Asymptote A

Average Accuracy: 76.8% → 99.9%

Time: 8.2s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (* 2.0 (/ 1.0 (- 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 / (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 / (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 / (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 / (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 / 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 / (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 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

2. Applied egg-rr 77.8%

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

   [Start] 76.8	\[ \frac{1}{x + 1} - \frac{1}{x - 1} \]
   frac-sub [=>] 77.8	\[ \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   associate-/r* [=>] 77.8	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x + 1}}{x - 1}} \]
   *-un-lft-identity [<=] 77.8	\[ \frac{\frac{\color{blue}{\left(x - 1\right)} - \left(x + 1\right) \cdot 1}{x + 1}}{x - 1} \]
   *-rgt-identity [=>] 77.8	\[ \frac{\frac{\left(x - 1\right) - \color{blue}{\left(x + 1\right)}}{x + 1}}{x - 1} \]
   associate--l- [=>] 77.8	\[ \frac{\frac{\color{blue}{x - \left(1 + \left(x + 1\right)\right)}}{x + 1}}{x - 1} \]
   +-commutative [=>] 77.8	\[ \frac{\frac{x - \left(1 + \color{blue}{\left(1 + x\right)}\right)}{x + 1}}{x - 1} \]
   +-commutative [=>] 77.8	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{\color{blue}{1 + x}}}{x - 1} \]
   sub-neg [=>] 77.8	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{\color{blue}{x + \left(-1\right)}} \]
   metadata-eval [=>] 77.8	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{x + \color{blue}{-1}} \]

3. Applied egg-rr 77.8%

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

   [Start] 77.8	\[ \frac{\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{x + -1} \]
   frac-2neg [=>] 77.8	\[ \color{blue}{\frac{-\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}}{-\left(x + -1\right)}} \]
   div-inv [=>] 77.8	\[ \color{blue}{\left(-\frac{x - \left(1 + \left(1 + x\right)\right)}{1 + x}\right) \cdot \frac{1}{-\left(x + -1\right)}} \]
   distribute-neg-frac [=>] 77.8	\[ \color{blue}{\frac{-\left(x - \left(1 + \left(1 + x\right)\right)\right)}{1 + x}} \cdot \frac{1}{-\left(x + -1\right)} \]
   associate-+r+ [=>] 77.8	\[ \frac{-\left(x - \color{blue}{\left(\left(1 + 1\right) + x\right)}\right)}{1 + x} \cdot \frac{1}{-\left(x + -1\right)} \]
   +-commutative [=>] 77.8	\[ \frac{-\left(x - \color{blue}{\left(x + \left(1 + 1\right)\right)}\right)}{1 + x} \cdot \frac{1}{-\left(x + -1\right)} \]
   metadata-eval [=>] 77.8	\[ \frac{-\left(x - \left(x + \color{blue}{2}\right)\right)}{1 + x} \cdot \frac{1}{-\left(x + -1\right)} \]
   +-commutative [=>] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{\color{blue}{x + 1}} \cdot \frac{1}{-\left(x + -1\right)} \]
   +-commutative [=>] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{x + 1} \cdot \frac{1}{-\color{blue}{\left(-1 + x\right)}} \]
   distribute-neg-in [=>] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{x + 1} \cdot \frac{1}{\color{blue}{\left(--1\right) + \left(-x\right)}} \]
   metadata-eval [=>] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{x + 1} \cdot \frac{1}{\color{blue}{1} + \left(-x\right)} \]
   sub-neg [<=] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{x + 1} \cdot \frac{1}{\color{blue}{1 - x}} \]

4. Simplified 77.8%

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

   [Start] 77.8	\[ \frac{-\left(x - \left(x + 2\right)\right)}{x + 1} \cdot \frac{1}{1 - x} \]
   associate-*l/ [=>] 77.8	\[ \color{blue}{\frac{\left(-\left(x - \left(x + 2\right)\right)\right) \cdot \frac{1}{1 - x}}{x + 1}} \]
   +-commutative [<=] 77.8	\[ \frac{\left(-\left(x - \color{blue}{\left(2 + x\right)}\right)\right) \cdot \frac{1}{1 - x}}{x + 1} \]

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	585

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

Alternative 2
Accuracy	98.7%
Cost	584

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

Alternative 3
Accuracy	99.4%
Cost	576

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

Alternative 4
Accuracy	99.9%
Cost	576

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

Alternative 5
Accuracy	99.4%
Cost	448

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

Alternative 6
Accuracy	50.9%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))