Average Error: 61.5 → 0.4
Time: 15.5s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)}{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)}{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}
double f(double x) {
        double r72093 = 1.0;
        double r72094 = x;
        double r72095 = r72093 - r72094;
        double r72096 = log(r72095);
        double r72097 = r72093 + r72094;
        double r72098 = log(r72097);
        double r72099 = r72096 / r72098;
        return r72099;
}
↓
double f(double x) {
        double r72100 = 1.0;
        double r72101 = x;
        double r72102 = 2.0;
        double r72103 = pow(r72101, r72102);
        double r72104 = 1.0;
        double r72105 = pow(r72104, r72102);
        double r72106 = r72103 / r72105;
        double r72107 = -0.5;
        double r72108 = log(r72104);
        double r72109 = fma(r72104, r72101, r72108);
        double r72110 = fma(r72106, r72107, r72109);
        double r72111 = 0.5;
        double r72112 = r72111 * r72106;
        double r72113 = fma(r72104, r72101, r72112);
        double r72114 = r72108 - r72113;
        double r72115 = r72110 / r72114;
        double r72116 = r72100 / r72115;
        return r72116;
}

Error

Bits error versus x

Target

Original61.5
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 61.5

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

  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

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied clear-num0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 +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)) (* 0.416666666666666685 (pow x 3))))

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