Asymptote A

Average Accuracy: 77.4% → 99.9%

Time: 6.7s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{\frac{2}{1 - x}}{1 + x} \]

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (1.0d0 - x)) / (1.0d0 + x)
end function

Java

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

↓

public static double code(double x) {
	return (2.0 / (1.0 - x)) / (1.0 + x);
}

Python

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

↓

def code(x):
	return (2.0 / (1.0 - x)) / (1.0 + x)

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = (2.0 / (1.0 - x)) / (1.0 + x);
end

Wolfram

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

↓

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

Derivation

1. Initial program 77.4%

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

2. Applied egg-rr 78.3%

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

   [Start] 77.4	\[ \frac{1}{x + 1} - \frac{1}{x - 1} \]
   *-un-lft-identity [=>] 77.4	\[ \color{blue}{1 \cdot \left(\frac{1}{x + 1} - \frac{1}{x - 1}\right)} \]
   *-commutative [=>] 77.4	\[ \color{blue}{\left(\frac{1}{x + 1} - \frac{1}{x - 1}\right) \cdot 1} \]
   frac-sub [=>] 78.3	\[ \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}} \cdot 1 \]
   associate-/r* [=>] 78.3	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{x - 1}} \cdot 1 \]
   associate-/r/ [<=] 78.3	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{\frac{x - 1}{1}}} \]
   frac-2neg [=>] 78.3	\[ \frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{\color{blue}{\frac{-\left(x - 1\right)}{-1}}} \]
   associate-/r/ [=>] 78.3	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{-\left(x - 1\right)} \cdot \left(-1\right)} \]

3. Simplified 78.3%

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

   [Start] 78.3	\[ \frac{\frac{x + \left(-2 - x\right)}{1 + x}}{1 - x} \cdot -1 \]
   *-commutative [=>] 78.3	\[ \color{blue}{-1 \cdot \frac{\frac{x + \left(-2 - x\right)}{1 + x}}{1 - x}} \]
   associate-/r* [<=] 78.3	\[ -1 \cdot \color{blue}{\frac{x + \left(-2 - x\right)}{\left(1 + x\right) \cdot \left(1 - x\right)}} \]
   associate-*r/ [=>] 78.3	\[ \color{blue}{\frac{-1 \cdot \left(x + \left(-2 - x\right)\right)}{\left(1 + x\right) \cdot \left(1 - x\right)}} \]
   neg-mul-1 [<=] 78.3	\[ \frac{\color{blue}{-\left(x + \left(-2 - x\right)\right)}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   sub0-neg [<=] 78.3	\[ \frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   +-commutative [=>] 78.3	\[ \frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   associate--r+ [=>] 78.3	\[ \frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   neg-sub0 [<=] 78.3	\[ \frac{\color{blue}{\left(-\left(-2 - x\right)\right)} - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   sub-neg [=>] 78.3	\[ \frac{\left(-\color{blue}{\left(-2 + \left(-x\right)\right)}\right) - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   mul-1-neg [<=] 78.3	\[ \frac{\left(-\left(-2 + \color{blue}{-1 \cdot x}\right)\right) - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   distribute-neg-in [=>] 78.3	\[ \frac{\color{blue}{\left(\left(--2\right) + \left(--1 \cdot x\right)\right)} - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   metadata-eval [=>] 78.3	\[ \frac{\left(\color{blue}{2} + \left(--1 \cdot x\right)\right) - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   mul-1-neg [=>] 78.3	\[ \frac{\left(2 + \left(-\color{blue}{\left(-x\right)}\right)\right) - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   remove-double-neg [=>] 78.3	\[ \frac{\left(2 + \color{blue}{x}\right) - x}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   +-commutative [=>] 78.3	\[ \frac{\left(2 + x\right) - x}{\color{blue}{\left(x + 1\right)} \cdot \left(1 - x\right)} \]

4. Applied egg-rr 91.3%

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

   [Start] 78.3	\[ \frac{\left(2 + x\right) - x}{\left(x + 1\right) \cdot \left(1 - x\right)} \]
   associate-/r* [=>] 78.3	\[ \color{blue}{\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 - x}} \]
   flip-- [=>] 78.3	\[ \frac{\frac{\left(2 + x\right) - x}{x + 1}}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}}} \]
   +-commutative [<=] 78.3	\[ \frac{\frac{\left(2 + x\right) - x}{x + 1}}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{x + 1}}} \]
   associate-/r/ [=>] 78.3	\[ \color{blue}{\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 \cdot 1 - x \cdot x} \cdot \left(x + 1\right)} \]

5. Applied egg-rr 99.9%

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

   [Start] 91.3	\[ \frac{2}{\left(1 - x \cdot x\right) \cdot \left(x + 1\right)} \cdot \left(x + 1\right) \]
   associate-*l/ [=>] 91.3	\[ \color{blue}{\frac{2 \cdot \left(x + 1\right)}{\left(1 - x \cdot x\right) \cdot \left(x + 1\right)}} \]
   associate-/r* [=>] 99.2	\[ \color{blue}{\frac{\frac{2 \cdot \left(x + 1\right)}{1 - x \cdot x}}{x + 1}} \]
   associate-/l* [=>] 99.3	\[ \frac{\color{blue}{\frac{2}{\frac{1 - x \cdot x}{x + 1}}}}{x + 1} \]
   metadata-eval [<=] 99.3	\[ \frac{\frac{2}{\frac{\color{blue}{1 \cdot 1} - x \cdot x}{x + 1}}}{x + 1} \]
   +-commutative [=>] 99.3	\[ \frac{\frac{2}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + x}}}}{x + 1} \]
   flip-- [<=] 99.9	\[ \frac{\frac{2}{\color{blue}{1 - x}}}{x + 1} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{2}{1 - x}}{\color{blue}{1 + x}} \]

6. Final simplification 99.9%

   \[\leadsto \frac{\frac{2}{1 - x}}{1 + x} \]

Alternatives

Alternative 1
Accuracy	98.2%
Cost	585

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

Alternative 2
Accuracy	98.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;2 + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	448

\[\frac{2}{1 - x \cdot x} \]

Alternative 4
Accuracy	50.4%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))