Asymptote A

Average Error: 14.7 → 0.1

Time: 7.4s

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 14.7

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

2. Applied egg-rr 14.2

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

3. Simplified 14.2

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

   [Start] 14.2	\[ \frac{x + \left(-2 - x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]
   associate-/r* [=>] 14.2	\[ \color{blue}{\frac{\frac{x + \left(-2 - x\right)}{-1 - x}}{1 - x}} \]
   associate-+r- [=>] 14.2	\[ \frac{\frac{\color{blue}{\left(x + -2\right) - x}}{-1 - x}}{1 - x} \]

4. Applied egg-rr 16.0

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + \left(-2 - x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}\right)} - 1} \]

5. Simplified 0.1

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

   [Start] 16.0	\[ e^{\mathsf{log1p}\left(\frac{x + \left(-2 - x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}\right)} - 1 \]
   expm1-def [=>] 15.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + \left(-2 - x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}\right)\right)} \]
   expm1-log1p [=>] 14.2	\[ \color{blue}{\frac{x + \left(-2 - x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]
   *-commutative [=>] 14.2	\[ \frac{x + \left(-2 - x\right)}{\color{blue}{\left(1 - x\right) \cdot \left(-1 - x\right)}} \]
   associate-/r* [=>] 14.2	\[ \color{blue}{\frac{\frac{x + \left(-2 - x\right)}{1 - x}}{-1 - x}} \]
   associate-+r- [=>] 14.2	\[ \frac{\frac{\color{blue}{\left(x + -2\right) - x}}{1 - x}}{-1 - x} \]
   +-commutative [=>] 14.2	\[ \frac{\frac{\color{blue}{\left(-2 + x\right)} - x}{1 - x}}{-1 - x} \]
   associate-+r- [<=] 0.1	\[ \frac{\frac{\color{blue}{-2 + \left(x - x\right)}}{1 - x}}{-1 - x} \]
   +-commutative [<=] 0.1	\[ \frac{\frac{\color{blue}{\left(x - x\right) + -2}}{1 - x}}{-1 - x} \]
   +-inverses [=>] 0.1	\[ \frac{\frac{\color{blue}{0} + -2}{1 - x}}{-1 - x} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{-2}}{1 - x}}{-1 - x} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
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
Error	0.7
Cost	585

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

Alternative 3
Error	0.4
Cost	448

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

Alternative 4
Error	31.4
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))