Average Error: 58.5 → 0.2
Time: 5.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r87064 = 1.0;
        double r87065 = eps;
        double r87066 = r87064 - r87065;
        double r87067 = r87064 + r87065;
        double r87068 = r87066 / r87067;
        double r87069 = log(r87068);
        return r87069;
}
↓
double f(double eps) {
        double r87070 = 2.0;
        double r87071 = eps;
        double r87072 = r87070 * r87071;
        double r87073 = -r87072;
        double r87074 = 0.6666666666666666;
        double r87075 = 3.0;
        double r87076 = pow(r87071, r87075);
        double r87077 = 0.4;
        double r87078 = 5.0;
        double r87079 = pow(r87071, r87078);
        double r87080 = r87077 * r87079;
        double r87081 = fma(r87074, r87076, r87080);
        double r87082 = r87073 - r87081;
        return r87082;
}

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 log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.66666666666666663 \cdot {\varepsilon}^{3} + 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)}\]

  8. Final simplification0.2

    \[\leadsto \left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2020025 +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))))