Average Error: 58.7 → 0.2
Time: 20.2s
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 r2656218 = 1.0;
        double r2656219 = eps;
        double r2656220 = r2656218 - r2656219;
        double r2656221 = r2656218 + r2656219;
        double r2656222 = r2656220 / r2656221;
        double r2656223 = log(r2656222);
        return r2656223;
}
↓
double f(double eps) {
        double r2656224 = -0.6666666666666666;
        double r2656225 = eps;
        double r2656226 = r2656225 * r2656225;
        double r2656227 = r2656226 * r2656225;
        double r2656228 = -2.0;
        double r2656229 = 5.0;
        double r2656230 = pow(r2656225, r2656229);
        double r2656231 = -0.4;
        double r2656232 = r2656230 * r2656231;
        double r2656233 = fma(r2656228, r2656225, r2656232);
        double r2656234 = fma(r2656224, r2656227, r2656233);
        return r2656234;
}

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  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 2019144 +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))))