Average Error: 58.5 → 0.2
Time: 17.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r1976234 = 1.0;
        double r1976235 = eps;
        double r1976236 = r1976234 - r1976235;
        double r1976237 = r1976234 + r1976235;
        double r1976238 = r1976236 / r1976237;
        double r1976239 = log(r1976238);
        return r1976239;
}
↓
double f(double eps) {
        double r1976240 = -0.6666666666666666;
        double r1976241 = eps;
        double r1976242 = r1976240 * r1976241;
        double r1976243 = r1976241 * r1976241;
        double r1976244 = r1976242 * r1976243;
        double r1976245 = -2.0;
        double r1976246 = r1976245 * r1976241;
        double r1976247 = 5.0;
        double r1976248 = pow(r1976241, r1976247);
        double r1976249 = 0.4;
        double r1976250 = r1976248 * r1976249;
        double r1976251 = r1976246 - r1976250;
        double r1976252 = r1976244 + r1976251;
        return r1976252;
}

Error

Bits error versus eps

Try it out

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. 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 
(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))))