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

(FPCore (x)
 :precision binary64
 (/ (- (log1p (* x (- x))) (log1p x)) (log1p x)))

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

double code(double x) {
	return (log1p((x * -x)) - log1p(x)) / log1p(x);
}

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

public static double code(double x) {
	return (Math.log1p((x * -x)) - Math.log1p(x)) / Math.log1p(x);
}

def code(x):
	return math.log((1.0 - x)) / math.log((1.0 + x))

def code(x):
	return (math.log1p((x * -x)) - math.log1p(x)) / math.log1p(x)

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

function code(x)
	return Float64(Float64(log1p(Float64(x * Float64(-x))) - log1p(x)) / log1p(x))
end

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

code[x_] := N[(N[(N[Log[1 + N[(x * (-x)), $MachinePrecision]], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision] / N[Log[1 + x], $MachinePrecision]), $MachinePrecision]

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

\frac{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - \mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}

Error?

Try it out?

Derivation?

1. Initial program 2.2

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

2. Simplified2.6

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

   [Start]2.2	\[ \frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
   log1p-def [=>]2.6	\[ \frac{\log \left(1 - x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Applied egg-rr3.6

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

4. Simplified0.2

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

   [Start]3.6	\[ \frac{\left(\log \left(1 - x \cdot x\right) - e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)}\right) + 1}{\mathsf{log1p}\left(x\right)} \]
   associate-+l- [=>]3.3	\[ \frac{\color{blue}{\log \left(1 - x \cdot x\right) - \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1\right)}}{\mathsf{log1p}\left(x\right)} \]
   sub-neg [=>]3.3	\[ \frac{\log \color{blue}{\left(1 + \left(-x \cdot x\right)\right)} - \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1\right)}{\mathsf{log1p}\left(x\right)} \]
   log1p-def [=>]3.3	\[ \frac{\color{blue}{\mathsf{log1p}\left(-x \cdot x\right)} - \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1\right)}{\mathsf{log1p}\left(x\right)} \]
   distribute-rgt-neg-in [=>]3.3	\[ \frac{\mathsf{log1p}\left(\color{blue}{x \cdot \left(-x\right)}\right) - \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1\right)}{\mathsf{log1p}\left(x\right)} \]
   expm1-def [=>]0.3	\[ \frac{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)\right)}}{\mathsf{log1p}\left(x\right)} \]
   expm1-log1p [=>]0.2	\[ \frac{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - \color{blue}{\mathsf{log1p}\left(x\right)}}{\mathsf{log1p}\left(x\right)} \]

5. Final simplification0.2

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

Alternatives

Error

Reproduce?

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