Average Error: 58.6 → 0.6
Time: 12.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \left(\varepsilon - \frac{\varepsilon}{1 \cdot 1}\right) - \varepsilon\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \left(\varepsilon - \frac{\varepsilon}{1 \cdot 1}\right) - \varepsilon\right) + \log 1
double f(double eps) {
        double r78317 = 1.0;
        double r78318 = eps;
        double r78319 = r78317 - r78318;
        double r78320 = r78317 + r78318;
        double r78321 = r78319 / r78320;
        double r78322 = log(r78321);
        return r78322;
}
↓
double f(double eps) {
        double r78323 = 2.0;
        double r78324 = eps;
        double r78325 = 1.0;
        double r78326 = r78325 * r78325;
        double r78327 = r78324 / r78326;
        double r78328 = r78324 - r78327;
        double r78329 = r78324 * r78328;
        double r78330 = r78329 - r78324;
        double r78331 = r78323 * r78330;
        double r78332 = log(r78325);
        double r78333 = r78331 + r78332;
        return r78333;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(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))))