Asymptote A

Average Accuracy: 77.0% → 99.9%

Time: 5.7s

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 77.0%

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

2. Applied egg-rr 77.9%

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

   [Start] 77.0	\[ \frac{1}{x + 1} - \frac{1}{x - 1} \]
   frac-sub [=>] 77.9	\[ \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   *-commutative [=>] 77.9	\[ \frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}} \]
   associate-/r* [=>] 77.9	\[ \color{blue}{\frac{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x - 1}}{x + 1}} \]
   *-un-lft-identity [<=] 77.9	\[ \frac{\frac{\color{blue}{\left(x - 1\right)} - \left(x + 1\right) \cdot 1}{x - 1}}{x + 1} \]
   sub-neg [=>] 77.9	\[ \frac{\frac{\color{blue}{\left(x + \left(-1\right)\right)} - \left(x + 1\right) \cdot 1}{x - 1}}{x + 1} \]
   *-rgt-identity [=>] 77.9	\[ \frac{\frac{\left(x + \left(-1\right)\right) - \color{blue}{\left(x + 1\right)}}{x - 1}}{x + 1} \]
   associate--l+ [=>] 77.9	\[ \frac{\frac{\color{blue}{x + \left(\left(-1\right) - \left(x + 1\right)\right)}}{x - 1}}{x + 1} \]
   +-commutative [=>] 77.9	\[ \frac{\frac{x + \left(\left(-1\right) - \color{blue}{\left(1 + x\right)}\right)}{x - 1}}{x + 1} \]
   associate--r+ [=>] 77.9	\[ \frac{\frac{x + \color{blue}{\left(\left(\left(-1\right) - 1\right) - x\right)}}{x - 1}}{x + 1} \]
   metadata-eval [=>] 77.9	\[ \frac{\frac{x + \left(\left(\color{blue}{-1} - 1\right) - x\right)}{x - 1}}{x + 1} \]
   metadata-eval [=>] 77.9	\[ \frac{\frac{x + \left(\color{blue}{-2} - x\right)}{x - 1}}{x + 1} \]
   sub-neg [=>] 77.9	\[ \frac{\frac{x + \left(-2 - x\right)}{\color{blue}{x + \left(-1\right)}}}{x + 1} \]
   metadata-eval [=>] 77.9	\[ \frac{\frac{x + \left(-2 - x\right)}{x + \color{blue}{-1}}}{x + 1} \]
   +-commutative [=>] 77.9	\[ \frac{\frac{x + \left(-2 - x\right)}{x + -1}}{\color{blue}{1 + x}} \]

3. Applied egg-rr 99.9%

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

   [Start] 77.9	\[ \frac{\frac{x + \left(-2 - x\right)}{x + -1}}{1 + x} \]
   frac-2neg [=>] 77.9	\[ \frac{\color{blue}{\frac{-\left(x + \left(-2 - x\right)\right)}{-\left(x + -1\right)}}}{1 + x} \]
   div-inv [=>] 77.9	\[ \frac{\color{blue}{\left(-\left(x + \left(-2 - x\right)\right)\right) \cdot \frac{1}{-\left(x + -1\right)}}}{1 + x} \]
   neg-sub0 [=>] 77.9	\[ \frac{\color{blue}{\left(0 - \left(x + \left(-2 - x\right)\right)\right)} \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   metadata-eval [<=] 77.9	\[ \frac{\left(\color{blue}{\log 1} - \left(x + \left(-2 - x\right)\right)\right) \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   +-commutative [=>] 77.9	\[ \frac{\left(\log 1 - \color{blue}{\left(\left(-2 - x\right) + x\right)}\right) \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   associate-+l- [=>] 99.9	\[ \frac{\left(\log 1 - \color{blue}{\left(-2 - \left(x - x\right)\right)}\right) \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   associate--r- [=>] 99.9	\[ \frac{\color{blue}{\left(\left(\log 1 - -2\right) + \left(x - x\right)\right)} \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   metadata-eval [=>] 99.9	\[ \frac{\left(\left(\color{blue}{0} - -2\right) + \left(x - x\right)\right) \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   metadata-eval [=>] 99.9	\[ \frac{\left(\color{blue}{2} + \left(x - x\right)\right) \cdot \frac{1}{-\left(x + -1\right)}}{1 + x} \]
   +-commutative [=>] 99.9	\[ \frac{\left(2 + \left(x - x\right)\right) \cdot \frac{1}{-\color{blue}{\left(-1 + x\right)}}}{1 + x} \]
   distribute-neg-in [=>] 99.9	\[ \frac{\left(2 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\left(--1\right) + \left(-x\right)}}}{1 + x} \]
   metadata-eval [=>] 99.9	\[ \frac{\left(2 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{1} + \left(-x\right)}}{1 + x} \]
   sub-neg [<=] 99.9	\[ \frac{\left(2 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{1 - x}}}{1 + x} \]

4. Simplified 99.9%

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

   [Start] 99.9	\[ \frac{\left(2 + \left(x - x\right)\right) \cdot \frac{1}{1 - x}}{1 + x} \]
   associate-*r/ [=>] 99.9	\[ \frac{\color{blue}{\frac{\left(2 + \left(x - x\right)\right) \cdot 1}{1 - x}}}{1 + x} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\left(x - x\right) + 2\right)} \cdot 1}{1 - x}}{1 + x} \]
   +-inverses [=>] 99.9	\[ \frac{\frac{\left(\color{blue}{0} + 2\right) \cdot 1}{1 - x}}{1 + x} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{\color{blue}{2} \cdot 1}{1 - x}}{1 + x} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{\color{blue}{2}}{1 - x}}{1 + x} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.3%
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

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

Alternative 3
Accuracy	99.4%
Cost	448

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

Alternative 4
Accuracy	50.6%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))