Asymptote C

Percentage Accurate: 54.5% → 99.8%

Time: 5.5s

Alternatives: 6

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-3 - x}{x \cdot x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (- x -1.0)) (/ (- x -1.0) (- x 1.0))) 5e-8)
   (/ (- (/ (- -3.0 x) (* x x)) 3.0) x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x - -1.0)) - ((x - -1.0) / (x - 1.0))) <= 5e-8) {
		tmp = (((-3.0 - x) / (x * x)) - 3.0) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x - -1.0)) - Float64(Float64(x - -1.0) / Float64(x - 1.0))) <= 5e-8)
		tmp = Float64(Float64(Float64(Float64(-3.0 - x) / Float64(x * x)) - 3.0) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / fma(x, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(N[(-3.0 - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 4.9999999999999998e-8

   1. Initial program 7.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. div-add N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. unpow2 N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1 \cdot x - 3}{{x}^{2}} - 3}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\frac{-3 - x}{x \cdot x} - 3}{x} \]

   if 4.9999999999999998e-8 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 99.5

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

      18. lift-+.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. frac-add N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   8. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-3, x, -1\right)}}{\mathsf{fma}\left(x, x, -1\right)} \]

   9. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-3 - x}{x \cdot x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (- x -1.0)) (/ (- x -1.0) (- x 1.0))) 5e-8)
   (/ (- (/ -1.0 x) 3.0) x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x - -1.0)) - ((x - -1.0) / (x - 1.0))) <= 5e-8) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x - -1.0)) - Float64(Float64(x - -1.0) / Float64(x - 1.0))) <= 5e-8)
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / fma(x, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 4.9999999999999998e-8

   1. Initial program 7.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.9999999999999998e-8 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 99.5

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

      18. lift-+.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. frac-add N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   8. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-3, x, -1\right)}}{\mathsf{fma}\left(x, x, -1\right)} \]

   9. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 0:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (- x -1.0)) (/ (- x -1.0) (- x 1.0))) 0.0)
   (/ -3.0 x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x - -1.0)) - ((x - -1.0) / (x - 1.0))) <= 0.0) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x - -1.0)) - Float64(Float64(x - -1.0) / Float64(x - 1.0))) <= 0.0)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / fma(x, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-3.0 / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 6.2%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 98.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 98.1

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

      18. lift-+.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. frac-add N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   8. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-3, x, -1\right)}}{\mathsf{fma}\left(x, x, -1\right)} \]

   9. Applied rewrites 100.0%

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

4. Final simplification 99.7%

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

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (- x -1.0)) (/ (- x -1.0) (- x 1.0))) 5e-5)
   (/ -3.0 x)
   (* (fma 3.0 x 1.0) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x - -1.0)) - ((x - -1.0) / (x - 1.0))) <= 5e-5) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(3.0, x, 1.0) * fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x - -1.0)) - Float64(Float64(x - -1.0) / Float64(x - 1.0))) <= 5e-5)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(3.0, x, 1.0) * fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], N[(-3.0 / x), $MachinePrecision], N[(N[(3.0 * x + 1.0), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000024e-5

   1. Initial program 8.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \cdot \left(1 + 3 \cdot x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{\left(3 \cdot x + 1\right)} \]

      11. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{\mathsf{fma}\left(3, x, 1\right)} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (- x -1.0)) (/ (- x -1.0) (- x 1.0))) 5e-5)
   (/ -3.0 x)
   (fma (+ 3.0 x) x 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x - -1.0)) - ((x - -1.0) / (x - 1.0))) <= 5e-5) {
		tmp = -3.0 / x;
	} else {
		tmp = fma((3.0 + x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x - -1.0)) - Float64(Float64(x - -1.0) / Float64(x - 1.0))) <= 5e-5)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(Float64(3.0 + x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], N[(-3.0 / x), $MachinePrecision], N[(N[(3.0 + x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000024e-5

   1. Initial program 8.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(3 + x, x, 1\right)} \]

      4. lower-+.f64 99.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{3 + x}, x, 1\right) \]

   5. Applied rewrites 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x - -1} - \frac{x - -1}{x - 1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.5%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))