Asymptote C

Percentage Accurate: 55.0% → 99.7%

Time: 6.0s

Alternatives: 5

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: 55.0% 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: 99.7% 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) / Float64(-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 7.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. lft-mult-inverse N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lift-fma.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. lift-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-out N/A

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

      13. neg-mul-1 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 N/A

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

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

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

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.4% accurate, 0.2× speedup

Math

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

FPCore

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

   1. Initial program 7.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.4%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 1e-9)
   (/ (- (/ -1.0 x) 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))) <= 1e-9) {
		tmp = ((-1.0 / x) - 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))) <= 1e-9)
		tmp = Float64(Float64(Float64(-1.0 / x) - 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], 1e-9], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / 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)))) < 1.00000000000000006e-9

   1. Initial program 7.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 4: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

   1. Initial program 7.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 5: 51.6% 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 50.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 47.0%

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

Reproduce

herbie shell --seed 2024311 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))