Average Error: 58.7 → 0.2
Time: 32.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right))_*\]
double f(double eps) {
        double r4380469 = 1.0;
        double r4380470 = eps;
        double r4380471 = r4380469 - r4380470;
        double r4380472 = r4380469 + r4380470;
        double r4380473 = r4380471 / r4380472;
        double r4380474 = log(r4380473);
        return r4380474;
}
↓
double f(double eps) {
        double r4380475 = eps;
        double r4380476 = 5.0;
        double r4380477 = pow(r4380475, r4380476);
        double r4380478 = -0.4;
        double r4380479 = -2.0;
        double r4380480 = r4380475 * r4380479;
        double r4380481 = -0.6666666666666666;
        double r4380482 = r4380481 * r4380475;
        double r4380483 = r4380482 * r4380475;
        double r4380484 = r4380475 * r4380483;
        double r4380485 = r4380480 + r4380484;
        double r4380486 = fma(r4380477, r4380478, r4380485);
        return r4380486;
}


\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right))_*

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

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019102 +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))))