Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x -1.0)) (/ x (+ 1.0 x))) -1.0)
   (/ (- -1.0 (* x x)) (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 = (-1.0 - (x * x)) / 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(Float64(-1.0 - Float64(x * x)) / fma(x, Float64(-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[(-1.0 - N[(x * x), $MachinePrecision]), $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. frac-2neg N/A

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

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

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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 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 99.6%

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

Reproduce

herbie shell --seed 2024225 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))