Average Error: 58.5 → 0.3
Time: 13.5s
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 r43994 = 1.0;
        double r43995 = eps;
        double r43996 = r43994 - r43995;
        double r43997 = r43994 + r43995;
        double r43998 = r43996 / r43997;
        double r43999 = log(r43998);
        return r43999;
}
↓
double f(double eps) {
        double r44000 = eps;
        double r44001 = 1.0;
        double r44002 = r44000 / r44001;
        double r44003 = 3.0;
        double r44004 = pow(r44002, r44003);
        double r44005 = -0.6666666666666666;
        double r44006 = r44004 * r44005;
        double r44007 = 0.4;
        double r44008 = 5.0;
        double r44009 = pow(r44000, r44008);
        double r44010 = pow(r44001, r44008);
        double r44011 = r44009 / r44010;
        double r44012 = r44007 * r44011;
        double r44013 = 2.0;
        double r44014 = r44013 * r44000;
        double r44015 = r44012 + r44014;
        double r44016 = r44006 - r44015;
        return r44016;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied div-inv58.5

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

  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 0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(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))))