Average Error: 58.5 → 0.2
Time: 7.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r443 = 1.0;
        double r444 = eps;
        double r445 = r443 - r444;
        double r446 = r443 + r444;
        double r447 = r445 / r446;
        double r448 = log(r447);
        return r448;
}
↓
double f(double eps) {
        double r449 = 2.0;
        double r450 = eps;
        double r451 = r449 * r450;
        double r452 = -r451;
        double r453 = 0.6666666666666666;
        double r454 = 3.0;
        double r455 = pow(r450, r454);
        double r456 = 0.4;
        double r457 = 5.0;
        double r458 = pow(r450, r457);
        double r459 = r456 * r458;
        double r460 = fma(r453, r455, r459);
        double r461 = r452 - r460;
        return r461;
}

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 log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.66666666666666663 \cdot {\varepsilon}^{3} + 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)}\]

  8. Final simplification0.2

    \[\leadsto \left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2020025 +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))))