qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 24.0s

Precision: 64

\[-1 \lt x \land x \lt 1\]

Math

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

↓

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

C

double f(double x) {
        double r4109906 = 1.0;
        double r4109907 = x;
        double r4109908 = r4109906 - r4109907;
        double r4109909 = log(r4109908);
        double r4109910 = r4109906 + r4109907;
        double r4109911 = log(r4109910);
        double r4109912 = r4109909 / r4109911;
        return r4109912;
}

↓

double f(double x) {
        double r4109913 = 1.0;
        double r4109914 = log(r4109913);
        double r4109915 = x;
        double r4109916 = r4109915 / r4109913;
        double r4109917 = r4109916 * r4109916;
        double r4109918 = 0.5;
        double r4109919 = r4109917 * r4109918;
        double r4109920 = r4109914 - r4109919;
        double r4109921 = r4109913 * r4109915;
        double r4109922 = r4109920 - r4109921;
        double r4109923 = -0.5;
        double r4109924 = r4109923 * r4109917;
        double r4109925 = r4109914 + r4109921;
        double r4109926 = r4109924 + r4109925;
        double r4109927 = r4109922 / r4109926;
        double r4109928 = r4109927 * r4109927;
        double r4109929 = cbrt(r4109928);
        double r4109930 = r4109929 * r4109929;
        double r4109931 = r4109930 * r4109929;
        double r4109932 = r4109927 * r4109931;
        double r4109933 = cbrt(r4109932);
        return r4109933;
}

Error

Bits error versus x

Target

Original	61.4
Target	0.3
Herbie	0.4

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

1. Initial program 61.4

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

2. Taylor expanded around 0 60.5

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

3. Simplified 60.5

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

7. Applied add-cbrt-cube 43.2

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

8. Applied add-cbrt-cube 42.6

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

9. Applied cbrt-undiv 42.6

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

10. Simplified 0.4

    \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\log 1 - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) - 1 \cdot x}{\frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + \left(\log 1 + 1 \cdot x\right)} \cdot \left(\frac{\left(\log 1 - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) - 1 \cdot x}{\frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + \left(\log 1 + 1 \cdot x\right)} \cdot \frac{\left(\log 1 - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) - 1 \cdot x}{\frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + \left(\log 1 + 1 \cdot x\right)}\right)}}\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.4

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

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))