Average Error: 58.5 → 0.2
Time: 8.7s
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 r54959 = 1.0;
        double r54960 = eps;
        double r54961 = r54959 - r54960;
        double r54962 = r54959 + r54960;
        double r54963 = r54961 / r54962;
        double r54964 = log(r54963);
        return r54964;
}
↓
double f(double eps) {
        double r54965 = 0.6666666666666666;
        double r54966 = eps;
        double r54967 = 3.0;
        double r54968 = pow(r54966, r54967);
        double r54969 = 1.0;
        double r54970 = pow(r54969, r54967);
        double r54971 = r54968 / r54970;
        double r54972 = r54965 * r54971;
        double r54973 = 0.4;
        double r54974 = 5.0;
        double r54975 = pow(r54966, r54974);
        double r54976 = pow(r54969, r54974);
        double r54977 = r54975 / r54976;
        double r54978 = r54973 * r54977;
        double r54979 = 2.0;
        double r54980 = r54979 * r54966;
        double r54981 = r54978 + r54980;
        double r54982 = r54972 + r54981;
        double r54983 = -r54982;
        return r54983;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied log-div58.5

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

  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 2019304 
(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))))