Average Error: 58.6 → 0.2
Time: 9.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} + \left(\frac{\frac{-2}{5}}{\frac{{1}^{5}}{{\varepsilon}^{5}}} - 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} + \left(\frac{\frac{-2}{5}}{\frac{{1}^{5}}{{\varepsilon}^{5}}} - 2 \cdot \varepsilon\right)
double f(double eps) {
        double r130578 = 1.0;
        double r130579 = eps;
        double r130580 = r130578 - r130579;
        double r130581 = r130578 + r130579;
        double r130582 = r130580 / r130581;
        double r130583 = log(r130582);
        return r130583;
}
↓
double f(double eps) {
        double r130584 = -0.6666666666666666;
        double r130585 = eps;
        double r130586 = 1.0;
        double r130587 = r130585 / r130586;
        double r130588 = 3.0;
        double r130589 = pow(r130587, r130588);
        double r130590 = r130584 * r130589;
        double r130591 = -0.4;
        double r130592 = 5.0;
        double r130593 = pow(r130586, r130592);
        double r130594 = pow(r130585, r130592);
        double r130595 = r130593 / r130594;
        double r130596 = r130591 / r130595;
        double r130597 = 2.0;
        double r130598 = r130597 * r130585;
        double r130599 = r130596 - r130598;
        double r130600 = r130590 + r130599;
        return r130600;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 58.6

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

  3. Applied div-inv58.6

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

  4. Applied log-prod58.6

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

  5. Simplified58.6

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

  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

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 
(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))))