Asymptote A

Average Error: 14.5 → 0.4

Time: 3.8s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 2.0 (- 1.0 (* x 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 * 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 * 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 * 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 * 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(2.0 / Float64(1.0 - Float64(x * 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 * 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[(2.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.5

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

2. Applied egg-rr 13.8

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

3. Simplified 13.8

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

   [Start] 13.8	\[ \frac{-\left(x + \left(-2 - x\right)\right)}{-\mathsf{fma}\left(x, x, -1\right)} \]
   sub0-neg [<=] 13.8	\[ \frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   +-commutative [=>] 13.8	\[ \frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   associate--r+ [=>] 13.8	\[ \frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   neg-sub0 [<=] 13.8	\[ \frac{\color{blue}{\left(-\left(-2 - x\right)\right)} - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   sub-neg [=>] 13.8	\[ \frac{\left(-\color{blue}{\left(-2 + \left(-x\right)\right)}\right) - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   mul-1-neg [<=] 13.8	\[ \frac{\left(-\left(-2 + \color{blue}{-1 \cdot x}\right)\right) - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   distribute-neg-in [=>] 13.8	\[ \frac{\color{blue}{\left(\left(--2\right) + \left(--1 \cdot x\right)\right)} - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   metadata-eval [=>] 13.8	\[ \frac{\left(\color{blue}{2} + \left(--1 \cdot x\right)\right) - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   mul-1-neg [=>] 13.8	\[ \frac{\left(2 + \left(-\color{blue}{\left(-x\right)}\right)\right) - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   remove-double-neg [=>] 13.8	\[ \frac{\left(2 + \color{blue}{x}\right) - x}{-\mathsf{fma}\left(x, x, -1\right)} \]
   neg-mul-1 [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{-1 \cdot \mathsf{fma}\left(x, x, -1\right)}} \]
   fma-udef [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{-1 \cdot \color{blue}{\left(x \cdot x + -1\right)}} \]
   distribute-lft-in [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{-1 \cdot \left(x \cdot x\right) + -1 \cdot -1}} \]
   associate-*r* [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{\left(-1 \cdot x\right) \cdot x} + -1 \cdot -1} \]
   mul-1-neg [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{\left(-x\right)} \cdot x + -1 \cdot -1} \]
   metadata-eval [=>] 13.8	\[ \frac{\left(2 + x\right) - x}{\left(-x\right) \cdot x + \color{blue}{1}} \]
   +-commutative [<=] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{1 + \left(-x\right) \cdot x}} \]
   cancel-sign-sub-inv [<=] 13.8	\[ \frac{\left(2 + x\right) - x}{\color{blue}{1 - x \cdot x}} \]

4. Taylor expanded in x around 0 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.1
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
Error	0.8
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 3
Error	31.4
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))