Average Error: 61.4 → 0.4
Time: 21.6s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)}^{3}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)}^{3}}
double f(double x) {
        double r73976 = 1.0;
        double r73977 = x;
        double r73978 = r73976 - r73977;
        double r73979 = log(r73978);
        double r73980 = r73976 + r73977;
        double r73981 = log(r73980);
        double r73982 = r73979 / r73981;
        return r73982;
}
↓
double f(double x) {
        double r73983 = 1.0;
        double r73984 = log(r73983);
        double r73985 = x;
        double r73986 = r73983 * r73985;
        double r73987 = 0.5;
        double r73988 = 2.0;
        double r73989 = pow(r73985, r73988);
        double r73990 = pow(r73983, r73988);
        double r73991 = r73989 / r73990;
        double r73992 = r73987 * r73991;
        double r73993 = r73986 + r73992;
        double r73994 = r73984 - r73993;
        double r73995 = r73986 + r73984;
        double r73996 = r73995 - r73992;
        double r73997 = r73994 / r73996;
        double r73998 = 3.0;
        double r73999 = pow(r73997, r73998);
        double r74000 = cbrt(r73999);
        return r74000;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.4

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Taylor expanded around 0 60.5

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube42.3

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

  6. Applied add-cbrt-cube41.7

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

  7. Applied cbrt-undiv41.7

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

  8. Simplified0.4

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

  9. Final simplification0.4

    \[\leadsto \sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)}^{3}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))