Average Error: 58.7 → 0.2
Time: 5.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r90141 = 1.0;
        double r90142 = eps;
        double r90143 = r90141 - r90142;
        double r90144 = r90141 + r90142;
        double r90145 = r90143 / r90144;
        double r90146 = log(r90145);
        return r90146;
}
↓
double f(double eps) {
        double r90147 = 0.6666666666666666;
        double r90148 = eps;
        double r90149 = 3.0;
        double r90150 = pow(r90148, r90149);
        double r90151 = 1.0;
        double r90152 = pow(r90151, r90149);
        double r90153 = r90150 / r90152;
        double r90154 = r90147 * r90153;
        double r90155 = 0.4;
        double r90156 = 5.0;
        double r90157 = pow(r90148, r90156);
        double r90158 = pow(r90151, r90156);
        double r90159 = r90157 / r90158;
        double r90160 = r90155 * r90159;
        double r90161 = r90154 + r90160;
        double r90162 = 2.0;
        double r90163 = r90162 * r90148;
        double r90164 = r90161 + r90163;
        double r90165 = -r90164;
        return r90165;
}

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-div58.6

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

  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. Using strategy rm

  6. Applied associate-+r+0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020002 
(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))))