Average Error: 58.4 → 0.2
Time: 11.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right), \frac{2}{3}, \frac{{\varepsilon}^{5} \cdot \frac{2}{5}}{{1}^{5}}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right), \frac{2}{3}, \frac{{\varepsilon}^{5} \cdot \frac{2}{5}}{{1}^{5}}\right)\right)
double f(double eps) {
        double r3847260 = 1.0;
        double r3847261 = eps;
        double r3847262 = r3847260 - r3847261;
        double r3847263 = r3847260 + r3847261;
        double r3847264 = r3847262 / r3847263;
        double r3847265 = log(r3847264);
        return r3847265;
}
↓
double f(double eps) {
        double r3847266 = 2.0;
        double r3847267 = eps;
        double r3847268 = 1.0;
        double r3847269 = r3847267 / r3847268;
        double r3847270 = r3847269 * r3847269;
        double r3847271 = r3847269 * r3847270;
        double r3847272 = 0.6666666666666666;
        double r3847273 = 5.0;
        double r3847274 = pow(r3847267, r3847273);
        double r3847275 = 0.4;
        double r3847276 = r3847274 * r3847275;
        double r3847277 = pow(r3847268, r3847273);
        double r3847278 = r3847276 / r3847277;
        double r3847279 = fma(r3847271, r3847272, r3847278);
        double r3847280 = fma(r3847266, r3847267, r3847279);
        double r3847281 = -r3847280;
        return r3847281;
}

Error

Bits error versus eps

Target

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.4

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right), \frac{2}{3}, \frac{\frac{2}{5} \cdot {\varepsilon}^{5}}{{1}^{5}}\right)\right)}\]

  6. Final simplification0.2

    \[\leadsto -\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right), \frac{2}{3}, \frac{{\varepsilon}^{5} \cdot \frac{2}{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2019174 +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))))