Average Error: 58.7 → 0.2
Time: 5.4s
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 r84371 = 1.0;
        double r84372 = eps;
        double r84373 = r84371 - r84372;
        double r84374 = r84371 + r84372;
        double r84375 = r84373 / r84374;
        double r84376 = log(r84375);
        return r84376;
}
↓
double f(double eps) {
        double r84377 = 0.6666666666666666;
        double r84378 = eps;
        double r84379 = 3.0;
        double r84380 = pow(r84378, r84379);
        double r84381 = 1.0;
        double r84382 = pow(r84381, r84379);
        double r84383 = r84380 / r84382;
        double r84384 = r84377 * r84383;
        double r84385 = 0.4;
        double r84386 = 5.0;
        double r84387 = pow(r84378, r84386);
        double r84388 = pow(r84381, r84386);
        double r84389 = r84387 / r84388;
        double r84390 = r84385 * r84389;
        double r84391 = 2.0;
        double r84392 = r84391 * r84378;
        double r84393 = r84390 + r84392;
        double r84394 = r84384 + r84393;
        double r84395 = -r84394;
        return r84395;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  3. Applied log-div58.7

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

  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 2020033 
(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))))