Average Error: 58.6 → 0.2
Time: 13.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - {\varepsilon}^{5} \cdot \frac{2}{5}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - {\varepsilon}^{5} \cdot \frac{2}{5}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r756232 = 1.0;
        double r756233 = eps;
        double r756234 = r756232 - r756233;
        double r756235 = r756232 + r756233;
        double r756236 = r756234 / r756235;
        double r756237 = log(r756236);
        return r756237;
}
↓
double f(double eps) {
        double r756238 = eps;
        double r756239 = r756238 * r756238;
        double r756240 = r756239 * r756238;
        double r756241 = -0.6666666666666666;
        double r756242 = r756240 * r756241;
        double r756243 = 5.0;
        double r756244 = pow(r756238, r756243);
        double r756245 = 0.4;
        double r756246 = r756244 * r756245;
        double r756247 = r756242 - r756246;
        double r756248 = 2.0;
        double r756249 = r756248 * r756238;
        double r756250 = r756247 - r756249;
        return r756250;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  5. Applied associate--r+0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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