Asymptote A

Average Error: 22.8% → 0.11%

Time: 6.7s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return (-2.0 / (x + 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

↓

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

↓

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

Python

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

↓

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

↓

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

↓

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

Derivation

1. Initial program 22.8

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

2. Applied egg-rr 21.83

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

3. Applied egg-rr 21.83

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

4. Simplified 0.11

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

   [Start] 21.83	\[ \left(x + \left(-2 - x\right)\right) \cdot \left(\frac{1}{x + -1} \cdot \frac{1}{x + 1}\right) \]
   associate-*l/ [=>] 21.83	\[ \left(x + \left(-2 - x\right)\right) \cdot \color{blue}{\frac{1 \cdot \frac{1}{x + 1}}{x + -1}} \]
   associate-*r/ [=>] 21.83	\[ \color{blue}{\frac{\left(x + \left(-2 - x\right)\right) \cdot \left(1 \cdot \frac{1}{x + 1}\right)}{x + -1}} \]
   *-lft-identity [=>] 21.83	\[ \frac{\left(x + \left(-2 - x\right)\right) \cdot \color{blue}{\frac{1}{x + 1}}}{x + -1} \]
   associate-*r/ [=>] 21.83	\[ \frac{\color{blue}{\frac{\left(x + \left(-2 - x\right)\right) \cdot 1}{x + 1}}}{x + -1} \]
   *-rgt-identity [=>] 21.83	\[ \frac{\frac{\color{blue}{x + \left(-2 - x\right)}}{x + 1}}{x + -1} \]
   +-commutative [=>] 21.83	\[ \frac{\frac{\color{blue}{\left(-2 - x\right) + x}}{x + 1}}{x + -1} \]
   associate--r- [<=] 0.11	\[ \frac{\frac{\color{blue}{-2 - \left(x - x\right)}}{x + 1}}{x + -1} \]
   sub-neg [=>] 0.11	\[ \frac{\frac{\color{blue}{-2 + \left(-\left(x - x\right)\right)}}{x + 1}}{x + -1} \]
   +-inverses [=>] 0.11	\[ \frac{\frac{-2 + \left(-\color{blue}{0}\right)}{x + 1}}{x + -1} \]
   metadata-eval [=>] 0.11	\[ \frac{\frac{-2 + \color{blue}{0}}{x + 1}}{x + -1} \]
   metadata-eval [=>] 0.11	\[ \frac{\frac{\color{blue}{-2}}{x + 1}}{x + -1} \]

5. Final simplification 0.11

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

Alternatives

Alternative 1
Error	1.14%
Cost	841

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

Alternative 2
Error	1.68%
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
Error	1.14%
Cost	585

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

Alternative 4
Error	49.66%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))