Average Error: 58.7 → 0.2
Time: 13.6s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r70202 = 1.0;
        double r70203 = eps;
        double r70204 = r70202 - r70203;
        double r70205 = r70202 + r70203;
        double r70206 = r70204 / r70205;
        double r70207 = log(r70206);
        return r70207;
}
↓
double f(double eps) {
        double r70208 = 2.0;
        double r70209 = eps;
        double r70210 = 0.6666666666666666;
        double r70211 = 3.0;
        double r70212 = pow(r70209, r70211);
        double r70213 = 0.4;
        double r70214 = 5.0;
        double r70215 = pow(r70209, r70214);
        double r70216 = r70213 * r70215;
        double r70217 = fma(r70210, r70212, r70216);
        double r70218 = fma(r70208, r70209, r70217);
        double r70219 = -r70218;
        return r70219;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.7

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

  3. Applied div-inv58.7

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

  4. Applied log-prod58.7

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

  5. Simplified58.6

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  9. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  10. Final simplification0.2

    \[\leadsto -\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019323 +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))))