Average Error: 58.5 → 0.2
Time: 13.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r69653 = 1.0;
        double r69654 = eps;
        double r69655 = r69653 - r69654;
        double r69656 = r69653 + r69654;
        double r69657 = r69655 / r69656;
        double r69658 = log(r69657);
        return r69658;
}
↓
double f(double eps) {
        double r69659 = eps;
        double r69660 = 1.0;
        double r69661 = r69659 / r69660;
        double r69662 = 3.0;
        double r69663 = pow(r69661, r69662);
        double r69664 = -0.6666666666666666;
        double r69665 = r69663 * r69664;
        double r69666 = 0.4;
        double r69667 = 5.0;
        double r69668 = pow(r69659, r69667);
        double r69669 = pow(r69660, r69667);
        double r69670 = r69668 / r69669;
        double r69671 = r69666 * r69670;
        double r69672 = 2.0;
        double r69673 = r69672 * r69659;
        double r69674 = r69671 + r69673;
        double r69675 = r69665 - r69674;
        return r69675;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

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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. Using strategy rm

  3. Applied div-inv58.5

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

  4. Applied log-prod58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 
(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))))