Average Error: 58.7 → 0.2
Time: 6.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r65056 = 1.0;
        double r65057 = eps;
        double r65058 = r65056 - r65057;
        double r65059 = r65056 + r65057;
        double r65060 = r65058 / r65059;
        double r65061 = log(r65060);
        return r65061;
}
↓
double f(double eps) {
        double r65062 = 0.6666666666666666;
        double r65063 = eps;
        double r65064 = 3.0;
        double r65065 = pow(r65063, r65064);
        double r65066 = 1.0;
        double r65067 = pow(r65066, r65064);
        double r65068 = r65065 / r65067;
        double r65069 = r65062 * r65068;
        double r65070 = 0.4;
        double r65071 = 5.0;
        double r65072 = pow(r65063, r65071);
        double r65073 = pow(r65066, r65071);
        double r65074 = r65072 / r65073;
        double r65075 = r65070 * r65074;
        double r65076 = r65069 + r65075;
        double r65077 = 2.0;
        double r65078 = r65077 * r65063;
        double r65079 = r65076 + r65078;
        double r65080 = -r65079;
        return r65080;
}

Error

Bits error versus eps

Try it out

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-div58.7

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

  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. Using strategy rm

  6. Applied associate-+r+0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020081 
(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))))