Average Error: 58.7 → 0.2
Time: 13.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r64204 = 1.0;
        double r64205 = eps;
        double r64206 = r64204 - r64205;
        double r64207 = r64204 + r64205;
        double r64208 = r64206 / r64207;
        double r64209 = log(r64208);
        return r64209;
}
↓
double f(double eps) {
        double r64210 = 2.0;
        double r64211 = eps;
        double r64212 = r64210 * r64211;
        double r64213 = 0.6666666666666666;
        double r64214 = 3.0;
        double r64215 = pow(r64211, r64214);
        double r64216 = r64213 * r64215;
        double r64217 = 0.4;
        double r64218 = 5.0;
        double r64219 = pow(r64211, r64218);
        double r64220 = r64217 * r64219;
        double r64221 = r64216 + r64220;
        double r64222 = r64212 + r64221;
        double r64223 = -r64222;
        return r64223;
}

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 add-exp-log58.7

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

  4. Applied add-exp-log58.7

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

  5. Applied div-exp58.7

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

  6. Applied rem-log-exp58.6

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

  7. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}}\]

  9. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  10. Final simplification0.2

    \[\leadsto -\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(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))))