Asymptote B

Percentage Accurate: 100.0% → 99.6%

Time: 6.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x -1.0)) (/ x (+ 1.0 x))) -1.0)
   (/ (fma x x 1.0) (fma x x -1.0))
   (+ 1.0 (/ 2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / (x + -1.0)) + (x / (1.0 + x))) <= -1.0) {
		tmp = fma(x, x, 1.0) / fma(x, x, -1.0);
	} else {
		tmp = 1.0 + (2.0 / (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(x / Float64(1.0 + x))) <= -1.0)
		tmp = Float64(fma(x, x, 1.0) / fma(x, x, -1.0));
	else
		tmp = Float64(1.0 + Float64(2.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(x * x + 1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-add N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1 + \frac{2}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x -1.0)) (/ x (+ 1.0 x))) -1.0)
   (fma x (* -2.0 (fma x (* x x) x)) -1.0)
   (+ 1.0 (/ 2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / (x + -1.0)) + (x / (1.0 + x))) <= -1.0) {
		tmp = fma(x, (-2.0 * fma(x, (x * x), x)), -1.0);
	} else {
		tmp = 1.0 + (2.0 / (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(x / Float64(1.0 + x))) <= -1.0)
		tmp = fma(x, Float64(-2.0 * fma(x, Float64(x * x), x)), -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(x * N[(-2.0 * N[(x * N[(x * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(-2 \cdot {x}^{2} - 2\right) \cdot x, -1\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-2 \cdot {x}^{2} - 2\right)}, -1\right) \]

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft1-in N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-lft-in N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, -2 \cdot \color{blue}{\mathsf{fma}\left(x, {x}^{2}, x\right)}, -1\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, -2 \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot x}, x\right), -1\right) \]

      20. lower-*.f64 99.6

          \[\leadsto \mathsf{fma}\left(x, -2 \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot x}, x\right), -1\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2 \cdot \mathsf{fma}\left(x, x \cdot x, x\right), -1\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{1 + \frac{2}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))