qlog (example 3.10)

Average Accuracy: 4.2% → 100.0%

Time: 15.8s

Precision: binary64

Cost: 19520

\[-1 < x \land x < 1\]

Math

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

↓

\[{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (pow (/ (log1p x) (log1p (- x))) -1.0))

C

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

↓

double code(double x) {
	return pow((log1p(x) / log1p(-x)), -1.0);
}

Java

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.pow((Math.log1p(x) / Math.log1p(-x)), -1.0);
}

Python

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

↓

def code(x):
	return math.pow((math.log1p(x) / math.log1p(-x)), -1.0)

Julia

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

↓

function code(x)
	return Float64(log1p(x) / log1p(Float64(-x))) ^ -1.0
end

Wolfram

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

↓

code[x_] := N[Power[N[(N[Log[1 + x], $MachinePrecision] / N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Target

Original	4.2%
Target	99.5%
Herbie	100.0%

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right) \]

Derivation

1. Initial program 4.2%

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

2. Applied egg-rr 100.0%

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

   [Start] 4.2	\[ \frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
   clear-num [=>] 4.1	\[ \color{blue}{\frac{1}{\frac{\log \left(1 + x\right)}{\log \left(1 - x\right)}}} \]
   inv-pow [=>] 4.1	\[ \color{blue}{{\left(\frac{\log \left(1 + x\right)}{\log \left(1 - x\right)}\right)}^{-1}} \]
   log1p-def [=>] 6.2	\[ {\left(\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{\log \left(1 - x\right)}\right)}^{-1} \]
   sub-neg [=>] 6.2	\[ {\left(\frac{\mathsf{log1p}\left(x\right)}{\log \color{blue}{\left(1 + \left(-x\right)\right)}}\right)}^{-1} \]
   log1p-def [=>] 100.0	\[ {\left(\frac{\mathsf{log1p}\left(x\right)}{\color{blue}{\mathsf{log1p}\left(-x\right)}}\right)}^{-1} \]

3. Final simplification 100.0%

   \[\leadsto {\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13056

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

Alternative 2
Accuracy	99.3%
Cost	576

\[\left(-1 - x\right) + -0.5 \cdot \left(x \cdot x\right) \]

Alternative 3
Accuracy	98.9%
Cost	192

\[-1 - x \]

Alternative 4
Accuracy	97.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023126 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))