logq (problem 3.4.3)

Average Accuracy: 8.5% → 100.0%

Time: 9.2s

Precision: binary64

Cost: 13056

Math

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

↓

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

FPCore

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

↓

(FPCore (eps) :precision binary64 (- (log1p (- eps)) (log1p eps)))

C

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

↓

double code(double eps) {
	return log1p(-eps) - log1p(eps);
}

Java

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

↓

public static double code(double eps) {
	return Math.log1p(-eps) - Math.log1p(eps);
}

Python

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

↓

def code(eps):
	return math.log1p(-eps) - math.log1p(eps)

Julia

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

↓

function code(eps)
	return Float64(log1p(Float64(-eps)) - log1p(eps))
end

Wolfram

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

↓

code[eps_] := N[(N[Log[1 + (-eps)], $MachinePrecision] - N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]

Target

Original	8.5%
Target	99.7%
Herbie	100.0%

\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

1. Initial program 8.5%

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

2. Simplified 100.0%

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

   [Start] 8.5	\[ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
   log-div [=>] 8.5	\[ \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)} \]
   sub-neg [=>] 8.5	\[ \log \color{blue}{\left(1 + \left(-\varepsilon\right)\right)} - \log \left(1 + \varepsilon\right) \]
   log1p-def [=>] 21.1	\[ \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} - \log \left(1 + \varepsilon\right) \]
   log1p-def [=>] 100.0	\[ \mathsf{log1p}\left(-\varepsilon\right) - \color{blue}{\mathsf{log1p}\left(\varepsilon\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6912

\[\varepsilon \cdot -2 + -0.6666666666666666 \cdot {\varepsilon}^{3} \]

Alternative 2
Accuracy	99.0%
Cost	192

\[\varepsilon \cdot -2 \]

Reproduce

herbie shell --seed 2023152 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))