Average Error: 58.5 → 0.2
Time: 12.6s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r81261 = 1.0;
        double r81262 = eps;
        double r81263 = r81261 - r81262;
        double r81264 = r81261 + r81262;
        double r81265 = r81263 / r81264;
        double r81266 = log(r81265);
        return r81266;
}
↓
double f(double eps) {
        double r81267 = eps;
        double r81268 = 1.0;
        double r81269 = r81267 / r81268;
        double r81270 = 3.0;
        double r81271 = pow(r81269, r81270);
        double r81272 = -0.6666666666666666;
        double r81273 = r81271 * r81272;
        double r81274 = 0.4;
        double r81275 = 5.0;
        double r81276 = pow(r81267, r81275);
        double r81277 = pow(r81268, r81275);
        double r81278 = r81276 / r81277;
        double r81279 = r81274 * r81278;
        double r81280 = 2.0;
        double r81281 = r81280 * r81267;
        double r81282 = r81279 + r81281;
        double r81283 = r81273 - r81282;
        return r81283;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

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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. Using strategy rm

  3. Applied div-inv58.6

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

  4. Applied log-prod58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))