Average Error: 58.5 → 0.6
Time: 5.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log \left({1}^{3} - {\varepsilon}^{3}\right) - \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5 + 2\right) + \left(2 \cdot \log 1 - \frac{1}{2} \cdot \frac{{\varepsilon}^{2}}{{1}^{2}}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log \left({1}^{3} - {\varepsilon}^{3}\right) - \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5 + 2\right) + \left(2 \cdot \log 1 - \frac{1}{2} \cdot \frac{{\varepsilon}^{2}}{{1}^{2}}\right)\right)
double f(double eps) {
        double r88190 = 1.0;
        double r88191 = eps;
        double r88192 = r88190 - r88191;
        double r88193 = r88190 + r88191;
        double r88194 = r88192 / r88193;
        double r88195 = log(r88194);
        return r88195;
}
↓
double f(double eps) {
        double r88196 = 1.0;
        double r88197 = 3.0;
        double r88198 = pow(r88196, r88197);
        double r88199 = eps;
        double r88200 = pow(r88199, r88197);
        double r88201 = r88198 - r88200;
        double r88202 = log(r88201);
        double r88203 = 0.5;
        double r88204 = r88199 * r88203;
        double r88205 = 2.0;
        double r88206 = r88204 + r88205;
        double r88207 = r88199 * r88206;
        double r88208 = 2.0;
        double r88209 = log(r88196);
        double r88210 = r88208 * r88209;
        double r88211 = 0.5;
        double r88212 = pow(r88199, r88208);
        double r88213 = pow(r88196, r88208);
        double r88214 = r88212 / r88213;
        double r88215 = r88211 * r88214;
        double r88216 = r88210 - r88215;
        double r88217 = r88207 + r88216;
        double r88218 = r88202 - r88217;
        return r88218;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied div-inv58.5

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

  4. Applied log-prod58.5

    \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{1}{1 + \varepsilon}\right)}\]
  5. Using strategy rm

  6. Applied flip3--58.5

    \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} - {\varepsilon}^{3}}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon + 1 \cdot \varepsilon\right)}\right)} + \log \left(\frac{1}{1 + \varepsilon}\right)\]

  7. Applied log-div58.5

    \[\leadsto \color{blue}{\left(\log \left({1}^{3} - {\varepsilon}^{3}\right) - \log \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon + 1 \cdot \varepsilon\right)\right)\right)} + \log \left(\frac{1}{1 + \varepsilon}\right)\]

  8. Applied associate-+l-58.5

    \[\leadsto \color{blue}{\log \left({1}^{3} - {\varepsilon}^{3}\right) - \left(\log \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon + 1 \cdot \varepsilon\right)\right) - \log \left(\frac{1}{1 + \varepsilon}\right)\right)}\]

  9. Simplified58.5

    \[\leadsto \log \left({1}^{3} - {\varepsilon}^{3}\right) - \color{blue}{\left(\log \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon + 1 \cdot \varepsilon\right)\right) + \log \left(1 + \varepsilon\right)\right)}\]

  10. Taylor expanded around 0 0.6

    \[\leadsto \log \left({1}^{3} - {\varepsilon}^{3}\right) - \color{blue}{\left(\left(0.5 \cdot {\varepsilon}^{2} + \left(2 \cdot \log 1 + 2 \cdot \varepsilon\right)\right) - \frac{1}{2} \cdot \frac{{\varepsilon}^{2}}{{1}^{2}}\right)}\]

  11. Simplified0.6

    \[\leadsto \log \left({1}^{3} - {\varepsilon}^{3}\right) - \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot 0.5 + 2\right) + \left(2 \cdot \log 1 - \frac{1}{2} \cdot \frac{{\varepsilon}^{2}}{{1}^{2}}\right)\right)}\]

  12. Final simplification0.6

    \[\leadsto \log \left({1}^{3} - {\varepsilon}^{3}\right) - \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5 + 2\right) + \left(2 \cdot \log 1 - \frac{1}{2} \cdot \frac{{\varepsilon}^{2}}{{1}^{2}}\right)\right)\]

Reproduce

herbie shell --seed 2020046 
(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))))