Average Error: 58.6 → 0.3
Time: 4.4s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[\varepsilon \cdot -2 + {\varepsilon}^{3} \cdot -0.6666666666666666 \]
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (+ (* eps -2.0) (* (pow eps 3.0) -0.6666666666666666)))
double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}
↓
double code(double eps) {
	return (eps * -2.0) + (pow(eps, 3.0) * -0.6666666666666666);
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = log(((1.0d0 - eps) / (1.0d0 + eps)))
end function
↓
real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps * (-2.0d0)) + ((eps ** 3.0d0) * (-0.6666666666666666d0))
end function

public static double code(double eps) {
	return Math.log(((1.0 - eps) / (1.0 + eps)));
}
↓
public static double code(double eps) {
	return (eps * -2.0) + (Math.pow(eps, 3.0) * -0.6666666666666666);
}

def code(eps):
	return math.log(((1.0 - eps) / (1.0 + eps)))
↓
def code(eps):
	return (eps * -2.0) + (math.pow(eps, 3.0) * -0.6666666666666666)

function code(eps)
	return log(Float64(Float64(1.0 - eps) / Float64(1.0 + eps)))
end
↓
function code(eps)
	return Float64(Float64(eps * -2.0) + Float64((eps ^ 3.0) * -0.6666666666666666))
end

function tmp = code(eps)
	tmp = log(((1.0 - eps) / (1.0 + eps)));
end
↓
function tmp = code(eps)
	tmp = (eps * -2.0) + ((eps ^ 3.0) * -0.6666666666666666);
end

code[eps_] := N[Log[N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[eps_] := N[(N[(eps * -2.0), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]

\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot -2 + {\varepsilon}^{3} \cdot -0.6666666666666666

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Simplified0.0

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

  3. Taylor expanded in eps around 0 0.3

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)} \]

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2022131 
(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))))