Average Error: 58.6 → 0.0
Time: 7.1s
Precision: binary64
Cost: 13056
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \]
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps) :precision binary64 (- (log1p (- eps)) (log1p eps)))
double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}
↓
double code(double eps) {
	return log1p(-eps) - log1p(eps);
}

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);
}

def code(eps):
	return math.log(((1.0 - eps) / (1.0 + eps)))
↓
def code(eps):
	return math.log1p(-eps) - math.log1p(eps)

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

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]

\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.0
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

  1. Initial program 58.6

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
    Proof
    (-.f64 (log1p.f64 (neg.f64 eps)) (log1p.f64 eps)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 eps)))) (log1p.f64 eps)): 256 points increase in error, 0 points decrease in error
    (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 eps))) (log1p.f64 eps)): 0 points increase in error, 0 points decrease in error
    (-.f64 (log.f64 (-.f64 1 eps)) (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 eps)))): 253 points increase in error, 3 points decrease in error
    (Rewrite<= log-div_binary64 (log.f64 (/.f64 (-.f64 1 eps) (+.f64 1 eps)))): 5 points increase in error, 6 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.3
Cost704
\[\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right) + \varepsilon \cdot -2 \]
Alternative 2
Error0.6
Cost192
\[\varepsilon \cdot -2 \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))