Average Error: 58.4 → 0.3
Time: 19.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r2275087 = 1.0;
        double r2275088 = eps;
        double r2275089 = r2275087 - r2275088;
        double r2275090 = r2275087 + r2275088;
        double r2275091 = r2275089 / r2275090;
        double r2275092 = log(r2275091);
        return r2275092;
}
↓
double f(double eps) {
        double r2275093 = eps;
        double r2275094 = r2275093 * r2275093;
        double r2275095 = r2275093 * r2275094;
        double r2275096 = -0.6666666666666666;
        double r2275097 = -0.4;
        double r2275098 = 5.0;
        double r2275099 = pow(r2275093, r2275098);
        double r2275100 = -2.0;
        double r2275101 = r2275100 * r2275093;
        double r2275102 = fma(r2275097, r2275099, r2275101);
        double r2275103 = fma(r2275095, r2275096, r2275102);
        return r2275103;
}

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

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

  4. Final simplification0.3

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

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))))