3frac (problem 3.3.3)

Percentage Accurate: 84.0% → 99.5%

Time: 12.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 100000:\\ \;\;\;\;\frac{\left(x_m + 1\right) \cdot \left(\mathsf{fma}\left(x_m, 2, -2\right) - x_m\right) + x_m \cdot \left(1 - x_m\right)}{\left({x_m}^{2} - x_m\right) \cdot \left(-1 - x_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x_m}^{3}} + \frac{2}{{x_m}^{5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 100000.0)
    (/
     (+ (* (+ x_m 1.0) (- (fma x_m 2.0 -2.0) x_m)) (* x_m (- 1.0 x_m)))
     (* (- (pow x_m 2.0) x_m) (- -1.0 x_m)))
    (+ (/ 2.0 (pow x_m 3.0)) (/ 2.0 (pow x_m 5.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 100000.0) {
		tmp = (((x_m + 1.0) * (fma(x_m, 2.0, -2.0) - x_m)) + (x_m * (1.0 - x_m))) / ((pow(x_m, 2.0) - x_m) * (-1.0 - x_m));
	} else {
		tmp = (2.0 / pow(x_m, 3.0)) + (2.0 / pow(x_m, 5.0));
	}
	return x_s * tmp;
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 100000.0)
		tmp = Float64(Float64(Float64(Float64(x_m + 1.0) * Float64(fma(x_m, 2.0, -2.0) - x_m)) + Float64(x_m * Float64(1.0 - x_m))) / Float64(Float64((x_m ^ 2.0) - x_m) * Float64(-1.0 - x_m)));
	else
		tmp = Float64(Float64(2.0 / (x_m ^ 3.0)) + Float64(2.0 / (x_m ^ 5.0)));
	end
	return Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 100000.0], N[(N[(N[(N[(x$95$m + 1.0), $MachinePrecision] * N[(N[(x$95$m * 2.0 + -2.0), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] - x$95$m), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e5

   1. Initial program 86.9%

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

   3. Simplified 86.9%

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

      1. frac-sub 73.8%

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

      2. associate-/r* 87.0%

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

      3. /-rgt-identity 87.0%

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

      4. clear-num 86.9%

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

      5. associate-/r/ 86.9%

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

      6. +-commutative 86.9%

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

      7. distribute-lft-in 86.9%

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

      8. metadata-eval 86.9%

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

      9. metadata-eval 86.9%

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

      10. *-rgt-identity 86.9%

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

      11. associate--l+ 86.9%

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

      12. metadata-eval 86.9%

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

   5. Applied egg-rr 86.9%

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

      1. *-commutative 86.9%

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

      2. /-rgt-identity 86.9%

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

      3. associate-+r- 86.9%

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

      4. +-commutative 86.9%

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

      5. associate--l+ 86.9%

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

      6. *-commutative 86.9%

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

   7. Simplified 86.9%

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

      1. frac-2neg 86.9%

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

      2. metadata-eval 86.9%

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

      3. distribute-neg-in 86.9%

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

      4. metadata-eval 86.9%

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

      5. sub-neg 86.9%

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

      6. frac-times 73.8%

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

      7. *-un-lft-identity 73.8%

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

      8. *-commutative 73.8%

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

      9. frac-sub 73.4%

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

   9. Applied egg-rr 73.3%

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

   11. Simplified 73.4%

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

   if 1e5 < x

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100000:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right) + x \cdot \left(1 - x\right)}{\left({x}^{2} - x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{-1}{1 - x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 520:\\ \;\;\;\;t_0 + \left(\left(\left(\frac{1}{x_m + 1} + \frac{-2}{x_m}\right) + \left(\frac{-2}{x_m} + t_0\right)\right) + \left(\frac{2}{x_m} + \frac{1}{1 - x_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x_m}^{3}} + \frac{2}{{x_m}^{5}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (/ -1.0 (- 1.0 x_m))))
   (*
    x_s
    (if (<= x_m 520.0)
      (+
       t_0
       (+
        (+ (+ (/ 1.0 (+ x_m 1.0)) (/ -2.0 x_m)) (+ (/ -2.0 x_m) t_0))
        (+ (/ 2.0 x_m) (/ 1.0 (- 1.0 x_m)))))
      (+ (/ 2.0 (pow x_m 3.0)) (/ 2.0 (pow x_m 5.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = -1.0 / (1.0 - x_m);
	double tmp;
	if (x_m <= 520.0) {
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	} else {
		tmp = (2.0 / pow(x_m, 3.0)) + (2.0 / pow(x_m, 5.0));
	}
	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) / (1.0d0 - x_m)
    if (x_m <= 520.0d0) then
        tmp = t_0 + ((((1.0d0 / (x_m + 1.0d0)) + ((-2.0d0) / x_m)) + (((-2.0d0) / x_m) + t_0)) + ((2.0d0 / x_m) + (1.0d0 / (1.0d0 - x_m))))
    else
        tmp = (2.0d0 / (x_m ** 3.0d0)) + (2.0d0 / (x_m ** 5.0d0))
    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 = -1.0 / (1.0 - x_m);
	double tmp;
	if (x_m <= 520.0) {
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	} else {
		tmp = (2.0 / Math.pow(x_m, 3.0)) + (2.0 / Math.pow(x_m, 5.0));
	}
	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 / (1.0 - x_m)
	tmp = 0
	if x_m <= 520.0:
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))))
	else:
		tmp = (2.0 / math.pow(x_m, 3.0)) + (2.0 / math.pow(x_m, 5.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(-1.0 / Float64(1.0 - x_m))
	tmp = 0.0
	if (x_m <= 520.0)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) + Float64(-2.0 / x_m)) + Float64(Float64(-2.0 / x_m) + t_0)) + Float64(Float64(2.0 / x_m) + Float64(1.0 / Float64(1.0 - x_m)))));
	else
		tmp = Float64(Float64(2.0 / (x_m ^ 3.0)) + Float64(2.0 / (x_m ^ 5.0)));
	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 / (1.0 - x_m);
	tmp = 0.0;
	if (x_m <= 520.0)
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	else
		tmp = (2.0 / (x_m ^ 3.0)) + (2.0 / (x_m ^ 5.0));
	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[(-1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 520.0], N[(t$95$0 + N[(N[(N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. div-inv 87.1%

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

      2. *-un-lft-identity 87.1%

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

      3. prod-diff 87.1%

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

      4. *-commutative 87.1%

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

      5. *-un-lft-identity 87.1%

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

      6. fma-def 87.1%

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

      7. div-inv 87.1%

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

      8. associate-+l+ 87.1%

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

   5. Applied egg-rr 87.1%

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

   7. Simplified 87.1%

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

   if 520 < x

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;\frac{-1}{1 - x} + \left(\left(\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \left(\frac{-2}{x} + \frac{-1}{1 - x}\right)\right) + \left(\frac{2}{x} + \frac{1}{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{-1}{1 - x_m}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 15500:\\ \;\;\;\;t_0 + \left(\left(\left(\frac{1}{x_m + 1} + \frac{-2}{x_m}\right) + \left(\frac{-2}{x_m} + t_0\right)\right) + \left(\frac{2}{x_m} + \frac{1}{1 - x_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x_m}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (/ -1.0 (- 1.0 x_m))))
   (*
    x_s
    (if (<= x_m 15500.0)
      (+
       t_0
       (+
        (+ (+ (/ 1.0 (+ x_m 1.0)) (/ -2.0 x_m)) (+ (/ -2.0 x_m) t_0))
        (+ (/ 2.0 x_m) (/ 1.0 (- 1.0 x_m)))))
      (/ 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) {
	double t_0 = -1.0 / (1.0 - x_m);
	double tmp;
	if (x_m <= 15500.0) {
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	} else {
		tmp = 2.0 / pow(x_m, 3.0);
	}
	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) / (1.0d0 - x_m)
    if (x_m <= 15500.0d0) then
        tmp = t_0 + ((((1.0d0 / (x_m + 1.0d0)) + ((-2.0d0) / x_m)) + (((-2.0d0) / x_m) + t_0)) + ((2.0d0 / x_m) + (1.0d0 / (1.0d0 - x_m))))
    else
        tmp = 2.0d0 / (x_m ** 3.0d0)
    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 = -1.0 / (1.0 - x_m);
	double tmp;
	if (x_m <= 15500.0) {
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	} else {
		tmp = 2.0 / Math.pow(x_m, 3.0);
	}
	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 / (1.0 - x_m)
	tmp = 0
	if x_m <= 15500.0:
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))))
	else:
		tmp = 2.0 / math.pow(x_m, 3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(-1.0 / Float64(1.0 - x_m))
	tmp = 0.0
	if (x_m <= 15500.0)
		tmp = Float64(t_0 + Float64(Float64(Float64(Float64(1.0 / Float64(x_m + 1.0)) + Float64(-2.0 / x_m)) + Float64(Float64(-2.0 / x_m) + t_0)) + Float64(Float64(2.0 / x_m) + Float64(1.0 / Float64(1.0 - x_m)))));
	else
		tmp = Float64(2.0 / (x_m ^ 3.0));
	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 / (1.0 - x_m);
	tmp = 0.0;
	if (x_m <= 15500.0)
		tmp = t_0 + ((((1.0 / (x_m + 1.0)) + (-2.0 / x_m)) + ((-2.0 / x_m) + t_0)) + ((2.0 / x_m) + (1.0 / (1.0 - x_m))));
	else
		tmp = 2.0 / (x_m ^ 3.0);
	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[(-1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 15500.0], N[(t$95$0 + N[(N[(N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 15500

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. div-inv 87.1%

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

      2. *-un-lft-identity 87.1%

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

      3. prod-diff 87.1%

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

      4. *-commutative 87.1%

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

      5. *-un-lft-identity 87.1%

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

      6. fma-def 87.1%

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

      7. div-inv 87.1%

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

      8. associate-+l+ 87.1%

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

   5. Applied egg-rr 87.1%

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

   7. Simplified 87.1%

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

   if 15500 < x

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in x around inf 99.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15500:\\ \;\;\;\;\frac{-1}{1 - x} + \left(\left(\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \left(\frac{-2}{x} + \frac{-1}{1 - x}\right)\right) + \left(\frac{2}{x} + \frac{1}{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 4: 84.0% accurate, 0.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   (/ 1.0 (+ x_m 1.0))
   (/ (/ (- (- x_m (* x_m 2.0)) -2.0) x_m) (+ 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)) + ((((x_m - (x_m * 2.0)) - -2.0) / x_m) / (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)) + ((((x_m - (x_m * 2.0d0)) - (-2.0d0)) / x_m) / (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)) + ((((x_m - (x_m * 2.0)) - -2.0) / x_m) / (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)) + ((((x_m - (x_m * 2.0)) - -2.0) / x_m) / (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(1.0 / Float64(x_m + 1.0)) + Float64(Float64(Float64(Float64(x_m - Float64(x_m * 2.0)) - -2.0) / x_m) / 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)) + ((((x_m - (x_m * 2.0)) - -2.0) / x_m) / (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[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x$95$m - N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. frac-sub 59.6%

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

   2. associate-/r* 82.5%

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

   3. +-commutative 82.5%

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

   4. distribute-lft-in 82.5%

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

   5. metadata-eval 82.5%

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

   6. metadata-eval 82.5%

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

   7. *-rgt-identity 82.5%

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

   8. associate--l+ 82.5%

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

   9. metadata-eval 82.5%

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

5. Applied egg-rr 82.5%

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

6. Final simplification 82.5%

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

Alternative 5: 84.0% 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 1 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 82.4%

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

2. Final simplification 82.4%

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

Alternative 6: 84.0% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ 1.0 (+ x_m 1.0)) (/ (- -1.0 (/ -2.0 x_m)) (+ 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)) + ((-1.0 - (-2.0 / x_m)) / (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)) + (((-1.0d0) - ((-2.0d0) / x_m)) / (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)) + ((-1.0 - (-2.0 / x_m)) / (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)) + ((-1.0 - (-2.0 / x_m)) / (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(1.0 / Float64(x_m + 1.0)) + Float64(Float64(-1.0 - Float64(-2.0 / x_m)) / 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)) + ((-1.0 - (-2.0 / x_m)) / (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[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. frac-sub 59.6%

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

   2. associate-/r* 82.5%

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

   3. +-commutative 82.5%

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

   4. distribute-lft-in 82.5%

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

   5. metadata-eval 82.5%

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

   6. metadata-eval 82.5%

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

   7. *-rgt-identity 82.5%

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

   8. associate--l+ 82.5%

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

   9. metadata-eval 82.5%

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

5. Applied egg-rr 82.5%

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

7. Applied egg-rr 82.5%

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

8. Final simplification 82.5%

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

Alternative 7: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1:\\ \;\;\;\;x_m \cdot -2 - \frac{2}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \left(x_m + -2\right)}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.0) (- (* x_m -2.0) (/ 2.0 x_m)) (/ (- x_m (+ 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) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (x_m - (x_m + -2.0)) / 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) :: tmp
    if (x_m <= 1.0d0) then
        tmp = (x_m * (-2.0d0)) - (2.0d0 / x_m)
    else
        tmp = (x_m - (x_m + (-2.0d0))) / 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 tmp;
	if (x_m <= 1.0) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (x_m - (x_m + -2.0)) / 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):
	tmp = 0
	if x_m <= 1.0:
		tmp = (x_m * -2.0) - (2.0 / x_m)
	else:
		tmp = (x_m - (x_m + -2.0)) / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(x_m * -2.0) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(x_m - Float64(x_m + -2.0)) / 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)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = (x_m * -2.0) - (2.0 / x_m);
	else
		tmp = (x_m - (x_m + -2.0)) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], N[(N[(x$95$m * -2.0), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m - N[(x$95$m + -2.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. frac-sub 73.8%

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

      2. associate-/r* 87.1%

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

      3. +-commutative 87.1%

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

      4. distribute-lft-in 87.1%

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

      5. metadata-eval 87.1%

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

      6. metadata-eval 87.1%

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

      7. *-rgt-identity 87.1%

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

      8. associate--l+ 87.1%

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

      9. metadata-eval 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in x around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. metadata-eval 69.1%

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

   8. Simplified 69.1%

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

   if 1 < x

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

      1. frac-sub 20.5%

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

      2. associate-/r* 69.7%

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

      3. +-commutative 69.7%

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

      4. distribute-lft-in 69.7%

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

      5. metadata-eval 69.7%

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

      6. metadata-eval 69.7%

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

      7. *-rgt-identity 69.7%

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

      8. associate--l+ 69.7%

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

      9. metadata-eval 69.7%

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

   5. Applied egg-rr 69.7%

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

   7. Applied egg-rr 67.2%

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

4. Final simplification 68.6%

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

Alternative 8: 51.4% 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 1 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 82.4%

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

3. Simplified 82.4%

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

4. Taylor expanded in x around 0 52.7%

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

5. Final simplification 52.7%

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

Alternative 9: 2.9% accurate, 7.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (- x_m)))

C

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-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 * -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 * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

4. Taylor expanded in x around 0 51.3%

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

   1. mul-1-neg 51.3%

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

   2. sub-neg 51.3%

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

   3. metadata-eval 51.3%

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

   4. +-commutative 51.3%

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

   5. sub-neg 51.3%

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

6. Simplified 51.3%

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

7. Taylor expanded in x around inf 2.9%

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

   1. mul-1-neg 2.9%

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

9. Simplified 2.9%

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

10. Final simplification 2.9%

    \[\leadsto -x \]

Developer target: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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