Average Error: 58.7 → 0.2
Time: 11.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, 2 \cdot \varepsilon + \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, 2 \cdot \varepsilon + \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)
double f(double eps) {
        double r893433 = 1.0;
        double r893434 = eps;
        double r893435 = r893433 - r893434;
        double r893436 = r893433 + r893434;
        double r893437 = r893435 / r893436;
        double r893438 = log(r893437);
        return r893438;
}
↓
double f(double eps) {
        double r893439 = eps;
        double r893440 = 5.0;
        double r893441 = pow(r893439, r893440);
        double r893442 = 0.4;
        double r893443 = 2.0;
        double r893444 = r893443 * r893439;
        double r893445 = r893439 * r893439;
        double r893446 = 0.6666666666666666;
        double r893447 = r893445 * r893446;
        double r893448 = r893439 * r893447;
        double r893449 = r893444 + r893448;
        double r893450 = fma(r893441, r893442, r893449);
        double r893451 = -r893450;
        return r893451;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \varepsilon \cdot \left(2 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied distribute-lft-in0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))