3frac (problem 3.3.3)

Percentage Accurate: 69.1% → 99.8%

Time: 9.8s

Alternatives: 11

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 135000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \mathsf{fma}\left(x\_m, x\_m, x\_m\right), \left(x\_m - 1\right) \cdot \left(-2 - x\_m\right)\right)}{\left(x\_m - 1\right) \cdot \mathsf{fma}\left(x\_m, x\_m, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \frac{2}{x\_m}}{x\_m}}{\left(x\_m - 1\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 (<= x_m 135000000.0)
    (/
     (fma 1.0 (fma x_m x_m x_m) (* (- x_m 1.0) (- -2.0 x_m)))
     (* (- x_m 1.0) (fma x_m x_m x_m)))
    (/ (/ (- 2.0 (/ 2.0 x_m)) x_m) (* (- x_m 1.0) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 135000000.0) {
		tmp = fma(1.0, fma(x_m, x_m, x_m), ((x_m - 1.0) * (-2.0 - x_m))) / ((x_m - 1.0) * fma(x_m, x_m, x_m));
	} else {
		tmp = ((2.0 - (2.0 / x_m)) / x_m) / ((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 <= 135000000.0)
		tmp = Float64(fma(1.0, fma(x_m, x_m, x_m), Float64(Float64(x_m - 1.0) * Float64(-2.0 - x_m))) / Float64(Float64(x_m - 1.0) * fma(x_m, x_m, x_m)));
	else
		tmp = Float64(Float64(Float64(2.0 - Float64(2.0 / x_m)) / x_m) / Float64(Float64(x_m - 1.0) * x_m));
	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, 135000000.0], N[(N[(1.0 * N[(x$95$m * x$95$m + x$95$m), $MachinePrecision] + N[(N[(x$95$m - 1.0), $MachinePrecision] * N[(-2.0 - x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m - 1.0), $MachinePrecision] * N[(x$95$m * x$95$m + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(N[(x$95$m - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e8

   1. Initial program 70.0%

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

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 70.1

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

   7. Applied rewrites 70.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. frac-add N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 17.9%

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

   if 1.35e8 < x

   1. Initial program 71.6%

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

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (fma (pow x_m -2.0) 2.0 (fma 2.0 (+ (pow x_m -4.0) (pow x_m -6.0)) 2.0))
   (pow x_m -3.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (fma(pow(x_m, -2.0), 2.0, fma(2.0, (pow(x_m, -4.0) + pow(x_m, -6.0)), 2.0)) * pow(x_m, -3.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(fma((x_m ^ -2.0), 2.0, fma(2.0, Float64((x_m ^ -4.0) + (x_m ^ -6.0)), 2.0)) * (x_m ^ -3.0)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[Power[x$95$m, -2.0], $MachinePrecision] * 2.0 + N[(2.0 * N[(N[Power[x$95$m, -4.0], $MachinePrecision] + N[Power[x$95$m, -6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (((2.0d0 / (x_m * x_m)) - (-2.0d0)) * (x_m ** (-3.0d0)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (((2.0 / (x_m * x_m)) - -2.0) * Math.pow(x_m, -3.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((2.0 / (x_m * x_m)) - -2.0) * math.pow(x_m, -3.0))

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (((2.0 / (x_m * x_m)) - -2.0) * (x_m ^ -3.0));
end

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.2%

    \[\leadsto \left(\frac{2}{x \cdot x} - -2\right) \cdot {\color{blue}{x}}^{-3} \]
11. Add Preprocessing

Alternative 6: 99.3% accurate, 0.9× speedup

Math

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

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

Derivation

1. Initial program 70.7%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 70.8

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

4. Applied rewrites 70.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. *-lft-identity N/A

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

   9. frac-add N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. *-lft-identity N/A

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

   13. associate-/r* N/A

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

6. Applied rewrites 5.2%

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

7. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 7: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 135000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \left(x\_m + 1\right) \cdot x\_m, \mathsf{fma}\left(-1 - x\_m, x\_m, 2\right)\right)}{\mathsf{fma}\left(x\_m, x\_m, -1\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \frac{2}{x\_m}}{x\_m}}{\left(x\_m - 1\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 (<= x_m 135000000.0)
    (/
     (fma 1.0 (* (+ x_m 1.0) x_m) (fma (- -1.0 x_m) x_m 2.0))
     (* (fma x_m x_m -1.0) x_m))
    (/ (/ (- 2.0 (/ 2.0 x_m)) x_m) (* (- x_m 1.0) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 135000000.0) {
		tmp = fma(1.0, ((x_m + 1.0) * x_m), fma((-1.0 - x_m), x_m, 2.0)) / (fma(x_m, x_m, -1.0) * x_m);
	} else {
		tmp = ((2.0 - (2.0 / x_m)) / x_m) / ((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 <= 135000000.0)
		tmp = Float64(fma(1.0, Float64(Float64(x_m + 1.0) * x_m), fma(Float64(-1.0 - x_m), x_m, 2.0)) / Float64(fma(x_m, x_m, -1.0) * x_m));
	else
		tmp = Float64(Float64(Float64(2.0 - Float64(2.0 / x_m)) / x_m) / Float64(Float64(x_m - 1.0) * x_m));
	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, 135000000.0], N[(N[(1.0 * N[(N[(x$95$m + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] + N[(N[(-1.0 - x$95$m), $MachinePrecision] * x$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * x$95$m + -1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(N[(x$95$m - 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e8

   1. Initial program 70.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   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. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-sub N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 17.9

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

   6. Applied rewrites 17.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 17.9

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

   9. Applied rewrites 17.9%

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

   if 1.35e8 < x

   1. Initial program 71.6%

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

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

Alternative 8: 98.8% accurate, 1.0× speedup

Math

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

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 (/ 2.0 x_m)) x_m) (* (- x_m 1.0) x_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 70.8

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

4. Applied rewrites 70.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

6. Applied rewrites 70.9%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 98.2

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

9. Applied rewrites 98.2%

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

Alternative 9: 98.8% accurate, 1.6× speedup

Math

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

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)) 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)) / 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)) / 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)) / 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)) / x_m)

Julia

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 70.8

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

4. Applied rewrites 70.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. *-lft-identity N/A

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

   9. frac-add N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. *-lft-identity N/A

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

   13. associate-/r* N/A

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

6. Applied rewrites 5.2%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.2

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

9. Applied rewrites 98.2%

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

Alternative 10: 53.5% 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 70.7%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 70.8

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

4. Applied rewrites 70.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

6. Applied rewrites 70.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 55.1%

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

10. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 55.1

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

12. Applied rewrites 55.1%

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

Alternative 11: 5.1% 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 70.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.1

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

5. Applied rewrites 5.1%

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

Developer Target 1: 99.1% 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 2024323 
(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))))