Average Error: 58.6 → 0.2
Time: 5.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\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)
double f(double eps) {
        double r91174 = 1.0;
        double r91175 = eps;
        double r91176 = r91174 - r91175;
        double r91177 = r91174 + r91175;
        double r91178 = r91176 / r91177;
        double r91179 = log(r91178);
        return r91179;
}
↓
double f(double eps) {
        double r91180 = 0.6666666666666666;
        double r91181 = eps;
        double r91182 = 3.0;
        double r91183 = pow(r91181, r91182);
        double r91184 = 1.0;
        double r91185 = pow(r91184, r91182);
        double r91186 = r91183 / r91185;
        double r91187 = r91180 * r91186;
        double r91188 = 0.4;
        double r91189 = 5.0;
        double r91190 = pow(r91181, r91189);
        double r91191 = pow(r91184, r91189);
        double r91192 = r91190 / r91191;
        double r91193 = r91188 * r91192;
        double r91194 = 2.0;
        double r91195 = r91194 * r91181;
        double r91196 = r91193 + r91195;
        double r91197 = r91187 + r91196;
        double r91198 = -r91197;
        return r91198;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020059 
(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))))