1.0)), (1.0 + x), (t_0 + (-2.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -4e-8)
		tmp = Float64(Float64(Float64(Float64(x + x) * Float64(x / 2.0)) + Float64(Float64(x + -1.0) * Float64(-1.0 - x))) / Float64(Float64(x + -1.0) * Float64(Float64(1.0 + x) * Float64(x / 2.0))));
	elseif (t_1 <= 1e-22)
		tmp = Float64(2.0 * Float64((x ^ -5.0) + (x ^ -3.0)));
	else
		tmp = fma(Float64(1.0 / fma(x, x, -1.0)), Float64(1.0 + x), Float64(t_0 + Float64(-2.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-8], N[(N[(N[(N[(x + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-22], N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision] + N[(t$95$0 + N[(-2.0 / x), $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))) < -4.0000000000000001e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-add 99.5%

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

      2. clear-num 99.5%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-+r+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \frac{\left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{x}{2} - \left(x + -1\right) \cdot \left(x + 1\right)}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x}{2}\right)} \]

       4. +-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   if -4.0000000000000001e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-22

   1. Initial program 73.7%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   4. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\right)} \]

      2. expm1-udef 73.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)} - 1} \]

      3. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{5}}} + \frac{2}{{x}^{3}}\right)} - 1 \]

      4. fma-def 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)}\right)} - 1 \]

      5. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      6. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      7. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)\right)} - 1 \]

      8. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)\right)} - 1 \]

      9. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{\color{blue}{-3}}\right)\right)} - 1 \]

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)} \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{2 \cdot {x}^{-5} + 2 \cdot {x}^{-3}} \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   if 1e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-/r/ 99.9%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

      6. fma-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}, 1 + x, \frac{1}{1 + x} + \frac{-2}{x}\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := \left(x + -1\right) \cdot \left(-1 - x\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + t_1}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;t_0 \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + x\right) + \left(x + -1\right)\right) + 2 \cdot t_1}{\left(x + -1\right) \cdot \left(x + x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1 (* (+ x -1.0) (- -1.0 x))))
   (if (<= t_0 -4e-8)
     (/ (+ (* (+ x x) (/ x 2.0)) t_1) (* (+ x -1.0) (* (+ 1.0 x) (/ x 2.0))))
     (if (<= t_0 1e-22)
       (* 2.0 (+ (pow x -5.0) (pow x -3.0)))
       (/
        (+ (* x (+ (+ 1.0 x) (+ x -1.0))) (* 2.0 t_1))
        (* (+ x -1.0) (+ x (* x x))))))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x + -1.0) * (-1.0 - x);
	double tmp;
	if (t_0 <= -4e-8) {
		tmp = (((x + x) * (x / 2.0)) + t_1) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else if (t_0 <= 1e-22) {
		tmp = 2.0 * (pow(x, -5.0) + pow(x, -3.0));
	} else {
		tmp = ((x * ((1.0 + x) + (x + -1.0))) + (2.0 * t_1)) / ((x + -1.0) * (x + (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = (x + (-1.0d0)) * ((-1.0d0) - x)
    if (t_0 <= (-4d-8)) then
        tmp = (((x + x) * (x / 2.0d0)) + t_1) / ((x + (-1.0d0)) * ((1.0d0 + x) * (x / 2.0d0)))
    else if (t_0 <= 1d-22) then
        tmp = 2.0d0 * ((x ** (-5.0d0)) + (x ** (-3.0d0)))
    else
        tmp = ((x * ((1.0d0 + x) + (x + (-1.0d0)))) + (2.0d0 * t_1)) / ((x + (-1.0d0)) * (x + (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x + -1.0) * (-1.0 - x);
	double tmp;
	if (t_0 <= -4e-8) {
		tmp = (((x + x) * (x / 2.0)) + t_1) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else if (t_0 <= 1e-22) {
		tmp = 2.0 * (Math.pow(x, -5.0) + Math.pow(x, -3.0));
	} else {
		tmp = ((x * ((1.0 + x) + (x + -1.0))) + (2.0 * t_1)) / ((x + -1.0) * (x + (x * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_1 = (x + -1.0) * (-1.0 - x)
	tmp = 0
	if t_0 <= -4e-8:
		tmp = (((x + x) * (x / 2.0)) + t_1) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)))
	elif t_0 <= 1e-22:
		tmp = 2.0 * (math.pow(x, -5.0) + math.pow(x, -3.0))
	else:
		tmp = ((x * ((1.0 + x) + (x + -1.0))) + (2.0 * t_1)) / ((x + -1.0) * (x + (x * x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(Float64(x + -1.0) * Float64(-1.0 - x))
	tmp = 0.0
	if (t_0 <= -4e-8)
		tmp = Float64(Float64(Float64(Float64(x + x) * Float64(x / 2.0)) + t_1) / Float64(Float64(x + -1.0) * Float64(Float64(1.0 + x) * Float64(x / 2.0))));
	elseif (t_0 <= 1e-22)
		tmp = Float64(2.0 * Float64((x ^ -5.0) + (x ^ -3.0)));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 + x) + Float64(x + -1.0))) + Float64(2.0 * t_1)) / Float64(Float64(x + -1.0) * Float64(x + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_1 = (x + -1.0) * (-1.0 - x);
	tmp = 0.0;
	if (t_0 <= -4e-8)
		tmp = (((x + x) * (x / 2.0)) + t_1) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	elseif (t_0 <= 1e-22)
		tmp = 2.0 * ((x ^ -5.0) + (x ^ -3.0));
	else
		tmp = ((x * ((1.0 + x) + (x + -1.0))) + (2.0 * t_1)) / ((x + -1.0) * (x + (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-8], N[(N[(N[(N[(x + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-22], N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 + x), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(x * 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))) < -4.0000000000000001e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-add 99.5%

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

      2. clear-num 99.5%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

       2. associate-+r+ 100.0%

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

       3. metadata-eval 100.0%

          \[\leadsto \frac{\left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{x}{2} - \left(x + -1\right) \cdot \left(x + 1\right)}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x}{2}\right)} \]

       4. +-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   if -4.0000000000000001e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-22

   1. Initial program 73.7%

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

   2. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   4. Simplified 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\right)} \]

      2. expm1-udef 73.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)} - 1} \]

      3. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{5}}} + \frac{2}{{x}^{3}}\right)} - 1 \]

      4. fma-def 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)}\right)} - 1 \]

      5. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      6. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

      7. div-inv 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)\right)} - 1 \]

      8. pow-flip 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)\right)} - 1 \]

      9. metadata-eval 73.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{\color{blue}{-3}}\right)\right)} - 1 \]

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)} \]

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{2 \cdot {x}^{-5} + 2 \cdot {x}^{-3}} \]

      4. distribute-lft-out 100.0%

         \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

   if 1e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. frac-add 99.9%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. associate-*l/ 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*r/ 100.0%

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

      11. /-rgt-identity 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-*l* 99.9%

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

      14. +-commutative 99.9%

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

      15. distribute-rgt1-in 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-22}:\\ \;\;\;\;2 \cdot \left({x}^{-5} + {x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + x\right) + \left(x + -1\right)\right) + 2 \cdot \left(\left(x + -1\right) \cdot \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + x \cdot x\right)}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot \left(x + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -20000000000.0)
   (/ 2.0 (pow x 3.0))
   (if (<= x 100000000.0)
     (/
      (+ (* (+ x x) (/ x 2.0)) (* (+ x -1.0) (- -1.0 x)))
      (* (+ x -1.0) (* (+ 1.0 x) (/ x 2.0))))
     (/ (- (/ 2.0 x) (/ 4.0 (* x x))) (* x (+ x -2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -20000000000.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else if (x <= 100000000.0) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-20000000000.0d0)) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else if (x <= 100000000.0d0) then
        tmp = (((x + x) * (x / 2.0d0)) + ((x + (-1.0d0)) * ((-1.0d0) - x))) / ((x + (-1.0d0)) * ((1.0d0 + x) * (x / 2.0d0)))
    else
        tmp = ((2.0d0 / x) - (4.0d0 / (x * x))) / (x * (x + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -20000000000.0) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else if (x <= 100000000.0) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -20000000000.0:
		tmp = 2.0 / math.pow(x, 3.0)
	elif x <= 100000000.0:
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)))
	else:
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -20000000000.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	elseif (x <= 100000000.0)
		tmp = Float64(Float64(Float64(Float64(x + x) * Float64(x / 2.0)) + Float64(Float64(x + -1.0) * Float64(-1.0 - x))) / Float64(Float64(x + -1.0) * Float64(Float64(1.0 + x) * Float64(x / 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / x) - Float64(4.0 / Float64(x * x))) / Float64(x * Float64(x + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -20000000000.0)
		tmp = 2.0 / (x ^ 3.0);
	elseif (x <= 100000000.0)
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	else
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -20000000000.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(N[(N[(N[(x + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / x), $MachinePrecision] - N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e10

   1. Initial program 73.3%

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

   2. Taylor expanded in x around inf 99.9%

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

   if -2e10 < x < 1e8

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-+r- 98.7%

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

      3. sub-neg 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. frac-add 98.7%

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

      2. clear-num 98.7%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-+r+ 99.9%

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

       3. metadata-eval 99.9%

          \[\leadsto \frac{\left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{x}{2} - \left(x + -1\right) \cdot \left(x + 1\right)}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x}{2}\right)} \]

       4. +-lft-identity 99.9%

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

   11. Simplified 99.9%

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

   if 1e8 < x

   1. Initial program 75.0%

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

      1. frac-sub 16.8%

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

      2. clear-num 20.9%

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

      3. *-commutative 20.9%

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

      4. +-commutative 20.9%

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

      5. /-rgt-identity 20.9%

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

      6. *-un-lft-identity 20.9%

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

      7. /-rgt-identity 20.9%

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

      8. +-commutative 20.9%

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

   3. Applied egg-rr 20.9%

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

      1. +-commutative 20.9%

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

      2. frac-2neg 20.9%

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

      3. metadata-eval 20.9%

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

      4. frac-add 17.0%

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

   5. Applied egg-rr 72.3%

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

      1. distribute-rgt-neg-out 72.3%

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

      2. distribute-lft-neg-out 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. unpow2 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 99.9%

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

       1. unpow2 99.9%

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

       2. distribute-rgt-out 99.9%

          \[\leadsto \frac{\frac{2}{x} - \frac{4}{x \cdot x}}{\color{blue}{x \cdot \left(-2 + x\right)}} \]

   13. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20000000000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot \left(x + -2\right)}\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-16} \lor \neg \left(t_0 \leq 10^{-22}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot x}\\ \end{array} \end{array} \]

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-16) (not (<= t_0 1e-22)))
     t_0
     (/ (- (/ 2.0 x) (/ 4.0 (* x x))) (* x x)))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -1e-16) || !(t_0 <= 1e-22)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if ((t_0 <= (-1d-16)) .or. (.not. (t_0 <= 1d-22))) then
        tmp = t_0
    else
        tmp = ((2.0d0 / x) - (4.0d0 / (x * x))) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_0 <= -1e-16) || !(t_0 <= 1e-22)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if (t_0 <= -1e-16) or not (t_0 <= 1e-22):
		tmp = t_0
	else:
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if ((t_0 <= -1e-16) || !(t_0 <= 1e-22))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 / x) - Float64(4.0 / Float64(x * x))) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if ((t_0 <= -1e-16) || ~((t_0 <= 1e-22)))
		tmp = t_0;
	else
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-16], N[Not[LessEqual[t$95$0, 1e-22]], $MachinePrecision]], t$95$0, N[(N[(N[(2.0 / x), $MachinePrecision] - N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $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))) < -9.9999999999999998e-17 or 1e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.4%

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

   if -9.9999999999999998e-17 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-22

   1. Initial program 73.9%

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

      1. frac-sub 17.8%

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

      2. clear-num 20.9%

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

      3. *-commutative 20.9%

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

      4. +-commutative 20.9%

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

      5. /-rgt-identity 20.9%

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

      6. *-un-lft-identity 20.9%

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

      7. /-rgt-identity 20.9%

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

      8. +-commutative 20.9%

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

   3. Applied egg-rr 20.9%

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

      1. +-commutative 20.9%

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

      2. frac-2neg 20.9%

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

      3. metadata-eval 20.9%

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

      4. frac-add 17.2%

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

   5. Applied egg-rr 70.8%

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

      1. distribute-rgt-neg-out 70.8%

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

      2. distribute-lft-neg-out 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

      5. unpow2 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 99.2%

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

       1. unpow2 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-16} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-22}\right):\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot x}\\ \end{array} \]

Alternative 6: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-8} \lor \neg \left(t_0 \leq 10^{-22}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot \left(x + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (or (<= t_0 -4e-8) (not (<= t_0 1e-22)))
     t_0
     (/ (- (/ 2.0 x) (/ 4.0 (* x x))) (* x (+ x -2.0))))))

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 <= -4e-8) || !(t_0 <= 1e-22)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	}
	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 <= (-4d-8)) .or. (.not. (t_0 <= 1d-22))) then
        tmp = t_0
    else
        tmp = ((2.0d0 / x) - (4.0d0 / (x * x))) / (x * (x + (-2.0d0)))
    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 <= -4e-8) || !(t_0 <= 1e-22)) {
		tmp = t_0;
	} else {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	}
	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 <= -4e-8) or not (t_0 <= 1e-22):
		tmp = t_0
	else:
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0))
	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 <= -4e-8) || !(t_0 <= 1e-22))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 / x) - Float64(4.0 / Float64(x * x))) / Float64(x * Float64(x + -2.0)));
	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 <= -4e-8) || ~((t_0 <= 1e-22)))
		tmp = t_0;
	else
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * (x + -2.0));
	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, -4e-8], N[Not[LessEqual[t$95$0, 1e-22]], $MachinePrecision]], t$95$0, N[(N[(N[(2.0 / x), $MachinePrecision] - N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.0000000000000001e-8 or 1e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.8%

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

   if -4.0000000000000001e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-22

   1. Initial program 73.7%

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

      1. frac-sub 18.0%

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

      2. clear-num 21.1%

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

      3. *-commutative 21.1%

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

      4. +-commutative 21.1%

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

      5. /-rgt-identity 21.1%

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

      6. *-un-lft-identity 21.1%

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

      7. /-rgt-identity 21.1%

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

      8. +-commutative 21.1%

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

   3. Applied egg-rr 21.1%

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

      1. +-commutative 21.1%

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

      2. frac-2neg 21.1%

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

      3. metadata-eval 21.1%

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

      4. frac-add 17.5%

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

   5. Applied egg-rr 70.7%

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

      1. distribute-rgt-neg-out 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. associate-*r/ 99.5%

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

      4. metadata-eval 99.5%

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

      5. unpow2 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in x around inf 99.6%

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

       1. unpow2 99.6%

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

       2. distribute-rgt-out 99.6%

          \[\leadsto \frac{\frac{2}{x} - \frac{4}{x \cdot x}}{\color{blue}{x \cdot \left(-2 + x\right)}} \]

   13. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -4 \cdot 10^{-8} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-22}\right):\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot \left(x + -2\right)}\\ \end{array} \]

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x} - \frac{4}{x \cdot x}\\ \mathbf{if}\;x \leq -95000000:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 x) (/ 4.0 (* x x)))))
   (if (<= x -95000000.0)
     (/ t_0 (* x x))
     (if (<= x 100000000.0)
       (/
        (+ (* (+ x x) (/ x 2.0)) (* (+ x -1.0) (- -1.0 x)))
        (* (+ x -1.0) (* (+ 1.0 x) (/ x 2.0))))
       (/ t_0 (* x (+ x -2.0)))))))

C

double code(double x) {
	double t_0 = (2.0 / x) - (4.0 / (x * x));
	double tmp;
	if (x <= -95000000.0) {
		tmp = t_0 / (x * x);
	} else if (x <= 100000000.0) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else {
		tmp = t_0 / (x * (x + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / x) - (4.0d0 / (x * x))
    if (x <= (-95000000.0d0)) then
        tmp = t_0 / (x * x)
    else if (x <= 100000000.0d0) then
        tmp = (((x + x) * (x / 2.0d0)) + ((x + (-1.0d0)) * ((-1.0d0) - x))) / ((x + (-1.0d0)) * ((1.0d0 + x) * (x / 2.0d0)))
    else
        tmp = t_0 / (x * (x + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (2.0 / x) - (4.0 / (x * x));
	double tmp;
	if (x <= -95000000.0) {
		tmp = t_0 / (x * x);
	} else if (x <= 100000000.0) {
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	} else {
		tmp = t_0 / (x * (x + -2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (2.0 / x) - (4.0 / (x * x))
	tmp = 0
	if x <= -95000000.0:
		tmp = t_0 / (x * x)
	elif x <= 100000000.0:
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)))
	else:
		tmp = t_0 / (x * (x + -2.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / x) - Float64(4.0 / Float64(x * x)))
	tmp = 0.0
	if (x <= -95000000.0)
		tmp = Float64(t_0 / Float64(x * x));
	elseif (x <= 100000000.0)
		tmp = Float64(Float64(Float64(Float64(x + x) * Float64(x / 2.0)) + Float64(Float64(x + -1.0) * Float64(-1.0 - x))) / Float64(Float64(x + -1.0) * Float64(Float64(1.0 + x) * Float64(x / 2.0))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(x + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (2.0 / x) - (4.0 / (x * x));
	tmp = 0.0;
	if (x <= -95000000.0)
		tmp = t_0 / (x * x);
	elseif (x <= 100000000.0)
		tmp = (((x + x) * (x / 2.0)) + ((x + -1.0) * (-1.0 - x))) / ((x + -1.0) * ((1.0 + x) * (x / 2.0)));
	else
		tmp = t_0 / (x * (x + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / x), $MachinePrecision] - N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -95000000.0], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(N[(N[(N[(x + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5e7

   1. Initial program 73.3%

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

      1. frac-sub 18.9%

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

      2. clear-num 20.7%

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

      3. *-commutative 20.7%

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

      4. +-commutative 20.7%

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

      5. /-rgt-identity 20.7%

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

      6. *-un-lft-identity 20.7%

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

      7. /-rgt-identity 20.7%

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

      8. +-commutative 20.7%

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

   3. Applied egg-rr 20.7%

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

      1. +-commutative 20.7%

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

      2. frac-2neg 20.7%

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

      3. metadata-eval 20.7%

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

      4. frac-add 17.4%

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

   5. Applied egg-rr 69.9%

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

      1. distribute-rgt-neg-out 69.9%

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

      2. distribute-lft-neg-out 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

      5. unpow2 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 99.8%

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

       1. unpow2 99.8%

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

   13. Simplified 99.8%

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

   if -9.5e7 < x < 1e8

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-+r- 98.7%

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

      3. sub-neg 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. frac-add 98.7%

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

      2. clear-num 98.7%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-+r+ 99.9%

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

       3. metadata-eval 99.9%

          \[\leadsto \frac{\left(x + \left(\color{blue}{0} + x\right)\right) \cdot \frac{x}{2} - \left(x + -1\right) \cdot \left(x + 1\right)}{\left(x + -1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x}{2}\right)} \]

       4. +-lft-identity 99.9%

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

   11. Simplified 99.9%

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

   if 1e8 < x

   1. Initial program 75.0%

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

      1. frac-sub 16.8%

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

      2. clear-num 20.9%

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

      3. *-commutative 20.9%

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

      4. +-commutative 20.9%

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

      5. /-rgt-identity 20.9%

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

      6. *-un-lft-identity 20.9%

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

      7. /-rgt-identity 20.9%

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

      8. +-commutative 20.9%

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

   3. Applied egg-rr 20.9%

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

      1. +-commutative 20.9%

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

      2. frac-2neg 20.9%

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

      3. metadata-eval 20.9%

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

      4. frac-add 17.0%

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

   5. Applied egg-rr 72.3%

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

      1. distribute-rgt-neg-out 72.3%

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

      2. distribute-lft-neg-out 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. unpow2 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 99.9%

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

       1. unpow2 99.9%

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

       2. distribute-rgt-out 99.9%

          \[\leadsto \frac{\frac{2}{x} - \frac{4}{x \cdot x}}{\color{blue}{x \cdot \left(-2 + x\right)}} \]

   13. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95000000:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{\left(x + x\right) \cdot \frac{x}{2} + \left(x + -1\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(\left(1 + x\right) \cdot \frac{x}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot \left(x + -2\right)}\\ \end{array} \]

Alternative 8: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x} - \frac{4}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (- (/ 2.0 x) (/ 4.0 (* x x))) (* x x))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((2.0d0 / x) - (4.0d0 / (x * x))) / (x * x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(Float64(2.0 / x) - Float64(4.0 / Float64(x * x))) / Float64(x * x));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = ((2.0 / x) - (4.0 / (x * x))) / (x * x);
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[(2.0 / x), $MachinePrecision] - N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.9%

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

      1. frac-sub 19.1%

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

      2. clear-num 22.1%

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

      3. *-commutative 22.1%

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

      4. +-commutative 22.1%

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

      5. /-rgt-identity 22.1%

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

      6. *-un-lft-identity 22.1%

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

      7. /-rgt-identity 22.1%

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

      8. +-commutative 22.1%

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

   3. Applied egg-rr 22.1%

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

      1. +-commutative 22.1%

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

      2. frac-2neg 22.1%

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

      3. metadata-eval 22.1%

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

      4. frac-add 18.5%

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

   5. Applied egg-rr 70.9%

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

      1. distribute-rgt-neg-out 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in x around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

      5. unpow2 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 97.6%

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

       1. unpow2 97.6%

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

   13. Simplified 97.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 98.3%

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

Alternative 9: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ 1.0 (+ x -1.0)) (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = (1.0d0 / (x + (-1.0d0))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 73.0%

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

   2. Taylor expanded in x around inf 71.3%

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

   if -0.650000000000000022 < x < 0.650000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   4. Simplified 99.0%

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

   if 0.650000000000000022 < x

   1. Initial program 74.6%

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

   2. Taylor expanded in x around inf 73.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 83.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 x) (/ 2.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ 1.0 (+ x -1.0)) (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (2.0d0 / x) - (2.0d0 / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = (1.0d0 / (x + (-1.0d0))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (2.0 / x) - (2.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / x) - Float64(2.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (2.0 / x) - (2.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+r- 72.9%

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

      3. sub-neg 72.9%

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

      4. metadata-eval 72.9%

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

      5. +-commutative 72.9%

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

   3. Applied egg-rr 72.9%

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

   4. Taylor expanded in x around inf 71.0%

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

   if -1 < x < 0.650000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   4. Simplified 99.0%

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

   if 0.650000000000000022 < x

   1. Initial program 74.6%

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

   2. Taylor expanded in x around inf 73.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \]

Alternative 11: 82.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. associate-+r- 73.8%

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

      3. sub-neg 73.8%

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

      4. metadata-eval 73.8%

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

      5. +-commutative 73.8%

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

   3. Applied egg-rr 73.8%

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

   4. Taylor expanded in x around inf 72.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 85.3%

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

Alternative 12: 76.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0 / (x * x)
	elif x <= 1.0:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = 1.0 / (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+r- 72.9%

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

      3. sub-neg 72.9%

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

      4. metadata-eval 72.9%

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

      5. +-commutative 72.9%

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

   3. Applied egg-rr 72.9%

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

   4. Taylor expanded in x around inf 71.3%

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

   5. Taylor expanded in x around inf 53.2%

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

      1. unpow2 53.2%

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

   7. Simplified 53.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   4. Simplified 99.0%

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

   if 1 < x

   1. Initial program 74.6%

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

      1. frac-sub 18.0%

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

      2. clear-num 22.1%

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

      3. *-commutative 22.1%

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

      4. +-commutative 22.1%

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

      5. /-rgt-identity 22.1%

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

      6. *-un-lft-identity 22.1%

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

      7. /-rgt-identity 22.1%

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

      8. +-commutative 22.1%

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

   3. Applied egg-rr 22.1%

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

      1. +-commutative 22.1%

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

      2. frac-2neg 22.1%

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

      3. metadata-eval 22.1%

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

      4. frac-add 18.2%

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

   5. Applied egg-rr 71.9%

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

      1. distribute-rgt-neg-out 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in x around 0 59.9%

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

   9. Taylor expanded in x around inf 59.9%

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

       1. unpow2 59.9%

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

   11. Simplified 59.9%

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

4. Final simplification 77.6%

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

Alternative 13: 75.4% 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 73.9%

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

      1. +-commutative 73.9%

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

      2. associate-+r- 73.8%

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

      3. sub-neg 73.8%

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

      4. metadata-eval 73.8%

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

      5. +-commutative 73.8%

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

   3. Applied egg-rr 73.8%

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

   4. Taylor expanded in x around inf 72.2%

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

   5. Taylor expanded in x around inf 56.1%

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

      1. unpow2 56.1%

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

   7. Simplified 56.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 76.8%

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

Alternative 14: 75.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0 / (x * x)
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 1.0 / (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+r- 72.9%

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

      3. sub-neg 72.9%

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

      4. metadata-eval 72.9%

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

      5. +-commutative 72.9%

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

   3. Applied egg-rr 72.9%

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

   4. Taylor expanded in x around inf 71.3%

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

   5. Taylor expanded in x around inf 53.2%

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

      1. unpow2 53.2%

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

   7. Simplified 53.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.2%

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

   if 1 < x

   1. Initial program 74.6%

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

      1. frac-sub 18.0%

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

      2. clear-num 22.1%

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

      3. *-commutative 22.1%

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

      4. +-commutative 22.1%

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

      5. /-rgt-identity 22.1%

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

      6. *-un-lft-identity 22.1%

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

      7. /-rgt-identity 22.1%

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

      8. +-commutative 22.1%

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

   3. Applied egg-rr 22.1%

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

      1. +-commutative 22.1%

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

      2. frac-2neg 22.1%

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

      3. metadata-eval 22.1%

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

      4. frac-add 18.2%

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

   5. Applied egg-rr 71.9%

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

      1. distribute-rgt-neg-out 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in x around 0 59.9%

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

   9. Taylor expanded in x around inf 59.9%

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

       1. unpow2 59.9%

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

   11. Simplified 59.9%

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

4. Final simplification 77.2%

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

Alternative 15: 51.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

2. Taylor expanded in x around 0 50.9%

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

3. Final simplification 50.9%

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

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 2023255 
(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))))