?

Average Error: 2.9 → 0.2
Time: 12.0s
Precision: binary64
Cost: 13056

?

\[0.001 \leq x \land x \leq 1\]
\[\log \left(\frac{1 - x}{1 + x}\right) \]
\[\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right) \]
(FPCore (x) :precision binary64 (log (/ (- 1.0 x) (+ 1.0 x))))
(FPCore (x) :precision binary64 (- (log1p (- x)) (log1p x)))
double code(double x) {
	return log(((1.0 - x) / (1.0 + x)));
}
double code(double x) {
	return log1p(-x) - log1p(x);
}
public static double code(double x) {
	return Math.log(((1.0 - x) / (1.0 + x)));
}
public static double code(double x) {
	return Math.log1p(-x) - Math.log1p(x);
}
def code(x):
	return math.log(((1.0 - x) / (1.0 + x)))
def code(x):
	return math.log1p(-x) - math.log1p(x)
function code(x)
	return log(Float64(Float64(1.0 - x) / Float64(1.0 + x)))
end
function code(x)
	return Float64(log1p(Float64(-x)) - log1p(x))
end
code[x_] := N[Log[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
code[x_] := N[(N[Log[1 + (-x)], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]
\log \left(\frac{1 - x}{1 + x}\right)
\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 2.9

    \[\log \left(\frac{1 - x}{1 + x}\right) \]
  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)} \]
    Proof

    [Start]2.9

    \[ \log \left(\frac{1 - x}{1 + x}\right) \]

    log-div [=>]2.9

    \[ \color{blue}{\log \left(1 - x\right) - \log \left(1 + x\right)} \]

    sub-neg [=>]2.9

    \[ \log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 + x\right) \]

    log1p-def [=>]2.8

    \[ \color{blue}{\mathsf{log1p}\left(-x\right)} - \log \left(1 + x\right) \]

    log1p-def [=>]0.2

    \[ \mathsf{log1p}\left(-x\right) - \color{blue}{\mathsf{log1p}\left(x\right)} \]
  3. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right) \]

Alternatives

Alternative 1
Error2.9
Cost6848
\[\log \left(\frac{1 - x}{x + 1}\right) \]
Alternative 2
Error41.1
Cost192
\[x \cdot -2 \]
Alternative 3
Error62.0
Cost64
\[0 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "log((1 - x) / (1 + x))"
  :precision binary64
  :pre (and (<= 0.001 x) (<= x 1.0))
  (log (/ (- 1.0 x) (+ 1.0 x))))