3frac (problem 3.3.3)

Percentage Accurate: 69.5% → 99.6%

Time: 19.7s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left|x\right| > 1\]

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: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(-1 - x\_m\right) \cdot \left(x\_m + -1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) + \frac{1}{x\_m + -1} \leq 10^{-27}:\\ \;\;\;\;2 \cdot {x\_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot t\_0 - x\_m \cdot \left(x\_m + x\_m\right)}{x\_m \cdot t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (- -1.0 x_m) (+ x_m -1.0))))
   (*
    x_s
    (if (<= (+ (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0))) 1e-27)
      (* 2.0 (pow x_m -3.0))
      (/ (- (* -2.0 t_0) (* x_m (+ x_m x_m))) (* x_m t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (-1.0 - x_m) * (x_m + -1.0);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 1e-27) {
		tmp = 2.0 * pow(x_m, -3.0);
	} else {
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x_m) * (x_m + (-1.0d0))
    if ((((1.0d0 / (x_m + 1.0d0)) - (2.0d0 / x_m)) + (1.0d0 / (x_m + (-1.0d0)))) <= 1d-27) then
        tmp = 2.0d0 * (x_m ** (-3.0d0))
    else
        tmp = (((-2.0d0) * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = (-1.0 - x_m) * (x_m + -1.0);
	double tmp;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 1e-27) {
		tmp = 2.0 * Math.pow(x_m, -3.0);
	} else {
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = (-1.0 - x_m) * (x_m + -1.0)
	tmp = 0
	if (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 1e-27:
		tmp = 2.0 * math.pow(x_m, -3.0)
	else:
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(-1.0 - x_m) * Float64(x_m + -1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m)) + Float64(1.0 / Float64(x_m + -1.0))) <= 1e-27)
		tmp = Float64(2.0 * (x_m ^ -3.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * t_0) - Float64(x_m * Float64(x_m + x_m))) / Float64(x_m * t_0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = (-1.0 - x_m) * (x_m + -1.0);
	tmp = 0.0;
	if ((((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0))) <= 1e-27)
		tmp = 2.0 * (x_m ^ -3.0);
	else
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(-1.0 - x$95$m), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-27], N[(2.0 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] - N[(x$95$m * N[(x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 1e-27

   1. Initial program 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-+r- 71.6%

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

      3. sub-neg 71.6%

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

      4. remove-double-neg 71.6%

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

      5. neg-sub0 71.6%

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

      6. associate-+l- 71.6%

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

      7. neg-sub0 71.6%

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

      8. distribute-neg-frac2 71.6%

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

      9. distribute-frac-neg2 71.6%

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

      10. associate-+r+ 71.7%

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

      11. +-commutative 71.7%

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

      12. remove-double-neg 71.7%

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

      13. distribute-neg-frac2 71.7%

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

      14. sub0-neg 71.7%

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

      15. associate-+l- 71.7%

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

      16. neg-sub0 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
   6. Step-by-step derivation

      1. associate-+r+ 98.8%

         \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

      2. +-commutative 98.8%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

      3. associate-+l+ 98.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

      4. associate-*r/ 98.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

      5. metadata-eval 98.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
   8. Step-by-step derivation

      1. div-inv 98.8%

         \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

      2. div-inv 98.8%

         \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

      3. fma-define 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 + \frac{2}{{x}^{4}}\right)} \cdot \frac{1}{{x}^{3}} \]

      4. pow-flip 98.8%

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

      5. metadata-eval 98.8%

         \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

      6. +-commutative 98.8%

         \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\frac{2}{{x}^{4}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

      7. div-inv 98.8%

         \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

      8. fma-define 98.8%

         \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{4}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

      9. pow-flip 98.8%

         \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-4\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

      10. metadata-eval 98.8%

          \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-4}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

      11. pow-flip 99.6%

          \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

      12. metadata-eval 99.6%

          \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3}} \]

   10. Taylor expanded in x around inf 99.1%

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

   if 1e-27 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 59.8%

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

      1. +-commutative 59.8%

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

      2. associate-+r- 58.6%

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

      3. sub-neg 58.6%

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

      4. remove-double-neg 58.6%

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

      5. neg-sub0 58.6%

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

      6. associate-+l- 58.6%

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

      7. neg-sub0 58.6%

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

      8. distribute-neg-frac2 58.6%

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

      9. distribute-frac-neg2 58.6%

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

      10. associate-+r+ 59.8%

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

      11. +-commutative 59.8%

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

      12. remove-double-neg 59.8%

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

      13. distribute-neg-frac2 59.8%

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

      14. sub0-neg 59.8%

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

      15. associate-+l- 59.8%

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

      16. neg-sub0 59.8%

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

   3. Simplified 59.8%

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

      1. +-commutative 59.8%

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

      2. associate-+l- 58.6%

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

   6. Applied egg-rr 58.6%

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

      1. frac-sub 57.3%

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

      2. frac-sub 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. div-inv 99.3%

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

      6. *-commutative 99.3%

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

      7. div-inv 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-rgt-identity 99.3%

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

      10. *-un-lft-identity 99.3%

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

      11. *-rgt-identity 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. associate--r+ 99.3%

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

      5. metadata-eval 99.3%

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

      6. neg-sub0 99.3%

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

      7. remove-double-neg 99.3%

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

      8. *-commutative 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-27}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right) - x \cdot \left(x + x\right)}{x \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\mathsf{fma}\left(2, {x\_m}^{-2}, \mathsf{fma}\left(2, {x\_m}^{-4}, 2\right)\right) \cdot {x\_m}^{-3}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (* (fma 2.0 (pow x_m -2.0) (fma 2.0 (pow x_m -4.0) 2.0)) (pow x_m -3.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (fma(2.0, pow(x_m, -2.0), fma(2.0, pow(x_m, -4.0), 2.0)) * pow(x_m, -3.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(fma(2.0, (x_m ^ -2.0), fma(2.0, (x_m ^ -4.0), 2.0)) * (x_m ^ -3.0)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(2.0 * N[Power[x$95$m, -2.0], $MachinePrecision] + N[(2.0 * N[Power[x$95$m, -4.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-+r+ 98.4%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 98.4%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 98.4%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 98.4%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 98.4%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 98.4%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. div-inv 98.4%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   3. fma-define 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 + \frac{2}{{x}^{4}}\right)} \cdot \frac{1}{{x}^{3}} \]

   4. pow-flip 98.4%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   5. metadata-eval 98.4%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   6. +-commutative 98.4%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\frac{2}{{x}^{4}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

   7. div-inv 98.4%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   8. fma-define 98.4%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{4}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 98.4%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-4\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 98.4%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-4}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   11. pow-flip 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   12. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3}} \]

10. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3} \]
11. Add Preprocessing

Alternative 3: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left({\left(\frac{1}{x\_m}\right)}^{3} \cdot \mathsf{fma}\left(2, {x\_m}^{-2}, 2\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (pow (/ 1.0 x_m) 3.0) (fma 2.0 (pow x_m -2.0) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (pow((1.0 / x_m), 3.0) * fma(2.0, pow(x_m, -2.0), 2.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64((Float64(1.0 / x_m) ^ 3.0) * fma(2.0, (x_m ^ -2.0), 2.0)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[Power[N[(1.0 / x$95$m), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[x$95$m, -2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 98.1%

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

   1. associate-*r/ 98.1%

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

   2. metadata-eval 98.1%

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

7. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. +-commutative 98.1%

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

   4. div-inv 98.1%

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

   5. fma-define 98.1%

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

   6. pow-flip 98.1%

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

   7. metadata-eval 98.1%

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

9. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{\left(\frac{{x}^{3}}{\mathsf{fma}\left(2, {x}^{-2}, 2\right)}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 98.1%

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

    2. associate-/r/ 98.1%

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

    3. metadata-eval 98.1%

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

    4. cube-div 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

    \[\leadsto {\left(\frac{1}{x}\right)}^{3} \cdot \mathsf{fma}\left(2, {x}^{-2}, 2\right) \]
13. Add Preprocessing

Alternative 4: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(-1 - x\_m\right) \cdot \left(x\_m + -1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 145000000:\\ \;\;\;\;\frac{-2 \cdot t\_0 - x\_m \cdot \left(x\_m + x\_m\right)}{x\_m \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m + -1} + \frac{-1}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (- -1.0 x_m) (+ x_m -1.0))))
   (*
    x_s
    (if (<= x_m 145000000.0)
      (/ (- (* -2.0 t_0) (* x_m (+ x_m x_m))) (* x_m t_0))
      (+ (/ 1.0 (+ x_m -1.0)) (/ -1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (-1.0 - x_m) * (x_m + -1.0);
	double tmp;
	if (x_m <= 145000000.0) {
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	} else {
		tmp = (1.0 / (x_m + -1.0)) + (-1.0 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x_m) * (x_m + (-1.0d0))
    if (x_m <= 145000000.0d0) then
        tmp = (((-2.0d0) * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0)
    else
        tmp = (1.0d0 / (x_m + (-1.0d0))) + ((-1.0d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = (-1.0 - x_m) * (x_m + -1.0);
	double tmp;
	if (x_m <= 145000000.0) {
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	} else {
		tmp = (1.0 / (x_m + -1.0)) + (-1.0 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = (-1.0 - x_m) * (x_m + -1.0)
	tmp = 0
	if x_m <= 145000000.0:
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0)
	else:
		tmp = (1.0 / (x_m + -1.0)) + (-1.0 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(-1.0 - x_m) * Float64(x_m + -1.0))
	tmp = 0.0
	if (x_m <= 145000000.0)
		tmp = Float64(Float64(Float64(-2.0 * t_0) - Float64(x_m * Float64(x_m + x_m))) / Float64(x_m * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m + -1.0)) + Float64(-1.0 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = (-1.0 - x_m) * (x_m + -1.0);
	tmp = 0.0;
	if (x_m <= 145000000.0)
		tmp = ((-2.0 * t_0) - (x_m * (x_m + x_m))) / (x_m * t_0);
	else
		tmp = (1.0 / (x_m + -1.0)) + (-1.0 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(-1.0 - x$95$m), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 145000000.0], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] - N[(x$95$m * N[(x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e8

   1. Initial program 72.3%

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

      1. +-commutative 72.3%

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

      2. associate-+r- 72.3%

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

      3. sub-neg 72.3%

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

      4. remove-double-neg 72.3%

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

      5. neg-sub0 72.3%

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

      6. associate-+l- 72.3%

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

      7. neg-sub0 72.3%

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

      8. distribute-neg-frac2 72.3%

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

      9. distribute-frac-neg2 72.3%

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

      10. associate-+r+ 72.3%

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

      11. +-commutative 72.3%

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

      12. remove-double-neg 72.3%

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

      13. distribute-neg-frac2 72.3%

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

      14. sub0-neg 72.3%

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

      15. associate-+l- 72.3%

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

      16. neg-sub0 72.3%

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

   3. Simplified 72.3%

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

      1. +-commutative 72.3%

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

      2. associate-+l- 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. frac-sub 26.4%

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

      2. frac-sub 29.4%

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

      3. *-rgt-identity 29.4%

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

      4. metadata-eval 29.4%

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

      5. div-inv 29.4%

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

      6. *-commutative 29.4%

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

      7. div-inv 29.4%

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

      8. metadata-eval 29.4%

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

      9. *-rgt-identity 29.4%

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

      10. *-un-lft-identity 29.4%

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

      11. *-rgt-identity 29.4%

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

   8. Applied egg-rr 29.4%

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

      1. *-commutative 29.4%

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

      2. associate--l+ 29.3%

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

      3. sub-neg 29.3%

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

      4. associate--r+ 29.4%

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

      5. metadata-eval 29.4%

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

      6. neg-sub0 29.4%

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

      7. remove-double-neg 29.4%

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

      8. *-commutative 29.4%

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

   10. Simplified 29.4%

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

   if 1.45e8 < x

   1. Initial program 70.2%

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

      1. +-commutative 70.2%

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

      2. associate-+r- 70.1%

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

      3. sub-neg 70.1%

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

      4. remove-double-neg 70.1%

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

      5. neg-sub0 70.1%

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

      6. associate-+l- 70.1%

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

      7. neg-sub0 70.1%

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

      8. distribute-neg-frac2 70.1%

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

      9. distribute-frac-neg2 70.1%

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

      10. associate-+r+ 70.2%

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

      11. +-commutative 70.2%

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

      12. remove-double-neg 70.2%

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

      13. distribute-neg-frac2 70.2%

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

      14. sub0-neg 70.2%

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

      15. associate-+l- 70.2%

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

      16. neg-sub0 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around inf 70.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 145000000:\\ \;\;\;\;\frac{-2 \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right) - x \cdot \left(x + x\right)}{x \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{1}{x\_m + -1} + \frac{\frac{-1 - \left(x\_m + x\_m \cdot -0.5\right)}{x\_m \cdot -0.5}}{-1 - x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   (/ 1.0 (+ x_m -1.0))
   (/ (/ (- -1.0 (+ x_m (* x_m -0.5))) (* x_m -0.5)) (- -1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + (((-1.0 - (x_m + (x_m * -0.5))) / (x_m * -0.5)) / (-1.0 - x_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((1.0d0 / (x_m + (-1.0d0))) + ((((-1.0d0) - (x_m + (x_m * (-0.5d0)))) / (x_m * (-0.5d0))) / ((-1.0d0) - x_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + (((-1.0 - (x_m + (x_m * -0.5))) / (x_m * -0.5)) / (-1.0 - x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((1.0 / (x_m + -1.0)) + (((-1.0 - (x_m + (x_m * -0.5))) / (x_m * -0.5)) / (-1.0 - x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(1.0 / Float64(x_m + -1.0)) + Float64(Float64(Float64(-1.0 - Float64(x_m + Float64(x_m * -0.5))) / Float64(x_m * -0.5)) / Float64(-1.0 - x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((1.0 / (x_m + -1.0)) + (((-1.0 - (x_m + (x_m * -0.5))) / (x_m * -0.5)) / (-1.0 - x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 - N[(x$95$m + N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. clear-num 71.3%

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

   2. frac-sub 23.5%

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

   3. *-un-lft-identity 23.5%

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

   4. div-inv 23.5%

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

   5. metadata-eval 23.5%

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

   6. div-inv 23.5%

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

   7. metadata-eval 23.5%

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

6. Applied egg-rr 23.5%

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

   1. associate-/r* 71.4%

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

   2. *-rgt-identity 71.4%

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

   3. associate--r+ 71.4%

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

8. Simplified 71.4%

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

9. Final simplification 71.4%

   \[\leadsto \frac{1}{x + -1} + \frac{\frac{-1 - \left(x + x \cdot -0.5\right)}{x \cdot -0.5}}{-1 - x} \]
10. Add Preprocessing

Alternative 6: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) + \frac{1}{x\_m + -1}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m)) (/ 1.0 (+ x_m -1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (((1.0d0 / (x_m + 1.0d0)) - (2.0d0 / x_m)) + (1.0d0 / (x_m + (-1.0d0))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m)) + Float64(1.0 / Float64(x_m + -1.0))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (((1.0 / (x_m + 1.0)) - (2.0 / x_m)) + (1.0 / (x_m + -1.0)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

3. Final simplification 71.3%

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

Alternative 7: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{1}{x\_m + -1} + \frac{-1 + \frac{-1}{x\_m}}{x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ 1.0 (+ x_m -1.0)) (/ (+ -1.0 (/ -1.0 x_m)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + ((-1.0 + (-1.0 / x_m)) / x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((1.0d0 / (x_m + (-1.0d0))) + (((-1.0d0) + ((-1.0d0) / x_m)) / x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + ((-1.0 + (-1.0 / x_m)) / x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((1.0 / (x_m + -1.0)) + ((-1.0 + (-1.0 / x_m)) / x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(1.0 / Float64(x_m + -1.0)) + Float64(Float64(-1.0 + Float64(-1.0 / x_m)) / x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((1.0 / (x_m + -1.0)) + ((-1.0 + (-1.0 / x_m)) / x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 69.3%

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

   1. associate-*r/ 69.3%

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

   2. neg-mul-1 69.3%

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

   3. distribute-neg-in 69.3%

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

   4. metadata-eval 69.3%

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

   5. distribute-neg-frac 69.3%

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

   6. metadata-eval 69.3%

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

7. Simplified 69.3%

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

8. Final simplification 69.3%

   \[\leadsto \frac{1}{x + -1} + \frac{-1 + \frac{-1}{x}}{x} \]
9. Add Preprocessing

Alternative 8: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{1}{x\_m + -1} + \frac{-1}{x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ 1.0 (+ x_m -1.0)) (/ -1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + (-1.0 / x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((1.0d0 / (x_m + (-1.0d0))) + ((-1.0d0) / x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((1.0 / (x_m + -1.0)) + (-1.0 / x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((1.0 / (x_m + -1.0)) + (-1.0 / x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(1.0 / Float64(x_m + -1.0)) + Float64(-1.0 / x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((1.0 / (x_m + -1.0)) + (-1.0 / x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 68.8%

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

6. Final simplification 68.8%

   \[\leadsto \frac{1}{x + -1} + \frac{-1}{x} \]
7. Add Preprocessing

Alternative 9: 5.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-2}{x\_m} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ -2.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-2.0 / x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((-2.0d0) / x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (-2.0 / x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (-2.0 / x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-2.0 / x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (-2.0 / x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 10: 5.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-1}{x\_m} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ -1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-1.0 / x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((-1.0d0) / x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (-1.0 / x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (-1.0 / x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-1.0 / x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (-1.0 / x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.3%

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

   3. sub-neg 71.3%

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

   4. remove-double-neg 71.3%

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

   5. neg-sub0 71.3%

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

   6. associate-+l- 71.3%

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

   7. neg-sub0 71.3%

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

   8. distribute-neg-frac2 71.3%

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

   9. distribute-frac-neg2 71.3%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 69.3%

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

   1. associate-*r/ 69.3%

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

   2. neg-mul-1 69.3%

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

   3. distribute-neg-in 69.3%

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

   4. metadata-eval 69.3%

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

   5. distribute-neg-frac 69.3%

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

   6. metadata-eval 69.3%

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

7. Simplified 69.3%

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

8. Taylor expanded in x around 0 50.1%

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

   1. sub-neg 50.1%

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

   2. metadata-eval 50.1%

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

   3. +-commutative 50.1%

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

   4. mul-1-neg 50.1%

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

   5. sub-neg 50.1%

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

   6. unpow2 50.1%

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

   7. associate-/r* 5.1%

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

   8. sub-neg 5.1%

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

   9. mul-1-neg 5.1%

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

   10. +-commutative 5.1%

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

   11. metadata-eval 5.1%

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

   12. sub-neg 5.1%

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

   13. div-sub 5.1%

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

   14. mul-1-neg 5.1%

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

   15. distribute-frac-neg 5.1%

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

   16. *-inverses 5.1%

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

   17. metadata-eval 5.1%

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

10. Simplified 5.1%

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

11. Taylor expanded in x around inf 5.1%

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

12. Final simplification 5.1%

    \[\leadsto \frac{-1}{x} \]
13. Add Preprocessing

Developer target: 99.2% 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 2024076 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

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