Average Error: 58.5 → 0.2
Time: 21.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(-2, \varepsilon, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(-2, \varepsilon, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)
double f(double eps) {
        double r1560958 = 1.0;
        double r1560959 = eps;
        double r1560960 = r1560958 - r1560959;
        double r1560961 = r1560958 + r1560959;
        double r1560962 = r1560960 / r1560961;
        double r1560963 = log(r1560962);
        return r1560963;
}
↓
double f(double eps) {
        double r1560964 = -0.6666666666666666;
        double r1560965 = eps;
        double r1560966 = r1560965 * r1560965;
        double r1560967 = r1560966 * r1560965;
        double r1560968 = -2.0;
        double r1560969 = 5.0;
        double r1560970 = pow(r1560965, r1560969);
        double r1560971 = -0.4;
        double r1560972 = r1560970 * r1560971;
        double r1560973 = fma(r1560968, r1560965, r1560972);
        double r1560974 = fma(r1560964, r1560967, r1560973);
        return r1560974;
}

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. 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 +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))))