3frac (problem 3.3.3)

Percentage Accurate: 69.5% → 99.7%

Time: 11.1s

Alternatives: 12

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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(-0.5 \cdot \left(-1 - x\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) + \frac{1}{x\_m - 1} \leq 10^{-27}:\\ \;\;\;\;2 \cdot {x\_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-1 - x\_m\right) - x\_m \cdot -0.5\right) \cdot \left(x\_m + -1\right) + t\_0 \cdot 1}{t\_0 \cdot \left(x\_m + -1\right)}\\ \end{array} \end{array} \end{array} \]

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 (* x_m (* -0.5 (- -1.0 x_m)))))
   (*
    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))
      (/
       (+ (* (- (- -1.0 x_m) (* x_m -0.5)) (+ x_m -1.0)) (* t_0 1.0))
       (* t_0 (+ x_m -1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (-0.5 * (-1.0 - x_m));
	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 = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	}
	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 = x_m * ((-0.5d0) * ((-1.0d0) - x_m))
    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 = (((((-1.0d0) - x_m) - (x_m * (-0.5d0))) * (x_m + (-1.0d0))) + (t_0 * 1.0d0)) / (t_0 * (x_m + (-1.0d0)))
    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 = x_m * (-0.5 * (-1.0 - x_m));
	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 = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	}
	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 = x_m * (-0.5 * (-1.0 - x_m))
	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 = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(-0.5 * Float64(-1.0 - x_m)))
	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(Float64(Float64(-1.0 - x_m) - Float64(x_m * -0.5)) * Float64(x_m + -1.0)) + Float64(t_0 * 1.0)) / Float64(t_0 * Float64(x_m + -1.0)));
	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 = x_m * (-0.5 * (-1.0 - x_m));
	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 = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	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[(x$95$m * N[(-0.5 * N[(-1.0 - x$95$m), $MachinePrecision]), $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[(N[(N[(-1.0 - x$95$m), $MachinePrecision] - N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x$95$m + -1.0), $MachinePrecision]), $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 98.2%

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

      1. div-inv 98.2%

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

      2. pow-flip 99.1%

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

      3. metadata-eval 99.1%

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

   7. Applied egg-rr 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. clear-num 59.8%

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

      2. frac-sub 59.9%

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

      3. *-un-lft-identity 59.9%

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

      4. div-inv 59.9%

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

      5. metadata-eval 59.9%

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

      6. div-inv 59.9%

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

      7. metadata-eval 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. +-commutative 59.9%

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

      2. frac-add 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. associate-*l* 99.4%

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

      5. associate-*l* 99.4%

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

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-1 - x\right) - x \cdot -0.5\right) \cdot \left(x + -1\right) + \left(x \cdot \left(-0.5 \cdot \left(-1 - x\right)\right)\right) \cdot 1}{\left(x \cdot \left(-0.5 \cdot \left(-1 - x\right)\right)\right) \cdot \left(x + -1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(2 + \mathsf{fma}\left(2, {x\_m}^{-2}, \mathsf{fma}\left(2, {x\_m}^{-6}, 2 \cdot {x\_m}^{-4}\right)\right)\right) \cdot {x\_m}^{-3}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (+
    2.0
    (fma 2.0 (pow x_m -2.0) (fma 2.0 (pow x_m -6.0) (* 2.0 (pow x_m -4.0)))))
   (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 * ((2.0 + fma(2.0, pow(x_m, -2.0), fma(2.0, pow(x_m, -6.0), (2.0 * pow(x_m, -4.0))))) * 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(Float64(2.0 + fma(2.0, (x_m ^ -2.0), fma(2.0, (x_m ^ -6.0), Float64(2.0 * (x_m ^ -4.0))))) * (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[(2.0 * N[Power[x$95$m, -2.0], $MachinePrecision] + N[(2.0 * N[Power[x$95$m, -6.0], $MachinePrecision] + N[(2.0 * N[Power[x$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 98.6%

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

   1. div-inv 98.6%

      \[\leadsto \color{blue}{\left(2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. fma-define 98.6%

      \[\leadsto \left(2 + \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   3. pow-flip 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   4. metadata-eval 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   5. fma-define 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{6}}, \frac{2}{{x}^{4}}\right)}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   6. pow-flip 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-6\right)}}, \frac{2}{{x}^{4}}\right)\right)\right) \cdot \frac{1}{{x}^{3}} \]

   7. metadata-eval 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-6}}, \frac{2}{{x}^{4}}\right)\right)\right) \cdot \frac{1}{{x}^{3}} \]

   8. div-inv 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}}\right)\right)\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 98.6%

      \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot \color{blue}{{x}^{\left(-4\right)}}\right)\right)\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 98.6%

       \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot {x}^{\color{blue}{-4}}\right)\right)\right) \cdot \frac{1}{{x}^{3}} \]

   11. pow-flip 99.4%

       \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot {x}^{-4}\right)\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   12. metadata-eval 99.4%

       \[\leadsto \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot {x}^{-4}\right)\right)\right) \cdot {x}^{\color{blue}{-3}} \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot {x}^{-4}\right)\right)\right) \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 3: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{2 + \left(2 \cdot \frac{\frac{1}{x\_m}}{x\_m} + \left(2 \cdot \frac{1}{{x\_m}^{6}} + \frac{2}{{x\_m}^{4}}\right)\right)}{{x\_m}^{3}} \end{array} \]

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
    (+
     (* 2.0 (/ (/ 1.0 x_m) x_m))
     (+ (* 2.0 (/ 1.0 (pow x_m 6.0))) (/ 2.0 (pow x_m 4.0)))))
   (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 * ((2.0 + ((2.0 * ((1.0 / x_m) / x_m)) + ((2.0 * (1.0 / pow(x_m, 6.0))) + (2.0 / pow(x_m, 4.0))))) / pow(x_m, 3.0));
}

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 * ((2.0d0 + ((2.0d0 * ((1.0d0 / x_m) / x_m)) + ((2.0d0 * (1.0d0 / (x_m ** 6.0d0))) + (2.0d0 / (x_m ** 4.0d0))))) / (x_m ** 3.0d0))
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 * ((2.0 + ((2.0 * ((1.0 / x_m) / x_m)) + ((2.0 * (1.0 / Math.pow(x_m, 6.0))) + (2.0 / Math.pow(x_m, 4.0))))) / Math.pow(x_m, 3.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((2.0 + ((2.0 * ((1.0 / x_m) / x_m)) + ((2.0 * (1.0 / math.pow(x_m, 6.0))) + (2.0 / math.pow(x_m, 4.0))))) / math.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(Float64(2.0 + Float64(Float64(2.0 * Float64(Float64(1.0 / x_m) / x_m)) + Float64(Float64(2.0 * Float64(1.0 / (x_m ^ 6.0))) + Float64(2.0 / (x_m ^ 4.0))))) / (x_m ^ 3.0)))
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 + ((2.0 * ((1.0 / x_m) / x_m)) + ((2.0 * (1.0 / (x_m ^ 6.0))) + (2.0 / (x_m ^ 4.0))))) / (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[(N[(2.0 * N[(N[(1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 98.6%

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

   1. inv-pow 98.6%

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

   2. unpow2 98.6%

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

   3. unpow-prod-down 98.6%

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

   4. inv-pow 98.6%

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

   5. inv-pow 98.6%

      \[\leadsto \frac{2 + \left(2 \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) + \left(2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right)}{{x}^{3}} \]

7. Applied egg-rr 98.6%

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

   1. un-div-inv 98.6%

      \[\leadsto \frac{2 + \left(2 \cdot \color{blue}{\frac{\frac{1}{x}}{x}} + \left(2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right)}{{x}^{3}} \]

9. Applied egg-rr 98.6%

   \[\leadsto \frac{2 + \left(2 \cdot \color{blue}{\frac{\frac{1}{x}}{x}} + \left(2 \cdot \frac{1}{{x}^{6}} + \frac{2}{{x}^{4}}\right)\right)}{{x}^{3}} \]
10. Add Preprocessing

Alternative 4: 98.9% accurate, 0.0× speedup

Math

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

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 (+ (* 2.0 (/ 1.0 (pow x_m 2.0))) (/ 2.0 (pow x_m 4.0))))
   (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 * ((2.0 + ((2.0 * (1.0 / pow(x_m, 2.0))) + (2.0 / pow(x_m, 4.0)))) / pow(x_m, 3.0));
}

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 * ((2.0d0 + ((2.0d0 * (1.0d0 / (x_m ** 2.0d0))) + (2.0d0 / (x_m ** 4.0d0)))) / (x_m ** 3.0d0))
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 * ((2.0 + ((2.0 * (1.0 / Math.pow(x_m, 2.0))) + (2.0 / Math.pow(x_m, 4.0)))) / Math.pow(x_m, 3.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((2.0 + ((2.0 * (1.0 / math.pow(x_m, 2.0))) + (2.0 / math.pow(x_m, 4.0)))) / math.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(Float64(2.0 + Float64(Float64(2.0 * Float64(1.0 / (x_m ^ 2.0))) + Float64(2.0 / (x_m ^ 4.0)))) / (x_m ^ 3.0)))
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 + ((2.0 * (1.0 / (x_m ^ 2.0))) + (2.0 / (x_m ^ 4.0)))) / (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[(N[(2.0 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing

Alternative 5: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(-0.5 \cdot \left(-1 - x\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 200000000:\\ \;\;\;\;\frac{\left(\left(-1 - x\_m\right) - x\_m \cdot -0.5\right) \cdot \left(x\_m + -1\right) + t\_0 \cdot 1}{t\_0 \cdot \left(x\_m + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + \left(x\_m + -1\right) \cdot -1}{\left(x\_m + -1\right) \cdot x\_m}\\ \end{array} \end{array} \end{array} \]

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 (* x_m (* -0.5 (- -1.0 x_m)))))
   (*
    x_s
    (if (<= x_m 200000000.0)
      (/
       (+ (* (- (- -1.0 x_m) (* x_m -0.5)) (+ x_m -1.0)) (* t_0 1.0))
       (* t_0 (+ x_m -1.0)))
      (/ (+ x_m (* (+ x_m -1.0) -1.0)) (* (+ x_m -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 = x_m * (-0.5 * (-1.0 - x_m));
	double tmp;
	if (x_m <= 200000000.0) {
		tmp = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	} else {
		tmp = (x_m + ((x_m + -1.0) * -1.0)) / ((x_m + -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 = x_m * ((-0.5d0) * ((-1.0d0) - x_m))
    if (x_m <= 200000000.0d0) then
        tmp = (((((-1.0d0) - x_m) - (x_m * (-0.5d0))) * (x_m + (-1.0d0))) + (t_0 * 1.0d0)) / (t_0 * (x_m + (-1.0d0)))
    else
        tmp = (x_m + ((x_m + (-1.0d0)) * (-1.0d0))) / ((x_m + (-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 = x_m * (-0.5 * (-1.0 - x_m));
	double tmp;
	if (x_m <= 200000000.0) {
		tmp = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	} else {
		tmp = (x_m + ((x_m + -1.0) * -1.0)) / ((x_m + -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 = x_m * (-0.5 * (-1.0 - x_m))
	tmp = 0
	if x_m <= 200000000.0:
		tmp = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0))
	else:
		tmp = (x_m + ((x_m + -1.0) * -1.0)) / ((x_m + -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(x_m * Float64(-0.5 * Float64(-1.0 - x_m)))
	tmp = 0.0
	if (x_m <= 200000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 - x_m) - Float64(x_m * -0.5)) * Float64(x_m + -1.0)) + Float64(t_0 * 1.0)) / Float64(t_0 * Float64(x_m + -1.0)));
	else
		tmp = Float64(Float64(x_m + Float64(Float64(x_m + -1.0) * -1.0)) / Float64(Float64(x_m + -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 = x_m * (-0.5 * (-1.0 - x_m));
	tmp = 0.0;
	if (x_m <= 200000000.0)
		tmp = ((((-1.0 - x_m) - (x_m * -0.5)) * (x_m + -1.0)) + (t_0 * 1.0)) / (t_0 * (x_m + -1.0));
	else
		tmp = (x_m + ((x_m + -1.0) * -1.0)) / ((x_m + -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[(x$95$m * N[(-0.5 * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 200000000.0], N[(N[(N[(N[(N[(-1.0 - x$95$m), $MachinePrecision] - N[(x$95$m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m + N[(N[(x$95$m + -1.0), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m + -1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e8

   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. clear-num 72.3%

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

      2. frac-sub 26.6%

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

      3. *-un-lft-identity 26.6%

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

      4. div-inv 26.6%

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

      5. metadata-eval 26.6%

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

      6. div-inv 26.6%

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

      7. metadata-eval 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. +-commutative 26.6%

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

      2. frac-add 29.4%

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

      3. *-rgt-identity 29.4%

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

      4. associate-*l* 29.4%

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

      5. associate-*l* 29.4%

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

   8. Applied egg-rr 29.4%

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

   if 2e8 < 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%

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

      1. frac-add 70.3%

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

      2. *-un-lft-identity 70.3%

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

   7. Applied egg-rr 70.3%

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

Alternative 7: 69.5% accurate, 1.0× speedup

Math

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

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)) (/ (/ (+ 2.0 x_m) x_m) (- -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)) + (((2.0 + x_m) / x_m) / (-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))) + (((2.0d0 + x_m) / x_m) / ((-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)) + (((2.0 + x_m) / x_m) / (-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)) + (((2.0 + x_m) / x_m) / (-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(2.0 + x_m) / x_m) / 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)) + (((2.0 + x_m) / x_m) / (-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[(2.0 + x$95$m), $MachinePrecision] / x$95$m), $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. frac-sub 23.5%

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

   2. associate-/r* 71.0%

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

   3. *-rgt-identity 71.0%

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

   4. fma-neg 71.4%

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

6. Applied egg-rr 71.4%

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

7. Taylor expanded in x around 0 71.4%

   \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\frac{2 + x}{x}}}{-1 - x} \]
8. Add Preprocessing

Alternative 8: 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. add-sqr-sqrt 22.5%

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

   2. fma-define 4.9%

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

   3. mul-1-neg 4.9%

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

   4. add-sqr-sqrt 4.8%

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

   5. sqrt-unprod 3.3%

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

   6. sqr-neg 3.3%

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

   7. mul-1-neg 3.3%

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

   8. mul-1-neg 3.3%

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

   9. sqrt-unprod 0.0%

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

   10. add-sqr-sqrt 3.1%

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

   11. fma-neg 3.1%

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

   12. add-sqr-sqrt 0.0%

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

7. Applied egg-rr 69.3%

   \[\leadsto \color{blue}{\frac{1}{x + -1} - \frac{1 + \frac{1}{x}}{x}} \]
8. Add Preprocessing

Alternative 9: 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. Add Preprocessing

Alternative 10: 68.0% accurate, 2.1× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (- (/ -2.0 x_m) (/ -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) - (-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) - ((-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) - (-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) - (-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(Float64(-2.0 / x_m) - 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) - (-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[(N[(-2.0 / x$95$m), $MachinePrecision] - N[(-2.0 / 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. Step-by-step derivation

   1. +-commutative 71.3%

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

   2. associate-+l- 71.3%

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

6. Applied egg-rr 71.3%

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

7. Taylor expanded in x around inf 68.5%

   \[\leadsto \frac{-2}{x} - \color{blue}{\frac{-2}{x}} \]
8. Add Preprocessing

Alternative 11: 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. Add Preprocessing

Alternative 12: 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 68.8%

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

6. Taylor expanded in x around 0 5.1%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
7. 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 -o generate:simplify
(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))))