Asymptote A

Average Accuracy: 77.0% → 99.9%

Time: 6.0s

Precision: binary64

Cost: 576

• Unsound rule application detected in e-graph. Results may not be sound.

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 77.0%

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

2. Applied egg-rr 78.2%

   \[\leadsto \color{blue}{\frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{x + -1}} \]
   Step-by-step derivation

   [Start] 77.0	\[ \frac{1}{x + 1} - \frac{1}{x - 1} \]
   frac-sub [=>] 78.3	\[ \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   associate-/r* [=>] 78.3	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{x - 1}} \]
   *-un-lft-identity [<=] 78.3	\[ \frac{\frac{\color{blue}{\left(x - 1\right)} - \left(x + 1\right) \cdot 1}{x + 1}}{x - 1} \]
   *-rgt-identity [=>] 78.3	\[ \frac{\frac{\left(x - 1\right) - \color{blue}{\left(x + 1\right)}}{x + 1}}{x - 1} \]
   associate--l- [=>] 78.2	\[ \frac{\frac{\color{blue}{x - \left(1 + \left(x + 1\right)\right)}}{x + 1}}{x - 1} \]
   +-commutative [=>] 78.2	\[ \frac{\frac{x - \left(1 + \color{blue}{\left(1 + x\right)}\right)}{x + 1}}{x - 1} \]
   +-commutative [=>] 78.2	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{\color{blue}{1 + x}}}{x - 1} \]
   sub-neg [=>] 78.2	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{\color{blue}{x + \left(-1\right)}} \]
   metadata-eval [=>] 78.2	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{x + \color{blue}{-1}} \]

3. Applied egg-rr 78.3%

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

   [Start] 78.2	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{x + -1} \]
   frac-2neg [=>] 78.2	\[ \frac{\color{blue}{\frac{-\left(x - \left(1 + \left(1 + x\right)\right)\right)}{-\left(1 + x\right)}}}{x + -1} \]
   div-inv [=>] 78.2	\[ \frac{\color{blue}{\left(-\left(x - \left(1 + \left(1 + x\right)\right)\right)\right) \cdot \frac{1}{-\left(1 + x\right)}}}{x + -1} \]
   associate-+r+ [=>] 78.3	\[ \frac{\left(-\left(x - \color{blue}{\left(\left(1 + 1\right) + x\right)}\right)\right) \cdot \frac{1}{-\left(1 + x\right)}}{x + -1} \]
   +-commutative [=>] 78.3	\[ \frac{\left(-\left(x - \color{blue}{\left(x + \left(1 + 1\right)\right)}\right)\right) \cdot \frac{1}{-\left(1 + x\right)}}{x + -1} \]
   metadata-eval [=>] 78.3	\[ \frac{\left(-\left(x - \left(x + \color{blue}{2}\right)\right)\right) \cdot \frac{1}{-\left(1 + x\right)}}{x + -1} \]
   distribute-neg-in [=>] 78.3	\[ \frac{\left(-\left(x - \left(x + 2\right)\right)\right) \cdot \frac{1}{\color{blue}{\left(-1\right) + \left(-x\right)}}}{x + -1} \]
   metadata-eval [=>] 78.3	\[ \frac{\left(-\left(x - \left(x + 2\right)\right)\right) \cdot \frac{1}{\color{blue}{-1} + \left(-x\right)}}{x + -1} \]

4. Simplified 99.9%

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

   [Start] 78.3	\[ \frac{\left(-\left(x - \left(x + 2\right)\right)\right) \cdot \frac{1}{-1 + \left(-x\right)}}{x + -1} \]
   associate-*r/ [=>] 78.3	\[ \frac{\color{blue}{\frac{\left(-\left(x - \left(x + 2\right)\right)\right) \cdot 1}{-1 + \left(-x\right)}}}{x + -1} \]
   *-rgt-identity [=>] 78.3	\[ \frac{\frac{\color{blue}{-\left(x - \left(x + 2\right)\right)}}{-1 + \left(-x\right)}}{x + -1} \]
   neg-sub0 [=>] 78.3	\[ \frac{\frac{\color{blue}{0 - \left(x - \left(x + 2\right)\right)}}{-1 + \left(-x\right)}}{x + -1} \]
   associate-+l- [<=] 78.3	\[ \frac{\frac{\color{blue}{\left(0 - x\right) + \left(x + 2\right)}}{-1 + \left(-x\right)}}{x + -1} \]
   associate-+r+ [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\left(0 - x\right) + x\right) + 2}}{-1 + \left(-x\right)}}{x + -1} \]
   associate-+l- [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(0 - \left(x - x\right)\right)} + 2}{-1 + \left(-x\right)}}{x + -1} \]
   +-inverses [=>] 99.9	\[ \frac{\frac{\left(0 - \color{blue}{0}\right) + 2}{-1 + \left(-x\right)}}{x + -1} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{\color{blue}{0} + 2}{-1 + \left(-x\right)}}{x + -1} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{\color{blue}{2}}{-1 + \left(-x\right)}}{x + -1} \]
   unsub-neg [=>] 99.9	\[ \frac{\frac{2}{\color{blue}{-1 - x}}}{x + -1} \]

5. Applied egg-rr 99.8%

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

   [Start] 99.9	\[ \frac{\frac{2}{-1 - x}}{x + -1} \]
   div-inv [=>] 99.8	\[ \color{blue}{\frac{2}{-1 - x} \cdot \frac{1}{x + -1}} \]
   +-commutative [=>] 99.8	\[ \frac{2}{-1 - x} \cdot \frac{1}{\color{blue}{-1 + x}} \]

6. Applied egg-rr 99.9%

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

   [Start] 99.8	\[ \frac{2}{-1 - x} \cdot \frac{1}{-1 + x} \]
   associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{2 \cdot \frac{1}{-1 + x}}{-1 - x}} \]
   un-div-inv [=>] 99.9	\[ \frac{\color{blue}{\frac{2}{-1 + x}}}{-1 - x} \]

7. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	49.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(x - -1\right)\\ \end{array} \]

Alternative 2
Accuracy	49.2%
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 3
Accuracy	51.0%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))