Asymptote C

Percentage Accurate: 54.2% → 99.8%

Time: 7.7s

Alternatives: 11

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: 54.2% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - 1\right) \cdot \frac{{x}^{-1} - -3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x + \mathsf{fma}\left(x, x, -1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (* (- (/ (/ -1.0 x) x) 1.0) (/ (- (pow x -1.0) -3.0) x))
   (/ (fma -3.0 x -1.0) (+ x (fma x x (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 1e-9) {
		tmp = (((-1.0 / x) / x) - 1.0) * ((pow(x, -1.0) - -3.0) / x);
	} else {
		tmp = fma(-3.0, x, -1.0) / (x + fma(x, x, (-1.0 - x)));
	}
	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))) <= 1e-9)
		tmp = Float64(Float64(Float64(Float64(-1.0 / x) / x) - 1.0) * Float64(Float64((x ^ -1.0) - -3.0) / x));
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(x + fma(x, x, Float64(-1.0 - x))));
	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], 1e-9], N[(N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Power[x, -1.0], $MachinePrecision] - -3.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x + N[(x * x + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $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)))) < 1.00000000000000006e-9

   1. Initial program 8.5%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-out N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if 1.00000000000000006e-9 < (-.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 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.8%

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

   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. 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%

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

   9. Applied rewrites 100.0%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+l+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - 1\right) \cdot \frac{{x}^{-1} - -3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x + \mathsf{fma}\left(x, x, -1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (pow (fma -0.3333333333333333 x 0.1111111111111111) -1.0)
   (/ (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))) <= 1e-9) {
		tmp = pow(fma(-0.3333333333333333, x, 0.1111111111111111), -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(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 1e-9)
		tmp = fma(-0.3333333333333333, x, 0.1111111111111111) ^ -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[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-9], N[Power[N[(-0.3333333333333333 * x + 0.1111111111111111), $MachinePrecision], -1.0], $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)))) < 1.00000000000000006e-9

   1. Initial program 8.5%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 5.7%

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

   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. metadata-eval N/A

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

      3. lower-fma.f64 57.2

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

   7. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lift-fma.f64 57.3

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

   9. Applied rewrites 57.3%

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

   10. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{3}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{-1}{3} + \frac{1}{9} \cdot \frac{1}{x}\right)}} \]

       4. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{-1}{3} \cdot x + \left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot x}} \]

       5. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9} \cdot \left(\frac{1}{x} \cdot x\right)}} \]

       6. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \frac{1}{9} \cdot \color{blue}{1}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9}}} \]

       8. lower-fma.f64 98.0

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   12. Applied rewrites 98.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   if 1.00000000000000006e-9 < (-.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 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.8%

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

   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. 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 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), 3 \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.002)
   (pow (fma -0.3333333333333333 x 0.1111111111111111) -1.0)
   (fma (fma x x 1.0) (* 3.0 x) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.002) {
		tmp = pow(fma(-0.3333333333333333, x, 0.1111111111111111), -1.0);
	} else {
		tmp = fma(fma(x, x, 1.0), (3.0 * x), 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))) <= 0.002)
		tmp = fma(-0.3333333333333333, x, 0.1111111111111111) ^ -1.0;
	else
		tmp = fma(fma(x, x, 1.0), Float64(3.0 * x), 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], 0.002], N[Power[N[(-0.3333333333333333 * x + 0.1111111111111111), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * x), $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)))) < 2e-3

   1. Initial program 9.0%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 6.3%

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

   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. metadata-eval N/A

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

      3. lower-fma.f64 57.5

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

   7. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lift-fma.f64 57.6

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

   9. Applied rewrites 57.6%

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

   10. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{3}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{-1}{3} + \frac{1}{9} \cdot \frac{1}{x}\right)}} \]

       4. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{-1}{3} \cdot x + \left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot x}} \]

       5. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9} \cdot \left(\frac{1}{x} \cdot x\right)}} \]

       6. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \frac{1}{9} \cdot \color{blue}{1}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9}}} \]

       8. lower-fma.f64 97.7

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   12. Applied rewrites 97.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   if 2e-3 < (-.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 99.9%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.9%

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

   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. 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%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x \cdot \left(1 + 3 \cdot x\right)\right)} \]
   9. 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. lft-mult-inverse N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. lft-mult-inverse N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt1-in N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

   10. Applied rewrites 99.9%

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

   12. Applied rewrites 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), 3 \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.002)
   (pow (fma -0.3333333333333333 x 0.1111111111111111) -1.0)
   (* (fma x x 1.0) (fma 3.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.002) {
		tmp = pow(fma(-0.3333333333333333, x, 0.1111111111111111), -1.0);
	} else {
		tmp = fma(x, x, 1.0) * fma(3.0, 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))) <= 0.002)
		tmp = fma(-0.3333333333333333, x, 0.1111111111111111) ^ -1.0;
	else
		tmp = Float64(fma(x, x, 1.0) * fma(3.0, 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], 0.002], N[Power[N[(-0.3333333333333333 * x + 0.1111111111111111), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * 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)))) < 2e-3

   1. Initial program 9.0%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 6.3%

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

   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. metadata-eval N/A

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

      3. lower-fma.f64 57.5

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

   7. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lift-fma.f64 57.6

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

   9. Applied rewrites 57.6%

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

   10. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{3}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{-1}{3} + \frac{1}{9} \cdot \frac{1}{x}\right)}} \]

       4. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{-1}{3} \cdot x + \left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot x}} \]

       5. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9} \cdot \left(\frac{1}{x} \cdot x\right)}} \]

       6. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \frac{1}{9} \cdot \color{blue}{1}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{-1}{3} \cdot x + \color{blue}{\frac{1}{9}}} \]

       8. lower-fma.f64 97.7

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   12. Applied rewrites 97.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)}} \]

   if 2e-3 < (-.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 99.9%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.1111111111111111\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{-3 - \frac{\frac{3}{x} - -1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x + \mathsf{fma}\left(x, x, -1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (/ (- -3.0 (/ (- (/ 3.0 x) -1.0) x)) x)
   (/ (fma -3.0 x -1.0) (+ x (fma x x (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 1e-9) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / (x + fma(x, x, (-1.0 - x)));
	}
	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))) <= 1e-9)
		tmp = Float64(Float64(-3.0 - Float64(Float64(Float64(3.0 / x) - -1.0) / x)) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(x + fma(x, x, Float64(-1.0 - x))));
	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], 1e-9], 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 + N[(x * x + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $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)))) < 1.00000000000000006e-9

   1. Initial program 8.5%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 1.00000000000000006e-9 < (-.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 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.8%

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

   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. 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%

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

   9. Applied rewrites 100.0%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+l+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 6: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x + \mathsf{fma}\left(x, x, -1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (/ (- (/ -1.0 x) 3.0) x)
   (/ (fma -3.0 x -1.0) (+ x (fma x x (- -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 1e-9) {
		tmp = ((-1.0 / x) - 3.0) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / (x + fma(x, x, (-1.0 - x)));
	}
	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))) <= 1e-9)
		tmp = Float64(Float64(Float64(-1.0 / x) - 3.0) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(x + fma(x, x, Float64(-1.0 - x))));
	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], 1e-9], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(x + N[(x * x + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $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)))) < 1.00000000000000006e-9

   1. Initial program 8.5%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.00000000000000006e-9 < (-.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 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.8%

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

   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. 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%

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

   9. Applied rewrites 100.0%

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

       1. lift-+.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+l+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (/ (- (/ -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))) <= 1e-9) {
		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))) <= 1e-9)
		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], 1e-9], 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)))) < 1.00000000000000006e-9

   1. Initial program 8.5%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.00000000000000006e-9 < (-.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 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. frac-add N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. 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 rewrites 99.8%

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

   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. 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. Add Preprocessing

Alternative 8: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.002)
   (/ -3.0 x)
   (* (fma x x 1.0) (fma 3.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.002) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, x, 1.0) * fma(3.0, 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))) <= 0.002)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(x, x, 1.0) * fma(3.0, 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], 0.002], N[(-3.0 / x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * 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)))) < 2e-3

   1. Initial program 9.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Applied rewrites 97.3%

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

   if 2e-3 < (-.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 99.9%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 9: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.002:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.002)
   (/ -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))) <= 0.002) {
		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))) <= 0.002)
		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], 0.002], 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-3

   1. Initial program 9.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Applied rewrites 97.3%

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

   if 2e-3 < (-.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 99.9%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 10: 50.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3 + x, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (+ 3.0 x) x 1.0))

C

double code(double x) {
	return fma((3.0 + x), x, 1.0);
}

Julia

function code(x)
	return fma(Float64(3.0 + x), x, 1.0)
end

Wolfram

code[x_] := N[(N[(3.0 + x), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 48.8%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
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 44.9

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

5. Applied rewrites 44.9%

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

Alternative 11: 50.7% 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 48.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.4%

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

Reproduce

herbie shell --seed 2024323 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))