Average Error: 58.9 → 0.5
Time: 29.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)
double f(double eps) {
        double r112204 = 1.0;
        double r112205 = eps;
        double r112206 = r112204 - r112205;
        double r112207 = r112204 + r112205;
        double r112208 = r112206 / r112207;
        double r112209 = log(r112208);
        return r112209;
}
↓
double f(double eps) {
        double r112210 = 2.0;
        double r112211 = -r112210;
        double r112212 = eps;
        double r112213 = 1.0;
        double r112214 = r112212 / r112213;
        double r112215 = fma(r112214, r112214, r112212);
        double r112216 = 2.0;
        double r112217 = pow(r112212, r112216);
        double r112218 = log(r112213);
        double r112219 = fma(r112210, r112217, r112218);
        double r112220 = fma(r112211, r112215, r112219);
        return r112220;
}

Error

Bits error versus eps

Target

Original58.9
Target0.1
Herbie0.5
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))