Asymptote C

Percentage Accurate: 53.5% → 99.8%

Time: 5.9s

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: 53.5% 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, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.0) {
		tmp = (((1.0 / x) - -3.0) / x) * ((-1.0 / (x * x)) - 1.0);
	} 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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / x) - -3.0) / x) * Float64(Float64(-1.0 / Float64(x * x)) - 1.0));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 / x), $MachinePrecision] - -3.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $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 7.7%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Applied rewrites 99.1%

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

   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 99.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. 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

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

   7. 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 99.5%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.0) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / 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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.0)
		tmp = Float64(Float64(-3.0 - Float64(Float64(Float64(3.0 / x) - -1.0) / x)) / 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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-3.0 - N[(N[(N[(3.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $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)))) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

   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 99.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. 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

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

   7. 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 99.4%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 0.0)
   (/ (- (/ -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 / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.0) {
		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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.0)
		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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], 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)))) < 0.0

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. neg-mul-1 N/A

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

      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 98.6

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

   5. Applied rewrites 98.6%

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

   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 99.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. 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

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

   7. 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 99.3%

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

Alternative 4: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- 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 / (1.0 + x)) - ((1.0 + x) / (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(1.0 + x)) - Float64(Float64(1.0 + x) / 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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $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 7.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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 99.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. metadata-eval N/A

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

      21. sub-neg N/A

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

      22. lower-fma.f64 N/A

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

      23. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. 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

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

   7. 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 99.1%

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

Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 2e-8)
   (/ -3.0 x)
   (fma (+ 3.0 x) x 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 2e-8) {
		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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 2e-8)
		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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], 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)))) < 2e-8

   1. Initial program 8.1%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 2e-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 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. +-commutative N/A

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

      5. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.9% 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 52.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.5%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))