Average Error: 58.4 → 0.8
Time: 16.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r79085 = 1.0;
        double r79086 = eps;
        double r79087 = r79085 - r79086;
        double r79088 = r79085 + r79086;
        double r79089 = r79087 / r79088;
        double r79090 = log(r79089);
        return r79090;
}
↓
double f(double eps) {
        double r79091 = 2.0;
        double r79092 = eps;
        double r79093 = 2.0;
        double r79094 = pow(r79092, r79093);
        double r79095 = 1.0;
        double r79096 = pow(r79095, r79093);
        double r79097 = r79094 / r79096;
        double r79098 = r79097 + r79092;
        double r79099 = r79094 - r79098;
        double r79100 = r79091 * r79099;
        double r79101 = log(r79095);
        double r79102 = r79100 + r79101;
        return r79102;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.8
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019297 
(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))))