Asymptote C

Average Error: 46.54% → 0.02%

Time: 8.7s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{3 + \frac{1}{x}}{\frac{1}{x} - x} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (+ 3.0 (/ 1.0 x)) (- (/ 1.0 x) x)))

C

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

↓

double code(double x) {
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 + (1.0d0 / x)) / ((1.0d0 / x) - x)
end function

Java

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

↓

public static double code(double x) {
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x);
}

Python

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

↓

def code(x):
	return (3.0 + (1.0 / x)) / ((1.0 / x) - x)

Julia

function code(x)
	return Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(Float64(3.0 + Float64(1.0 / x)) / Float64(Float64(1.0 / x) - x))
end

MATLAB

function tmp = code(x)
	tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = (3.0 + (1.0 / x)) / ((1.0 / x) - x);
end

Wolfram

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

↓

code[x_] := N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.54

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

2. Simplified 46.54

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

   [Start] 46.54	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   sub-neg [=>] 46.54	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
   +-commutative [=>] 46.54	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
   remove-double-neg [<=] 46.54	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
   sub-neg [<=] 46.54	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
   distribute-neg-frac [=>] 46.54	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
   neg-sub0 [=>] 46.54	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   +-commutative [=>] 46.54	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   associate--r+ [=>] 46.54	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   metadata-eval [=>] 46.54	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   sub-neg [=>] 46.54	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
   metadata-eval [=>] 46.54	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
   /-rgt-identity [<=] 46.54	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
   neg-mul-1 [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
   metadata-eval [<=] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
   *-commutative [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
   associate-/l* [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
   metadata-eval [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
   metadata-eval [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
   metadata-eval [<=] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
   associate-/l/ [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
   metadata-eval [=>] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
   neg-mul-1 [<=] 46.54	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

3. Applied egg-rr 46

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

4. Taylor expanded in x around 0 0.04

   \[\leadsto \frac{\color{blue}{3 + \frac{1}{x}}}{\left(-1 + x\right) \cdot \frac{-1 - x}{x}} \]

5. Taylor expanded in x around 0 0.02

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot x + \frac{1}{x}}} \]

6. Simplified 0.02

   \[\leadsto \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1}{x} - x}} \]
   Proof

   [Start] 0.02	\[ \frac{3 + \frac{1}{x}}{-1 \cdot x + \frac{1}{x}} \]
   +-commutative [=>] 0.02	\[ \frac{3 + \frac{1}{x}}{\color{blue}{\frac{1}{x} + -1 \cdot x}} \]
   metadata-eval [<=] 0.02	\[ \frac{3 + \frac{1}{x}}{\frac{\color{blue}{-1 \cdot -1}}{x} + -1 \cdot x} \]
   associate-*r/ [<=] 0.02	\[ \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot \frac{-1}{x}} + -1 \cdot x} \]
   mul-1-neg [=>] 0.02	\[ \frac{3 + \frac{1}{x}}{-1 \cdot \frac{-1}{x} + \color{blue}{\left(-x\right)}} \]
   unsub-neg [=>] 0.02	\[ \frac{3 + \frac{1}{x}}{\color{blue}{-1 \cdot \frac{-1}{x} - x}} \]
   associate-*r/ [=>] 0.02	\[ \frac{3 + \frac{1}{x}}{\color{blue}{\frac{-1 \cdot -1}{x}} - x} \]
   metadata-eval [=>] 0.02	\[ \frac{3 + \frac{1}{x}}{\frac{\color{blue}{1}}{x} - x} \]

7. Final simplification 0.02

   \[\leadsto \frac{3 + \frac{1}{x}}{\frac{1}{x} - x} \]

Alternatives

Alternative 1
Error	1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{-x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 2
Error	1.01%
Cost	777

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{3 + \frac{1}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot x\\ \end{array} \]

Alternative 3
Error	1.49%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 4
Error	2.05%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 5
Error	49.96%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))