Average Error: 58.5 → 0.2
Time: 14.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r29328 = 1.0;
        double r29329 = eps;
        double r29330 = r29328 - r29329;
        double r29331 = r29328 + r29329;
        double r29332 = r29330 / r29331;
        double r29333 = log(r29332);
        return r29333;
}
↓
double f(double eps) {
        double r29334 = eps;
        double r29335 = 1.0;
        double r29336 = r29334 / r29335;
        double r29337 = 3.0;
        double r29338 = pow(r29336, r29337);
        double r29339 = -0.6666666666666666;
        double r29340 = 0.4;
        double r29341 = 5.0;
        double r29342 = pow(r29334, r29341);
        double r29343 = pow(r29335, r29341);
        double r29344 = r29342 / r29343;
        double r29345 = 2.0;
        double r29346 = r29345 * r29334;
        double r29347 = fma(r29340, r29344, r29346);
        double r29348 = -r29347;
        double r29349 = fma(r29338, r29339, r29348);
        return r29349;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied flip-+58.6

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

  4. Applied associate-/r/58.6

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

  5. Applied log-prod58.6

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

  6. Simplified58.5

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

  7. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)}\]

  9. Final simplification0.2

    \[\leadsto \mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))))