Asymptote B

Average Error: 0.0 → 0.0

Time: 5.2s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 15.2

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

4. Simplified 0.0

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

   [Start] 15.2	\[ \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right) \cdot x + \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right) \cdot \left(-1 + \frac{x + 1}{x}\right) \]
   distribute-lft-out [=>] 15.2	\[ \color{blue}{\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right) \cdot \left(x + \left(-1 + \frac{x + 1}{x}\right)\right)} \]
   associate-+l+ [<=] 15.2	\[ \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot x\right) \cdot \color{blue}{\left(\left(x + -1\right) + \frac{x + 1}{x}\right)} \]
   associate-*l/ [=>] 15.2	\[ \color{blue}{\frac{1 \cdot x}{\mathsf{fma}\left(x, x, -1\right)}} \cdot \left(\left(x + -1\right) + \frac{x + 1}{x}\right) \]
   associate-/l* [=>] 15.2	\[ \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \cdot \left(\left(x + -1\right) + \frac{x + 1}{x}\right) \]
   associate-/r/ [<=] 15.1	\[ \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}{\left(x + -1\right) + \frac{x + 1}{x}}}} \]
   associate-/l* [<=] 15.1	\[ \color{blue}{\frac{1 \cdot \left(\left(x + -1\right) + \frac{x + 1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}}} \]
   *-lft-identity [=>] 15.1	\[ \frac{\color{blue}{\left(x + -1\right) + \frac{x + 1}{x}}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   metadata-eval [<=] 15.1	\[ \frac{\left(x + \color{blue}{\left(-1\right)}\right) + \frac{x + 1}{x}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   sub-neg [<=] 15.1	\[ \frac{\color{blue}{\left(x - 1\right)} + \frac{x + 1}{x}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   associate--r- [<=] 15.1	\[ \frac{\color{blue}{x - \left(1 - \frac{x + 1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   *-rgt-identity [<=] 15.1	\[ \frac{x - \left(1 - \color{blue}{\frac{x + 1}{x} \cdot 1}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   associate-*l/ [=>] 15.1	\[ \frac{x - \left(1 - \color{blue}{\frac{\left(x + 1\right) \cdot 1}{x}}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   associate-*r/ [<=] 15.1	\[ \frac{x - \left(1 - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x}}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   distribute-lft1-in [<=] 15.1	\[ \frac{x - \left(1 - \color{blue}{\left(x \cdot \frac{1}{x} + \frac{1}{x}\right)}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   rgt-mult-inverse [=>] 15.1	\[ \frac{x - \left(1 - \left(\color{blue}{1} + \frac{1}{x}\right)\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   associate--r+ [=>] 15.1	\[ \frac{x - \color{blue}{\left(\left(1 - 1\right) - \frac{1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   metadata-eval [=>] 15.1	\[ \frac{x - \left(\color{blue}{0} - \frac{1}{x}\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   neg-sub0 [<=] 15.1	\[ \frac{x - \color{blue}{\left(-\frac{1}{x}\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   distribute-neg-frac [=>] 15.1	\[ \frac{x - \color{blue}{\frac{-1}{x}}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]
   metadata-eval [=>] 15.1	\[ \frac{x - \frac{\color{blue}{-1}}{x}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{x}} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.48 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{x}{x + 1} - x\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	713

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

Alternative 3
Error	0.0
Cost	704

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

Alternative 4
Error	0.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1 - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	0.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	31.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))