Average Error: 61.4 → 0.5
Time: 9.1s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}} \cdot \frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}} \cdot \frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}
double f(double x) {
        double r105924 = 1.0;
        double r105925 = x;
        double r105926 = r105924 - r105925;
        double r105927 = log(r105926);
        double r105928 = r105924 + r105925;
        double r105929 = log(r105928);
        double r105930 = r105927 / r105929;
        return r105930;
}
↓
double f(double x) {
        double r105931 = 1.0;
        double r105932 = log(r105931);
        double r105933 = x;
        double r105934 = r105931 * r105933;
        double r105935 = 0.5;
        double r105936 = 2.0;
        double r105937 = pow(r105933, r105936);
        double r105938 = pow(r105931, r105936);
        double r105939 = r105937 / r105938;
        double r105940 = r105935 * r105939;
        double r105941 = r105934 + r105940;
        double r105942 = r105932 - r105941;
        double r105943 = cbrt(r105942);
        double r105944 = r105943 * r105943;
        double r105945 = r105932 - r105940;
        double r105946 = fma(r105933, r105931, r105945);
        double r105947 = cbrt(r105946);
        double r105948 = r105947 * r105947;
        double r105949 = r105944 / r105948;
        double r105950 = r105943 / r105947;
        double r105951 = r105949 * r105950;
        return r105951;
}

Error

Bits error versus x

Target

Original61.4
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.4

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

  2. Taylor expanded around 0 60.6

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

  3. Simplified60.6

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

  4. Taylor expanded around 0 0.5

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

  6. Applied add-cube-cbrt1.8

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied times-frac0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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))))