Average Error: 58.6 → 0.7
Time: 5.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r81060 = 1.0;
        double r81061 = eps;
        double r81062 = r81060 - r81061;
        double r81063 = r81060 + r81061;
        double r81064 = r81062 / r81063;
        double r81065 = log(r81064);
        return r81065;
}
↓
double f(double eps) {
        double r81066 = 2.0;
        double r81067 = eps;
        double r81068 = 2.0;
        double r81069 = pow(r81067, r81068);
        double r81070 = 1.0;
        double r81071 = r81067 / r81070;
        double r81072 = fma(r81071, r81071, r81067);
        double r81073 = r81069 - r81072;
        double r81074 = r81066 * r81073;
        double r81075 = log(r81070);
        double r81076 = r81074 + r81075;
        return r81076;
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019344 +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))))