3frac (problem 3.3.3)

Percentage Accurate: 69.7% → 99.2%

Time: 8.9s

Alternatives: 9

Speedup: 2.1×

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: 69.7% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{1}{x\_m - 1} + \left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) \leq 0:\\ \;\;\;\;{x\_m}^{-3} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1 - x\_m, x\_m, 2\right) + \mathsf{fma}\left(x\_m, x\_m, x\_m\right)}{\left(\left(x\_m + 1\right) \cdot x\_m\right) \cdot \left(x\_m - 1\right)}\\ \end{array} \end{array} \]

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 (<= (+ (/ 1.0 (- x_m 1.0)) (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m))) 0.0)
    (* (pow x_m -3.0) 2.0)
    (/
     (+ (fma (- -1.0 x_m) x_m 2.0) (fma x_m x_m x_m))
     (* (* (+ 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 (((1.0 / (x_m - 1.0)) + ((1.0 / (x_m + 1.0)) - (2.0 / x_m))) <= 0.0) {
		tmp = pow(x_m, -3.0) * 2.0;
	} else {
		tmp = (fma((-1.0 - x_m), x_m, 2.0) + fma(x_m, x_m, x_m)) / (((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(1.0 / Float64(x_m - 1.0)) + Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m))) <= 0.0)
		tmp = Float64((x_m ^ -3.0) * 2.0);
	else
		tmp = Float64(Float64(fma(Float64(-1.0 - x_m), x_m, 2.0) + fma(x_m, x_m, x_m)) / 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[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x$95$m, -3.0], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(-1.0 - x$95$m), $MachinePrecision] * x$95$m + 2.0), $MachinePrecision] + N[(x$95$m * x$95$m + x$95$m), $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)))) < 0.0

   1. Initial program 74.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 99.1%

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

   if 0.0 < (+.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 48.7%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   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. 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 52.0

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

   4. Applied rewrites 52.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 67.3

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. lower-fma.f64 67.3

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

   6. Applied rewrites 67.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 67.3

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

   9. Applied rewrites 67.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \leq 0:\\ \;\;\;\;{x}^{-3} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1 - x, x, 2\right) + \mathsf{fma}\left(x, x, x\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

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 (/ 1.0 (pow x_m 4.0)))
   (* (pow (- x_m) -3.0) (+ (/ (/ 2.0 x_m) x_m) 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-1.0 - (1.0 / pow(x_m, 4.0))) * (pow(-x_m, -3.0) * (((2.0 / x_m) / x_m) + 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 * (((-1.0d0) - (1.0d0 / (x_m ** 4.0d0))) * ((-x_m ** (-3.0d0)) * (((2.0d0 / x_m) / x_m) + 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 * ((-1.0 - (1.0 / Math.pow(x_m, 4.0))) * (Math.pow(-x_m, -3.0) * (((2.0 / x_m) / x_m) + 2.0)));
}

Python

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in x around -inf

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \left(-1 - \frac{1}{{x}^{4}}\right) \cdot \left({\left(-x\right)}^{-3} \cdot \left(\frac{\frac{2}{x}}{x} + 2\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

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

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 -2.0 (pow x_m -2.0) -2.0) (- -1.0 (pow x_m -4.0))) (* 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(-2.0, pow(x_m, -2.0), -2.0) * (-1.0 - pow(x_m, -4.0))) / (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(fma(-2.0, (x_m ^ -2.0), -2.0) * Float64(-1.0 - (x_m ^ -4.0))) / 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[(-2.0 * N[Power[x$95$m, -2.0], $MachinePrecision] + -2.0), $MachinePrecision] * N[(-1.0 - N[Power[x$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in x around -inf

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-2, {x}^{-2}, -2\right) \cdot \left(-1 - {x}^{-4}\right)}{x \cdot x}}{x} \]
9. Add Preprocessing

Alternative 4: 98.7% accurate, 0.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (- (/ 2.0 (* 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 73.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 98.5

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

5. Applied rewrites 98.5%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{1}{x\_m - 1} + \left(\frac{1}{x\_m + 1} - \frac{2}{x\_m}\right) \leq 0:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1 - x\_m, x\_m, 2\right) + \mathsf{fma}\left(x\_m, x\_m, x\_m\right)}{\left(\left(x\_m + 1\right) \cdot x\_m\right) \cdot \left(x\_m - 1\right)}\\ \end{array} \end{array} \]

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 (<= (+ (/ 1.0 (- x_m 1.0)) (- (/ 1.0 (+ x_m 1.0)) (/ 2.0 x_m))) 0.0)
    (/ (/ 2.0 x_m) (* x_m x_m))
    (/
     (+ (fma (- -1.0 x_m) x_m 2.0) (fma x_m x_m x_m))
     (* (* (+ 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 (((1.0 / (x_m - 1.0)) + ((1.0 / (x_m + 1.0)) - (2.0 / x_m))) <= 0.0) {
		tmp = (2.0 / x_m) / (x_m * x_m);
	} else {
		tmp = (fma((-1.0 - x_m), x_m, 2.0) + fma(x_m, x_m, x_m)) / (((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(1.0 / Float64(x_m - 1.0)) + Float64(Float64(1.0 / Float64(x_m + 1.0)) - Float64(2.0 / x_m))) <= 0.0)
		tmp = Float64(Float64(2.0 / x_m) / Float64(x_m * x_m));
	else
		tmp = Float64(Float64(fma(Float64(-1.0 - x_m), x_m, 2.0) + fma(x_m, x_m, x_m)) / 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[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / x$95$m), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 - x$95$m), $MachinePrecision] * x$95$m + 2.0), $MachinePrecision] + N[(x$95$m * x$95$m + x$95$m), $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)))) < 0.0

   1. Initial program 74.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 99.0%

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

   if 0.0 < (+.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 48.7%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   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. 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 52.0

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

   4. Applied rewrites 52.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 67.3

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. lower-fma.f64 67.3

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

   6. Applied rewrites 67.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 67.3

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

   9. Applied rewrites 67.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \leq 0:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1 - x, x, 2\right) + \mathsf{fma}\left(x, x, x\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.6× speedup

Math

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

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 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 / x_m) / 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 * x_m));
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.6

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 98.2%

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

Alternative 7: 98.1% accurate, 2.1× speedup

Math

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

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

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.6

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.6%

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

Alternative 8: 54.6% accurate, 2.3× speedup

Math

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

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

Derivation

1. Initial program 73.8%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 73.6%

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

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 58.0

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

7. Applied rewrites 58.0%

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

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 59.7%

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

11. Final simplification 59.7%

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

Alternative 9: 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 73.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.4

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

5. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024294 
(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))))