Asymptote C

Average Accuracy: 54.9% → 100.0%

Time: 12.8s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (+ -3.0 (/ -1.0 x)) (+ x (/ -1.0 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)) / (x + (-1.0 / 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)) / (x + ((-1.0d0) / 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)) / (x + (-1.0 / 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)) / (x + (-1.0 / 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(x + Float64(-1.0 / 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)) / (x + (-1.0 / 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[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

2. Applied egg-rr 55.4%

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

   [Start] 54.9	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   clear-num [=>] 54.9	\[ \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1} \]
   frac-sub [=>] 55.4	\[ \color{blue}{\frac{1 \cdot \left(x - 1\right) - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \left(x - 1\right)}} \]
   *-un-lft-identity [<=] 55.4	\[ \frac{\color{blue}{\left(x - 1\right)} - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \left(x - 1\right)} \]
   sub-neg [=>] 55.4	\[ \frac{\color{blue}{\left(x + \left(-1\right)\right)} - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \left(x - 1\right)} \]
   metadata-eval [=>] 55.4	\[ \frac{\left(x + \color{blue}{-1}\right) - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \left(x - 1\right)} \]
   sub-neg [=>] 55.4	\[ \frac{\left(x + -1\right) - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 55.4	\[ \frac{\left(x + -1\right) - \frac{x + 1}{x} \cdot \left(x + 1\right)}{\frac{x + 1}{x} \cdot \left(x + \color{blue}{-1}\right)} \]

3. Taylor expanded in x around 0 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{-\left(3 + \frac{1}{x}\right)}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
   distribute-neg-in [=>] 100.0	\[ \frac{\color{blue}{\left(-3\right) + \left(-\frac{1}{x}\right)}}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
   metadata-eval [=>] 100.0	\[ \frac{\color{blue}{-3} + \left(-\frac{1}{x}\right)}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
   distribute-neg-frac [=>] 100.0	\[ \frac{-3 + \color{blue}{\frac{-1}{x}}}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
   metadata-eval [=>] 100.0	\[ \frac{-3 + \frac{\color{blue}{-1}}{x}}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]

5. Taylor expanded in x around 0 100.0%

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

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + 1}{-1 + x}\\ \end{array} \]

Alternative 2
Accuracy	98.7%
Cost	713

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

Alternative 3
Accuracy	98.3%
Cost	584

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

Alternative 4
Accuracy	97.7%
Cost	456

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

Alternative 5
Accuracy	51.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023152 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))