Average Error: 58.5 → 0.2
Time: 13.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r49486 = 1.0;
        double r49487 = eps;
        double r49488 = r49486 - r49487;
        double r49489 = r49486 + r49487;
        double r49490 = r49488 / r49489;
        double r49491 = log(r49490);
        return r49491;
}
↓
double f(double eps) {
        double r49492 = eps;
        double r49493 = 1.0;
        double r49494 = r49492 / r49493;
        double r49495 = 3.0;
        double r49496 = pow(r49494, r49495);
        double r49497 = -0.6666666666666666;
        double r49498 = 0.4;
        double r49499 = 5.0;
        double r49500 = pow(r49492, r49499);
        double r49501 = pow(r49493, r49499);
        double r49502 = r49500 / r49501;
        double r49503 = 2.0;
        double r49504 = r49503 * r49492;
        double r49505 = fma(r49498, r49502, r49504);
        double r49506 = -r49505;
        double r49507 = fma(r49496, r49497, r49506);
        return r49507;
}

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 flip-+58.5

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

  4. Applied associate-/r/58.5

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

  5. Applied log-prod58.5

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

  6. Simplified58.5

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

  7. Taylor expanded around 0 0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +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))))