Average Error: 58.7 → 0.2
Time: 57.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)
double f(double eps) {
        double r7076376 = 1.0;
        double r7076377 = eps;
        double r7076378 = r7076376 - r7076377;
        double r7076379 = r7076376 + r7076377;
        double r7076380 = r7076378 / r7076379;
        double r7076381 = log(r7076380);
        return r7076381;
}
↓
double f(double eps) {
        double r7076382 = eps;
        double r7076383 = 5.0;
        double r7076384 = pow(r7076382, r7076383);
        double r7076385 = -0.4;
        double r7076386 = -2.0;
        double r7076387 = r7076382 * r7076386;
        double r7076388 = -0.6666666666666666;
        double r7076389 = r7076388 * r7076382;
        double r7076390 = r7076389 * r7076382;
        double r7076391 = r7076382 * r7076390;
        double r7076392 = r7076387 + r7076391;
        double r7076393 = fma(r7076384, r7076385, r7076392);
        return r7076393;
}

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

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +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))))