Average Error: 61.3 → 0.5
Time: 14.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)\right)
double f(double x) {
        double r109961 = 1.0;
        double r109962 = x;
        double r109963 = r109961 - r109962;
        double r109964 = log(r109963);
        double r109965 = r109961 + r109962;
        double r109966 = log(r109965);
        double r109967 = r109964 / r109966;
        return r109967;
}
↓
double f(double x) {
        double r109968 = 1.0;
        double r109969 = log(r109968);
        double r109970 = x;
        double r109971 = 0.5;
        double r109972 = 2.0;
        double r109973 = pow(r109970, r109972);
        double r109974 = pow(r109968, r109972);
        double r109975 = r109973 / r109974;
        double r109976 = r109971 * r109975;
        double r109977 = fma(r109968, r109970, r109976);
        double r109978 = r109969 - r109977;
        double r109979 = fma(r109968, r109970, r109969);
        double r109980 = r109979 - r109976;
        double r109981 = r109978 / r109980;
        double r109982 = expm1(r109981);
        double r109983 = log1p(r109982);
        return r109983;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 61.3

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

  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. Simplified60.5

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  7. Applied log1p-expm1-u0.5

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)\right)}\]

  8. Final simplification0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\right)\right)\]

Reproduce

herbie shell --seed 2020045 +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))))