Asymptote C

Percentage Accurate: 54.3% → 99.1%

Time: 8.1s

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: 54.3% 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0002)
   (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
   (fma x (+ x 3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = fma(x, (x + 3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = fma(x, Float64(x + 3.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + 3.0), $MachinePrecision] + 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)))) < 2.0000000000000001e-4

   1. Initial program 9.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. associate-*l/ N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 99.4%

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

   if 2.0000000000000001e-4 < (-.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. lower-fma.f64 N/A

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

      3. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = (-3.0 - ((x + 3.0) / (x * x))) / x;
	} else {
		tmp = fma(x, -3.0, -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(-1.0 - x) / Float64(x + -1.0))) <= 0.0)
		tmp = Float64(Float64(-3.0 - Float64(Float64(x + 3.0) / Float64(x * x))) / x);
	else
		tmp = Float64(fma(x, -3.0, -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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-3.0 - N[(N[(x + 3.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * -3.0 + -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 7.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lft-mult-inverse N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*l* N/A

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

      11. lft-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 99.3

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

   8. Simplified 99.3%

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

   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 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. frac-add N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

   4. Applied egg-rr 98.8%

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

   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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = fma(x, -3.0, -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(-1.0 - x) / Float64(x + -1.0))) <= 0.0)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(fma(x, -3.0, -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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * -3.0 + -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 7.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   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 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. frac-add N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

   4. Applied egg-rr 98.8%

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

   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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 99.3%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, -3.0, -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(-1.0 - x) / Float64(x + -1.0))) <= 0.0)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(x, -3.0, -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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-3.0 / x), $MachinePrecision], N[(N[(x * -3.0 + -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 7.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Simplified 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 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. frac-add N/A

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

   4. Applied egg-rr 98.8%

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

   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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

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

Alternative 5: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0002)
   (/ -3.0 x)
   (fma x (+ x 3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, (x + 3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(x, Float64(x + 3.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(-3.0 / x), $MachinePrecision], N[(x * N[(x + 3.0), $MachinePrecision] + 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)))) < 2.0000000000000001e-4

   1. Initial program 9.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.3

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

   5. Simplified 97.3%

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

   if 2.0000000000000001e-4 < (-.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. lower-fma.f64 N/A

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

      3. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.9% 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 51.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 47.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))