Average Error: 58.5 → 0.2
Time: 14.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r90304 = 1.0;
        double r90305 = eps;
        double r90306 = r90304 - r90305;
        double r90307 = r90304 + r90305;
        double r90308 = r90306 / r90307;
        double r90309 = log(r90308);
        return r90309;
}
↓
double f(double eps) {
        double r90310 = -0.6666666666666666;
        double r90311 = eps;
        double r90312 = 1.0;
        double r90313 = r90311 / r90312;
        double r90314 = 3.0;
        double r90315 = pow(r90313, r90314);
        double r90316 = 0.4;
        double r90317 = 5.0;
        double r90318 = pow(r90311, r90317);
        double r90319 = pow(r90312, r90317);
        double r90320 = r90318 / r90319;
        double r90321 = 2.0;
        double r90322 = r90321 * r90311;
        double r90323 = fma(r90316, r90320, r90322);
        double r90324 = -r90323;
        double r90325 = fma(r90310, r90315, r90324);
        return r90325;
}

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. Simplified58.5

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

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))))