Average Error: 58.4 → 0.7
Time: 16.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double f(double eps) {
        double r46078 = 1.0;
        double r46079 = eps;
        double r46080 = r46078 - r46079;
        double r46081 = r46078 + r46079;
        double r46082 = r46080 / r46081;
        double r46083 = log(r46082);
        return r46083;
}
↓
double f(double eps) {
        double r46084 = 2.0;
        double r46085 = eps;
        double r46086 = 2.0;
        double r46087 = pow(r46085, r46086);
        double r46088 = r46084 * r46087;
        double r46089 = 1.0;
        double r46090 = log(r46089);
        double r46091 = pow(r46089, r46086);
        double r46092 = r46087 / r46091;
        double r46093 = r46092 + r46085;
        double r46094 = r46084 * r46093;
        double r46095 = r46090 - r46094;
        double r46096 = r46088 + r46095;
        return r46096;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{\left(2 \cdot {\varepsilon}^{2} + \log 1\right) - \left(2 \cdot \frac{{\varepsilon}^{2}}{{1}^{2}} + 2 \cdot \varepsilon\right)}\]

  3. Simplified0.7

    \[\leadsto \color{blue}{2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto 2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019209 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))