logq (problem 3.4.3)

Average Error: 91.58% → 0.01%

Time: 5.4s

Precision: binary64

Cost: 13312

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(eps):
	return math.log1p((eps * -eps)) + (-2.0 * 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 * Float64(-eps))) + Float64(-2.0 * 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 + N[(eps * (-eps)), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	91.58%
Target	0.26%
Herbie	0.01%

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

Derivation

1. Initial program 91.58

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

2. Applied egg-rr 0.92

   \[\leadsto \color{blue}{\log \left(1 - \varepsilon \cdot \varepsilon\right) - \left(\mathsf{log1p}\left(\varepsilon\right) + \mathsf{log1p}\left(\varepsilon\right)\right)} \]

3. Simplified 0.01

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

   [Start] 0.92	\[ \log \left(1 - \varepsilon \cdot \varepsilon\right) - \left(\mathsf{log1p}\left(\varepsilon\right) + \mathsf{log1p}\left(\varepsilon\right)\right) \]
   count-2 [=>] 0.92	\[ \log \left(1 - \varepsilon \cdot \varepsilon\right) - \color{blue}{2 \cdot \mathsf{log1p}\left(\varepsilon\right)} \]
   cancel-sign-sub-inv [=>] 0.92	\[ \color{blue}{\log \left(1 - \varepsilon \cdot \varepsilon\right) + \left(-2\right) \cdot \mathsf{log1p}\left(\varepsilon\right)} \]
   sub-neg [=>] 0.92	\[ \log \color{blue}{\left(1 + \left(-\varepsilon \cdot \varepsilon\right)\right)} + \left(-2\right) \cdot \mathsf{log1p}\left(\varepsilon\right) \]
   log1p-def [=>] 0.01	\[ \color{blue}{\mathsf{log1p}\left(-\varepsilon \cdot \varepsilon\right)} + \left(-2\right) \cdot \mathsf{log1p}\left(\varepsilon\right) \]
   distribute-rgt-neg-in [=>] 0.01	\[ \mathsf{log1p}\left(\color{blue}{\varepsilon \cdot \left(-\varepsilon\right)}\right) + \left(-2\right) \cdot \mathsf{log1p}\left(\varepsilon\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{log1p}\left(\varepsilon \cdot \left(-\varepsilon\right)\right) + \color{blue}{-2} \cdot \mathsf{log1p}\left(\varepsilon\right) \]

4. Final simplification 0.01

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

Alternatives

Alternative 1
Error	0.01%
Cost	13056

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

Alternative 2
Error	0.41%
Cost	6912

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

Alternative 3
Error	0.86%
Cost	192

\[\varepsilon \cdot -2 \]

Reproduce

herbie shell --seed 2023125 
(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))))