3frac (problem 3.3.3)

Percentage Accurate: 84.3% → 100.0%

Time: 9.6s

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: 84.3% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 95000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x_m \cdot \left(\left(-x_m\right) - x_m\right), -0.5, \left(1 - x_m\right) \cdot \left(x_m + 1\right)\right)}{x_m + 1}}{\left(1 - x_m\right) \cdot \left(x_m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 95000000.0)
    (/
     (/
      (fma (* x_m (- (- x_m) x_m)) -0.5 (* (- 1.0 x_m) (+ x_m 1.0)))
      (+ x_m 1.0))
     (* (- 1.0 x_m) (* x_m -0.5)))
    (* 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 tmp;
	if (x_m <= 95000000.0) {
		tmp = (fma((x_m * (-x_m - x_m)), -0.5, ((1.0 - x_m) * (x_m + 1.0))) / (x_m + 1.0)) / ((1.0 - x_m) * (x_m * -0.5));
	} else {
		tmp = 2.0 * 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)
	tmp = 0.0
	if (x_m <= 95000000.0)
		tmp = Float64(Float64(fma(Float64(x_m * Float64(Float64(-x_m) - x_m)), -0.5, Float64(Float64(1.0 - x_m) * Float64(x_m + 1.0))) / Float64(x_m + 1.0)) / Float64(Float64(1.0 - x_m) * Float64(x_m * -0.5)));
	else
		tmp = Float64(2.0 * (x_m ^ -3.0));
	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, 95000000.0], N[(N[(N[(N[(x$95$m * N[((-x$95$m) - x$95$m), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(1.0 - x$95$m), $MachinePrecision] * N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x$95$m), $MachinePrecision] * N[(x$95$m * -0.5), $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 < 9.5e7

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

   3. Simplified 94.9%

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

      1. +-commutative 94.9%

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

      2. frac-add 74.9%

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

      3. clear-num 74.9%

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

      4. frac-add 74.4%

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

      5. fma-def 73.5%

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

      6. *-un-lft-identity 73.5%

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

      7. *-commutative 73.5%

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

      8. neg-mul-1 73.5%

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

      9. +-commutative 73.5%

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

      10. distribute-neg-in 73.5%

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

      11. neg-mul-1 73.5%

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

      12. metadata-eval 73.5%

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

      13. fma-def 73.5%

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

      14. div-inv 73.5%

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

      15. metadata-eval 73.5%

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

      16. div-inv 73.5%

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

      17. metadata-eval 73.5%

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

   5. Applied egg-rr 73.5%

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

      1. associate-*l* 73.5%

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

   7. Simplified 73.5%

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

      1. *-un-lft-identity 73.5%

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

      2. +-commutative 73.5%

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

      3. times-frac 68.8%

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

      4. +-commutative 68.8%

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

      5. *-commutative 68.8%

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

   9. Applied egg-rr 68.8%

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

       1. associate-*l/ 68.8%

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

       2. *-lft-identity 68.8%

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

       3. rem-square-sqrt 68.6%

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

       4. associate-*r/ 68.6%

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

       5. associate-*l/ 68.6%

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

       6. *-commutative 68.6%

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

       7. associate-*l/ 68.6%

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

   11. Simplified 74.4%

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

   if 9.5e7 < x

   1. Initial program 74.0%

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

      1. sub-neg 74.0%

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

      2. distribute-neg-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. associate-/r* 74.0%

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

      7. metadata-eval 74.0%

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

      8. neg-mul-1 74.0%

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

      9. +-commutative 74.0%

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

      10. associate-+l+ 74.0%

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

      11. +-commutative 74.0%

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

      12. neg-mul-1 74.0%

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

      13. metadata-eval 74.0%

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

      14. associate-/r* 74.0%

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

      15. metadata-eval 74.0%

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

      16. metadata-eval 74.0%

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

      17. +-commutative 74.0%

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

      18. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 74.0%

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

      3. div-inv 74.0%

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

      4. pow-flip 74.0%

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

      5. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 95000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot \left(\left(-x\right) - x\right), -0.5, \left(1 - x\right) \cdot \left(x + 1\right)\right)}{x + 1}}{\left(1 - x\right) \cdot \left(x \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(1 - x_m\right) \cdot \left(-1 - x_m\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 40000000:\\ \;\;\;\;\frac{-2 \cdot t_0 + x_m \cdot \left(x_m + x_m\right)}{x_m \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x_m}^{-3}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x_m) (- -1.0 x_m))))
   (*
    x_s
    (if (<= x_m 40000000.0)
      (/ (+ (* -2.0 t_0) (* x_m (+ x_m x_m))) (* x_m t_0))
      (* 2.0 (pow x_m -3.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (1.0 - x_m) * (-1.0 - x_m);
	double tmp;
	if (x_m <= 40000000.0) {
		tmp = ((-2.0 * t_0) + (x_m * (x_m + x_m))) / (x_m * t_0);
	} 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 - x_m) * ((-1.0d0) - x_m)
    if (x_m <= 40000000.0d0) then
        tmp = (((-2.0d0) * t_0) + (x_m * (x_m + x_m))) / (x_m * t_0)
    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 - x_m) * (-1.0 - x_m);
	double tmp;
	if (x_m <= 40000000.0) {
		tmp = ((-2.0 * t_0) + (x_m * (x_m + x_m))) / (x_m * t_0);
	} 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 - x_m) * (-1.0 - x_m)
	tmp = 0
	if x_m <= 40000000.0:
		tmp = ((-2.0 * t_0) + (x_m * (x_m + x_m))) / (x_m * t_0)
	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(Float64(1.0 - x_m) * Float64(-1.0 - x_m))
	tmp = 0.0
	if (x_m <= 40000000.0)
		tmp = Float64(Float64(Float64(-2.0 * t_0) + Float64(x_m * Float64(x_m + x_m))) / Float64(x_m * t_0));
	else
		tmp = Float64(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 - x_m) * (-1.0 - x_m);
	tmp = 0.0;
	if (x_m <= 40000000.0)
		tmp = ((-2.0 * t_0) + (x_m * (x_m + x_m))) / (x_m * t_0);
	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[(N[(1.0 - x$95$m), $MachinePrecision] * N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 40000000.0], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x$95$m * N[(x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e7

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

   3. Simplified 94.9%

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

      1. frac-add 74.9%

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

      2. div-inv 74.9%

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

      3. *-un-lft-identity 74.9%

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

      4. *-commutative 74.9%

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

      5. neg-mul-1 74.9%

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

      6. +-commutative 74.9%

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

      7. distribute-neg-in 74.9%

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

      8. neg-mul-1 74.9%

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

      9. metadata-eval 74.9%

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

      10. fma-def 74.9%

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

   5. Applied egg-rr 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l/ 74.9%

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

      3. *-lft-identity 74.9%

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

      4. fma-udef 74.9%

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

      5. neg-mul-1 74.9%

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

      6. metadata-eval 74.9%

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

      7. distribute-neg-in 74.9%

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

      8. +-commutative 74.9%

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

      9. distribute-neg-in 74.9%

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

      10. metadata-eval 74.9%

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

      11. unsub-neg 74.9%

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

      12. +-commutative 74.9%

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

   7. Simplified 74.9%

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

      1. +-commutative 74.9%

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

      2. frac-2neg 74.9%

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

      3. metadata-eval 74.9%

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

      4. frac-add 74.4%

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

      5. +-commutative 74.4%

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

      6. *-commutative 74.4%

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

      7. +-commutative 74.4%

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

      8. *-commutative 74.4%

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

   9. Applied egg-rr 74.4%

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

   11. Simplified 74.4%

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

   if 4e7 < x

   1. Initial program 74.0%

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

      1. sub-neg 74.0%

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

      2. distribute-neg-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. associate-/r* 74.0%

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

      7. metadata-eval 74.0%

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

      8. neg-mul-1 74.0%

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

      9. +-commutative 74.0%

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

      10. associate-+l+ 74.0%

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

      11. +-commutative 74.0%

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

      12. neg-mul-1 74.0%

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

      13. metadata-eval 74.0%

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

      14. associate-/r* 74.0%

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

      15. metadata-eval 74.0%

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

      16. metadata-eval 74.0%

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

      17. +-commutative 74.0%

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

      18. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 74.0%

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

      3. div-inv 74.0%

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

      4. pow-flip 74.0%

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

      5. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40000000:\\ \;\;\;\;\frac{-2 \cdot \left(\left(1 - x\right) \cdot \left(-1 - x\right)\right) + x \cdot \left(x + x\right)}{x \cdot \left(\left(1 - x\right) \cdot \left(-1 - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.65:\\ \;\;\;\;x_m \cdot -2 - \frac{2}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x_m} + \left(\frac{-1}{1 - x_m} + \frac{1}{x_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.65)
    (- (* x_m -2.0) (/ 2.0 x_m))
    (+ (/ -2.0 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 tmp;
	if (x_m <= 0.65) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-2.0 / 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) :: tmp
    if (x_m <= 0.65d0) then
        tmp = (x_m * (-2.0d0)) - (2.0d0 / x_m)
    else
        tmp = ((-2.0d0) / 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 tmp;
	if (x_m <= 0.65) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-2.0 / 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):
	tmp = 0
	if x_m <= 0.65:
		tmp = (x_m * -2.0) - (2.0 / x_m)
	else:
		tmp = (-2.0 / 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)
	tmp = 0.0
	if (x_m <= 0.65)
		tmp = Float64(Float64(x_m * -2.0) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(-2.0 / x_m) + Float64(Float64(-1.0 / Float64(1.0 - x_m)) + Float64(1.0 / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.65)
		tmp = (x_m * -2.0) - (2.0 / x_m);
	else
		tmp = (-2.0 / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.65], N[(N[(x$95$m * -2.0), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] + N[(N[(-1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 67.5%

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

      1. associate-*r/ 67.5%

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

      2. metadata-eval 67.5%

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

   6. Simplified 67.5%

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

   if 0.650000000000000022 < x

   1. Initial program 74.0%

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

      1. sub-neg 74.0%

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

      2. distribute-neg-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. associate-/r* 74.0%

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

      7. metadata-eval 74.0%

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

      8. neg-mul-1 74.0%

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

      9. +-commutative 74.0%

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

      10. associate-+l+ 74.0%

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

      11. +-commutative 74.0%

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

      12. neg-mul-1 74.0%

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

      13. metadata-eval 74.0%

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

      14. associate-/r* 74.0%

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

      15. metadata-eval 74.0%

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

      16. metadata-eval 74.0%

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

      17. +-commutative 74.0%

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

      18. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around inf 74.0%

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

4. Final simplification 68.8%

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

Alternative 4: 84.3% 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 90.8%

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

2. Final simplification 90.8%

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

Alternative 5: 84.3% accurate, 1.0× speedup

Math

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

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) (+ (/ -1.0 (- 1.0 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 * ((-2.0 / x_m) + ((-1.0 / (1.0 - 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 * (((-2.0d0) / x_m) + (((-1.0d0) / (1.0d0 - 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 * ((-2.0 / x_m) + ((-1.0 / (1.0 - 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 * ((-2.0 / x_m) + ((-1.0 / (1.0 - 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(-2.0 / x_m) + Float64(Float64(-1.0 / Float64(1.0 - 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 * ((-2.0 / x_m) + ((-1.0 / (1.0 - 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[(-2.0 / x$95$m), $MachinePrecision] + N[(N[(-1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

   1. sub-neg 90.8%

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

   2. distribute-neg-frac 90.8%

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

   3. metadata-eval 90.8%

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

   4. metadata-eval 90.8%

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

   5. metadata-eval 90.8%

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

   6. associate-/r* 90.8%

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

   7. metadata-eval 90.8%

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

   8. neg-mul-1 90.8%

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

   9. +-commutative 90.8%

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

   10. associate-+l+ 90.8%

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

   11. +-commutative 90.8%

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

   12. neg-mul-1 90.8%

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

   13. metadata-eval 90.8%

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

   14. associate-/r* 90.8%

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

   15. metadata-eval 90.8%

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

   16. metadata-eval 90.8%

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

   17. +-commutative 90.8%

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

   18. +-commutative 90.8%

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

3. Simplified 90.8%

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

4. Final simplification 90.8%

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

Alternative 6: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{-2}{x_m} + \left(1 + \frac{-1}{1 - x_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x_m} + \frac{2}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.6e+51)
    (+ (/ -2.0 x_m) (+ 1.0 (/ -1.0 (- 1.0 x_m))))
    (+ (/ -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) {
	double tmp;
	if (x_m <= 2.6e+51) {
		tmp = (-2.0 / x_m) + (1.0 + (-1.0 / (1.0 - x_m)));
	} else {
		tmp = (-2.0 / 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 <= 2.6d+51) then
        tmp = ((-2.0d0) / x_m) + (1.0d0 + ((-1.0d0) / (1.0d0 - x_m)))
    else
        tmp = ((-2.0d0) / 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 <= 2.6e+51) {
		tmp = (-2.0 / x_m) + (1.0 + (-1.0 / (1.0 - x_m)));
	} else {
		tmp = (-2.0 / 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 <= 2.6e+51:
		tmp = (-2.0 / x_m) + (1.0 + (-1.0 / (1.0 - x_m)))
	else:
		tmp = (-2.0 / 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 <= 2.6e+51)
		tmp = Float64(Float64(-2.0 / x_m) + Float64(1.0 + Float64(-1.0 / Float64(1.0 - x_m))));
	else
		tmp = Float64(Float64(-2.0 / x_m) + Float64(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 <= 2.6e+51)
		tmp = (-2.0 / x_m) + (1.0 + (-1.0 / (1.0 - x_m)));
	else
		tmp = (-2.0 / 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, 2.6e+51], N[(N[(-2.0 / x$95$m), $MachinePrecision] + N[(1.0 + N[(-1.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] + N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e51

   1. Initial program 92.8%

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

      1. sub-neg 92.8%

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

      2. distribute-neg-frac 92.8%

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

      3. metadata-eval 92.8%

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

      4. metadata-eval 92.8%

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

      5. metadata-eval 92.8%

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

      6. associate-/r* 92.8%

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

      7. metadata-eval 92.8%

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

      8. neg-mul-1 92.8%

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

      9. +-commutative 92.8%

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

      10. associate-+l+ 92.8%

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

      11. +-commutative 92.8%

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

      12. neg-mul-1 92.8%

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

      13. metadata-eval 92.8%

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

      14. associate-/r* 92.8%

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

      15. metadata-eval 92.8%

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

      16. metadata-eval 92.8%

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

      17. +-commutative 92.8%

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

      18. +-commutative 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 66.3%

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

   if 2.6000000000000001e51 < x

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. distribute-neg-frac 81.6%

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

      3. metadata-eval 81.6%

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

      4. metadata-eval 81.6%

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

      5. metadata-eval 81.6%

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

      6. associate-/r* 81.6%

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

      7. metadata-eval 81.6%

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

      8. neg-mul-1 81.6%

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

      9. +-commutative 81.6%

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

      10. associate-+l+ 81.6%

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

      11. +-commutative 81.6%

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

      12. neg-mul-1 81.6%

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

      13. metadata-eval 81.6%

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

      14. associate-/r* 81.6%

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

      15. metadata-eval 81.6%

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

      16. metadata-eval 81.6%

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

      17. +-commutative 81.6%

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

      18. +-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in x around inf 81.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{-2}{x} + \left(1 + \frac{-1}{1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + \frac{2}{x}\\ \end{array} \]

Alternative 7: 83.0% accurate, 1.7× speedup

Math

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

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 x_m)) (+ (/ -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) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = -x_m - (2.0 / x_m);
	} else {
		tmp = (-2.0 / 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 / x_m)
    else
        tmp = ((-2.0d0) / 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 / x_m);
	} else {
		tmp = (-2.0 / 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 / x_m)
	else:
		tmp = (-2.0 / 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) - Float64(2.0 / x_m));
	else
		tmp = Float64(Float64(-2.0 / x_m) + Float64(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 / x_m);
	else
		tmp = (-2.0 / 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[((-x$95$m) - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x$95$m), $MachinePrecision] + N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 67.7%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. neg-mul-1 67.4%

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

      2. associate-*r/ 67.4%

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

      3. metadata-eval 67.4%

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

   7. Simplified 67.4%

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

   if 1 < x

   1. Initial program 74.0%

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

      1. sub-neg 74.0%

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

      2. distribute-neg-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. associate-/r* 74.0%

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

      7. metadata-eval 74.0%

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

      8. neg-mul-1 74.0%

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

      9. +-commutative 74.0%

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

      10. associate-+l+ 74.0%

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

      11. +-commutative 74.0%

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

      12. neg-mul-1 74.0%

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

      13. metadata-eval 74.0%

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

      14. associate-/r* 74.0%

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

      15. metadata-eval 74.0%

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

      16. metadata-eval 74.0%

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

      17. +-commutative 74.0%

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

      18. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around inf 74.0%

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

4. Final simplification 68.7%

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

Alternative 8: 83.2% accurate, 1.7× speedup

Math

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

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)) (+ (/ -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) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = (x_m * -2.0) - (2.0 / x_m);
	} else {
		tmp = (-2.0 / 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 = ((-2.0d0) / 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 = (-2.0 / 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 = (-2.0 / 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(-2.0 / x_m) + Float64(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 = (-2.0 / 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[(-2.0 / x$95$m), $MachinePrecision] + N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 67.5%

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

      1. associate-*r/ 67.5%

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

      2. metadata-eval 67.5%

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

   6. Simplified 67.5%

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

   if 1 < x

   1. Initial program 74.0%

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

      1. sub-neg 74.0%

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

      2. distribute-neg-frac 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. metadata-eval 74.0%

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

      6. associate-/r* 74.0%

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

      7. metadata-eval 74.0%

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

      8. neg-mul-1 74.0%

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

      9. +-commutative 74.0%

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

      10. associate-+l+ 74.0%

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

      11. +-commutative 74.0%

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

      12. neg-mul-1 74.0%

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

      13. metadata-eval 74.0%

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

      14. associate-/r* 74.0%

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

      15. metadata-eval 74.0%

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

      16. metadata-eval 74.0%

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

      17. +-commutative 74.0%

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

      18. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in x around inf 74.0%

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

4. Final simplification 68.8%

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

Alternative 9: 51.8% 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 90.8%

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

   1. sub-neg 90.8%

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

   2. distribute-neg-frac 90.8%

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

   3. metadata-eval 90.8%

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

   4. metadata-eval 90.8%

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

   5. metadata-eval 90.8%

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

   6. associate-/r* 90.8%

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

   7. metadata-eval 90.8%

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

   8. neg-mul-1 90.8%

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

   9. +-commutative 90.8%

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

   10. associate-+l+ 90.8%

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

   11. +-commutative 90.8%

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

   12. neg-mul-1 90.8%

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

   13. metadata-eval 90.8%

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

   14. associate-/r* 90.8%

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

   15. metadata-eval 90.8%

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

   16. metadata-eval 90.8%

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

   17. +-commutative 90.8%

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

   18. +-commutative 90.8%

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

3. Simplified 90.8%

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

4. Taylor expanded in x around 0 56.1%

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

5. Final simplification 56.1%

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

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 2023322 
(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))))