Average Error: 58.6 → 0.2
Time: 17.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right), \left(\mathsf{fma}\left(\varepsilon, -2, \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right), \left(\mathsf{fma}\left(\varepsilon, -2, \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)\right)\right)
double f(double eps) {
        double r1587039 = 1.0;
        double r1587040 = eps;
        double r1587041 = r1587039 - r1587040;
        double r1587042 = r1587039 + r1587040;
        double r1587043 = r1587041 / r1587042;
        double r1587044 = log(r1587043);
        return r1587044;
}
↓
double f(double eps) {
        double r1587045 = -0.6666666666666666;
        double r1587046 = eps;
        double r1587047 = r1587046 * r1587046;
        double r1587048 = r1587047 * r1587046;
        double r1587049 = -2.0;
        double r1587050 = -0.4;
        double r1587051 = 5.0;
        double r1587052 = pow(r1587046, r1587051);
        double r1587053 = r1587050 * r1587052;
        double r1587054 = fma(r1587046, r1587049, r1587053);
        double r1587055 = fma(r1587045, r1587048, r1587054);
        return r1587055;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019129 +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))))