logq (problem 3.4.3)

Average Error: 58.6 → 0.0

Time: 9.2s

Precision: binary64

Cost: 13312

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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)) + (Math.log1p(eps) * -2.0);
}

Python

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

↓

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

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(log1p(eps) * -2.0))
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[(N[Log[1 + eps], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Target

Original	58.6
Target	0.2
Herbie	0.0

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

Derivation

1. Initial program 58.6

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

2. Applied egg-rr 58.6

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

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\varepsilon - \varepsilon \cdot \left(1 + \varepsilon\right)\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right)} \]

4. Taylor expanded in eps around 0 0.0

   \[\leadsto \mathsf{log1p}\left(\color{blue}{-1 \cdot {\varepsilon}^{2}}\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right) \]

5. Simplified 0.0

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\varepsilon \cdot \left(-\varepsilon\right)}\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right) \]
   Proof
   (*.f64 eps (neg.f64 eps)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 eps eps))): 0 points increase in error, 0 points decrease in error
   (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 eps 2))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13056

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

Alternative 2
Error	0.3
Cost	704

\[\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right) + \varepsilon \cdot -2 \]

Alternative 3
Error	0.6
Cost	192

\[\varepsilon \cdot -2 \]

Reproduce

herbie shell --seed 2022295 
(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))))