qlog (example 3.10)

Average Error: 95.77% → 0.04%

Time: 11.0s

Precision: binary64

Cost: 13056

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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.log1p(-x) / Math.log1p(x);
}

Python

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

↓

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

Julia

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

↓

function code(x)
	return Float64(log1p(Float64(-x)) / log1p(x))
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[(N[Log[1 + (-x)], $MachinePrecision] / N[Log[1 + x], $MachinePrecision]), $MachinePrecision]

Target

Original	95.77%
Target	0.5%
Herbie	0.04%

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

Derivation

1. Initial program 95.77

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

2. Simplified 0.04

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

   [Start] 95.77	\[ \frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
   sub-neg [=>] 95.77	\[ \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log \left(1 + x\right)} \]
   log1p-def [=>] 96.68	\[ \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]
   log1p-def [=>] 0.04	\[ \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Final simplification 0.04

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

Alternatives

Alternative 1
Error	0.67%
Cost	576

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

Alternative 2
Error	0.67%
Cost	576

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

Alternative 3
Error	1.02%
Cost	192

\[-1 - x \]

Alternative 4
Error	2.15%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023125 
(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))))