Average Error: 58.6 → 0.2
Time: 30.7s
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 r6607464 = 1.0;
        double r6607465 = eps;
        double r6607466 = r6607464 - r6607465;
        double r6607467 = r6607464 + r6607465;
        double r6607468 = r6607466 / r6607467;
        double r6607469 = log(r6607468);
        return r6607469;
}
↓
double f(double eps) {
        double r6607470 = eps;
        double r6607471 = 5.0;
        double r6607472 = pow(r6607470, r6607471);
        double r6607473 = -0.4;
        double r6607474 = -2.0;
        double r6607475 = r6607470 * r6607474;
        double r6607476 = -0.6666666666666666;
        double r6607477 = r6607476 * r6607470;
        double r6607478 = r6607477 * r6607470;
        double r6607479 = r6607470 * r6607478;
        double r6607480 = r6607475 + r6607479;
        double r6607481 = fma(r6607472, r6607473, r6607480);
        return r6607481;
}


\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

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  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-rgt-in0.2

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

  7. Simplified0.2

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

  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 2019101 +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))))