Average Error: 58.7 → 0.2
Time: 12.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r34107 = 1.0;
        double r34108 = eps;
        double r34109 = r34107 - r34108;
        double r34110 = r34107 + r34108;
        double r34111 = r34109 / r34110;
        double r34112 = log(r34111);
        return r34112;
}
↓
double f(double eps) {
        double r34113 = eps;
        double r34114 = 1.0;
        double r34115 = r34113 / r34114;
        double r34116 = 3.0;
        double r34117 = pow(r34115, r34116);
        double r34118 = -0.6666666666666666;
        double r34119 = r34117 * r34118;
        double r34120 = 0.4;
        double r34121 = 5.0;
        double r34122 = pow(r34113, r34121);
        double r34123 = pow(r34114, r34121);
        double r34124 = r34122 / r34123;
        double r34125 = r34120 * r34124;
        double r34126 = r34119 - r34125;
        double r34127 = 2.0;
        double r34128 = r34127 * r34113;
        double r34129 = r34126 - r34128;
        return r34129;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied log-div58.6

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

  4. Simplified58.6

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  8. Applied associate--r+0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 
(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))))