3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.7%

Time: 9.5s

Alternatives: 10

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left({\left(x\_m + 1\right)}^{-1} - \frac{2}{x\_m}\right) + {\left(x\_m - 1\right)}^{-1} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{2 - \frac{2}{x\_m}}{x\_m}}{x\_m \cdot \left(x\_m - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m, x\_m\right) + \left(-2 - x\_m\right) \cdot \left(x\_m - 1\right)}{\left(\left(x\_m + 1\right) \cdot x\_m\right) \cdot \left(x\_m - 1\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<=
       (+ (- (pow (+ x_m 1.0) -1.0) (/ 2.0 x_m)) (pow (- x_m 1.0) -1.0))
       2e-25)
    (/ (/ (- 2.0 (/ 2.0 x_m)) x_m) (* x_m (- x_m 1.0)))
    (/
     (+ (fma x_m x_m x_m) (* (- -2.0 x_m) (- x_m 1.0)))
     (* (* (+ x_m 1.0) 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) {
	double tmp;
	if (((pow((x_m + 1.0), -1.0) - (2.0 / x_m)) + pow((x_m - 1.0), -1.0)) <= 2e-25) {
		tmp = ((2.0 - (2.0 / x_m)) / x_m) / (x_m * (x_m - 1.0));
	} else {
		tmp = (fma(x_m, x_m, x_m) + ((-2.0 - x_m) * (x_m - 1.0))) / (((x_m + 1.0) * x_m) * (x_m - 1.0));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(Float64((Float64(x_m + 1.0) ^ -1.0) - Float64(2.0 / x_m)) + (Float64(x_m - 1.0) ^ -1.0)) <= 2e-25)
		tmp = Float64(Float64(Float64(2.0 - Float64(2.0 / x_m)) / x_m) / Float64(x_m * Float64(x_m - 1.0)));
	else
		tmp = Float64(Float64(fma(x_m, x_m, x_m) + Float64(Float64(-2.0 - x_m) * Float64(x_m - 1.0))) / Float64(Float64(Float64(x_m + 1.0) * x_m) * Float64(x_m - 1.0)));
	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[N[(N[(N[Power[N[(x$95$m + 1.0), $MachinePrecision], -1.0], $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x$95$m - 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], 2e-25], N[(N[(N[(2.0 - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(x$95$m * N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * x$95$m + x$95$m), $MachinePrecision] + N[(N[(-2.0 - x$95$m), $MachinePrecision] * N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 2.00000000000000008e-25

   1. Initial program 68.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 2.00000000000000008e-25 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 54.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. lift-/.f64 N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 57.5

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 57.5

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

   7. Applied rewrites 57.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 97.2

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

   9. Applied rewrites 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(x + 1\right)}^{-1} - \frac{2}{x}\right) + {\left(x - 1\right)}^{-1} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{2 - \frac{2}{x}}{x}}{x \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, x\right) + \left(-2 - x\right) \cdot \left(x - 1\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{\frac{2}{x\_m \cdot x\_m} - -2}{{x\_m}^{5}} - {x\_m}^{-3} \cdot -2\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (- (/ (- (/ 2.0 (* x_m x_m)) -2.0) (pow x_m 5.0)) (* (pow x_m -3.0) -2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((((2.0 / (x_m * x_m)) - -2.0) / pow(x_m, 5.0)) - (pow(x_m, -3.0) * -2.0));
}

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 * x_m)) - (-2.0d0)) / (x_m ** 5.0d0)) - ((x_m ** (-3.0d0)) * (-2.0d0)))
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 * x_m)) - -2.0) / Math.pow(x_m, 5.0)) - (Math.pow(x_m, -3.0) * -2.0));
}

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 * x_m)) - -2.0) / math.pow(x_m, 5.0)) - (math.pow(x_m, -3.0) * -2.0))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(Float64(2.0 / Float64(x_m * x_m)) - -2.0) / (x_m ^ 5.0)) - Float64((x_m ^ -3.0) * -2.0)))
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 * x_m)) - -2.0) / (x_m ^ 5.0)) - ((x_m ^ -3.0) * -2.0));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x$95$m, -3.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - {x}^{-3} \cdot \color{blue}{-2} \]
8. Add Preprocessing

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot {\left(0.5 \cdot \left({x\_m}^{3} - x\_m\right)\right)}^{-1} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (pow (* 0.5 (- (pow x_m 3.0) 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 * pow((0.5 * (pow(x_m, 3.0) - 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 * ((0.5d0 * ((x_m ** 3.0d0) - 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 * Math.pow((0.5 * (Math.pow(x_m, 3.0) - 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 * math.pow((0.5 * (math.pow(x_m, 3.0) - 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(0.5 * Float64((x_m ^ 3.0) - 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 * ((0.5 * ((x_m ^ 3.0) - 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[Power[N[(0.5 * N[(N[Power[x$95$m, 3.0], $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 97.9

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.2%

    \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \left({x}^{3} - x\right)}} \]

13. Final simplification 99.2%

    \[\leadsto {\left(0.5 \cdot \left({x}^{3} - x\right)\right)}^{-1} \]
14. Add Preprocessing

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{\frac{2}{x\_m \cdot x\_m} - -2}{{x\_m}^{5}} - \frac{\frac{-2}{x\_m \cdot x\_m}}{x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (-
   (/ (- (/ 2.0 (* x_m x_m)) -2.0) (pow x_m 5.0))
   (/ (/ -2.0 (* x_m x_m)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((((2.0 / (x_m * x_m)) - -2.0) / pow(x_m, 5.0)) - ((-2.0 / (x_m * x_m)) / x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((((2.0d0 / (x_m * x_m)) - (-2.0d0)) / (x_m ** 5.0d0)) - (((-2.0d0) / (x_m * x_m)) / x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((((2.0 / (x_m * x_m)) - -2.0) / Math.pow(x_m, 5.0)) - ((-2.0 / (x_m * x_m)) / x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((((2.0 / (x_m * x_m)) - -2.0) / math.pow(x_m, 5.0)) - ((-2.0 / (x_m * x_m)) / x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(Float64(2.0 / Float64(x_m * x_m)) - -2.0) / (x_m ^ 5.0)) - Float64(Float64(-2.0 / Float64(x_m * x_m)) / x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((((2.0 / (x_m * x_m)) - -2.0) / (x_m ^ 5.0)) - ((-2.0 / (x_m * x_m)) / x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{-2}{x \cdot x}}{\color{blue}{x}} \]
8. Add Preprocessing

Alternative 5: 67.6% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ (/ -1.0 x_m) (pow (- x_m 1.0) -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) + pow((x_m - 1.0), -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) + ((x_m - 1.0d0) ** (-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) + Math.pow((x_m - 1.0), -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) + math.pow((x_m - 1.0), -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 / x_m) + (Float64(x_m - 1.0) ^ -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) + ((x_m - 1.0) ^ -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 / x$95$m), $MachinePrecision] + N[Power[N[(x$95$m - 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Final simplification 66.3%

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

Alternative 6: 99.1% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/ (/ (fma (/ 2.0 (* x_m x_m)) (- 1.0 x_m) 2.0) x_m) (* 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 * ((fma((2.0 / (x_m * x_m)), (1.0 - x_m), 2.0) / x_m) / (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(fma(Float64(2.0 / Float64(x_m * x_m)), Float64(1.0 - x_m), 2.0) / x_m) / Float64(x_m * Float64(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[(N[(2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(x$95$m * N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

   \[\leadsto \frac{\frac{\frac{1 - x}{\left(x \cdot x\right) \cdot 0.5} - -2}{x}}{x \cdot \left(x - 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 98.6%

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

Alternative 7: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 98.1

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

7. Applied rewrites 98.1%

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

Alternative 8: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.9

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

7. Applied rewrites 96.9%

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

Alternative 9: 53.6% accurate, 2.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 2.0 (* x_m 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 * 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 * 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 * 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 * 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 / Float64(x_m * x_m)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 68.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 53.0

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

7. Applied rewrites 53.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 54.5%

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

11. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 54.5

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

13. Applied rewrites 54.5%

    \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]
14. Add Preprocessing

Alternative 10: 5.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.2

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

5. Applied rewrites 5.2%

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

Developer Target 1: 99.2% accurate, 1.8× 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 2024296 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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