logq (problem 3.4.3)

Average Accuracy: 8.2% → 100.0%

Time: 7.1s

Precision: binary64

Cost: 13376

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[2 \cdot \mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) \]

FPCore

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

↓

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

C

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

↓

double code(double eps) {
	return (2.0 * log1p(-eps)) - log1p(-(eps * eps));
}

Java

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

↓

public static double code(double eps) {
	return (2.0 * Math.log1p(-eps)) - Math.log1p(-(eps * eps));
}

Python

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

↓

def code(eps):
	return (2.0 * math.log1p(-eps)) - math.log1p(-(eps * eps))

Julia

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

↓

function code(eps)
	return Float64(Float64(2.0 * log1p(Float64(-eps))) - log1p(Float64(-Float64(eps * eps))))
end

Wolfram

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

↓

code[eps_] := N[(N[(2.0 * N[Log[1 + (-eps)], $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-N[(eps * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]

Target

Original	8.2%
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 9.3%

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

2. Applied egg-rr 9.3%

   \[\leadsto \color{blue}{-\log \left(\frac{\frac{1 + \varepsilon}{1 - \varepsilon}}{1}\right)} \]
   Proof

   [Start] 9.3	\[ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
   clear-num [=>] 9.2	\[ \log \color{blue}{\left(\frac{1}{\frac{1 + \varepsilon}{1 - \varepsilon}}\right)} \]
   clear-num [=>] 9.2	\[ \log \color{blue}{\left(\frac{1}{\frac{\frac{1 + \varepsilon}{1 - \varepsilon}}{1}}\right)} \]
   log-rec [=>] 9.3	\[ \color{blue}{-\log \left(\frac{\frac{1 + \varepsilon}{1 - \varepsilon}}{1}\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto -\color{blue}{\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - 2 \cdot \mathsf{log1p}\left(-\varepsilon\right)\right)} \]
   Proof

   [Start] 9.3	\[ -\log \left(\frac{\frac{1 + \varepsilon}{1 - \varepsilon}}{1}\right) \]
   /-rgt-identity [=>] 9.3	\[ -\log \color{blue}{\left(\frac{1 + \varepsilon}{1 - \varepsilon}\right)} \]
   flip-+ [=>] 9.2	\[ -\log \left(\frac{\color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}}{1 - \varepsilon}\right) \]
   associate-/l/ [=>] 9.2	\[ -\log \color{blue}{\left(\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}\right)} \]
   log-div [=>] 9.2	\[ -\color{blue}{\left(\log \left(1 \cdot 1 - \varepsilon \cdot \varepsilon\right) - \log \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right)} \]
   metadata-eval [=>] 9.2	\[ -\left(\log \left(\color{blue}{1} - \varepsilon \cdot \varepsilon\right) - \log \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right) \]
   sub-neg [=>] 9.2	\[ -\left(\log \color{blue}{\left(1 + \left(-\varepsilon \cdot \varepsilon\right)\right)} - \log \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right) \]
   log1p-def [=>] 9.6	\[ -\left(\color{blue}{\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right)} - \log \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right) \]
   pow1 [=>] 9.6	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - \log \left(\color{blue}{{\left(1 - \varepsilon\right)}^{1}} \cdot \left(1 - \varepsilon\right)\right)\right) \]
   pow-plus [=>] 9.6	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - \log \color{blue}{\left({\left(1 - \varepsilon\right)}^{\left(1 + 1\right)}\right)}\right) \]
   log-pow [=>] 9.6	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - \color{blue}{\left(1 + 1\right) \cdot \log \left(1 - \varepsilon\right)}\right) \]
   metadata-eval [=>] 9.6	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - \color{blue}{2} \cdot \log \left(1 - \varepsilon\right)\right) \]
   sub-neg [=>] 9.6	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - 2 \cdot \log \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right) \]
   log1p-def [=>] 100.0	\[ -\left(\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) - 2 \cdot \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)}\right) \]

4. Final simplification 100.0%

   \[\leadsto 2 \cdot \mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13056

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

Alternative 2
Accuracy	99.6%
Cost	6912

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

Alternative 3
Accuracy	99.2%
Cost	192

\[\varepsilon \cdot -2 \]

Reproduce

herbie shell --seed 2023159 
(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))))