Average Error: 58.4 → 0.3
Time: 13.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{2}{3}, 2\right), {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{2}{3}, 2\right), {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r4205675 = 1.0;
        double r4205676 = eps;
        double r4205677 = r4205675 - r4205676;
        double r4205678 = r4205675 + r4205676;
        double r4205679 = r4205677 / r4205678;
        double r4205680 = log(r4205679);
        return r4205680;
}
↓
double f(double eps) {
        double r4205681 = eps;
        double r4205682 = r4205681 * r4205681;
        double r4205683 = 0.6666666666666666;
        double r4205684 = 2.0;
        double r4205685 = fma(r4205682, r4205683, r4205684);
        double r4205686 = 5.0;
        double r4205687 = pow(r4205681, r4205686);
        double r4205688 = 0.4;
        double r4205689 = r4205687 * r4205688;
        double r4205690 = fma(r4205681, r4205685, r4205689);
        double r4205691 = -r4205690;
        return r4205691;
}

Error

Bits error versus eps

Target

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +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))))