Asymptote A

Percentage Accurate: 77.1% → 99.4%

Time: 7.7s

Alternatives: 5

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 (fma x x -1.0)))

C

double code(double x) {
	return -2.0 / fma(x, x, -1.0);
}

Julia

function code(x)
	return Float64(-2.0 / fma(x, x, -1.0))
end

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. *-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 77.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{-1 + x} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x \cdot x, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x 1.0)) (/ -1.0 (+ -1.0 x))) 0.0)
   (/ -2.0 (* x x))
   (fma 2.0 (* x x) 2.0)))

C

double code(double x) {
	double tmp;
	if (((1.0 / (x + 1.0)) + (-1.0 / (-1.0 + x))) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = fma(2.0, (x * x), 2.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 54.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{-1 + x} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x \cdot x, 2\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

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