Average Error: 58.8 → 0.6
Time: 15.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1
double f(double eps) {
        double r3697172 = 1.0;
        double r3697173 = eps;
        double r3697174 = r3697172 - r3697173;
        double r3697175 = r3697172 + r3697173;
        double r3697176 = r3697174 / r3697175;
        double r3697177 = log(r3697176);
        return r3697177;
}
↓
double f(double eps) {
        double r3697178 = 2.0;
        double r3697179 = eps;
        double r3697180 = r3697179 * r3697179;
        double r3697181 = 1.0;
        double r3697182 = r3697179 / r3697181;
        double r3697183 = r3697182 * r3697182;
        double r3697184 = r3697179 + r3697183;
        double r3697185 = r3697180 - r3697184;
        double r3697186 = r3697178 * r3697185;
        double r3697187 = log(r3697181);
        double r3697188 = r3697186 + r3697187;
        return r3697188;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019171 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))