Average Error: 58.7 → 0.2
Time: 22.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2\right)\right)\right)
double f(double eps) {
        double r4882592 = 1.0;
        double r4882593 = eps;
        double r4882594 = r4882592 - r4882593;
        double r4882595 = r4882592 + r4882593;
        double r4882596 = r4882594 / r4882595;
        double r4882597 = log(r4882596);
        return r4882597;
}
↓
double f(double eps) {
        double r4882598 = eps;
        double r4882599 = 5.0;
        double r4882600 = pow(r4882598, r4882599);
        double r4882601 = -0.4;
        double r4882602 = -0.6666666666666666;
        double r4882603 = r4882602 * r4882598;
        double r4882604 = r4882603 * r4882598;
        double r4882605 = 2.0;
        double r4882606 = r4882604 - r4882605;
        double r4882607 = r4882598 * r4882606;
        double r4882608 = fma(r4882600, r4882601, r4882607);
        return r4882608;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019121 +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))))