qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 21.1s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r67902 = 1.0;
        double r67903 = x;
        double r67904 = r67902 - r67903;
        double r67905 = log(r67904);
        double r67906 = r67902 + r67903;
        double r67907 = log(r67906);
        double r67908 = r67905 / r67907;
        return r67908;
}

↓

double f(double x) {
        double r67909 = 1.0;
        double r67910 = log(r67909);
        double r67911 = x;
        double r67912 = 0.5;
        double r67913 = 2.0;
        double r67914 = pow(r67911, r67913);
        double r67915 = pow(r67909, r67913);
        double r67916 = r67914 / r67915;
        double r67917 = r67912 * r67916;
        double r67918 = fma(r67909, r67911, r67917);
        double r67919 = r67910 - r67918;
        double r67920 = -0.5;
        double r67921 = fma(r67909, r67911, r67910);
        double r67922 = fma(r67920, r67916, r67921);
        double r67923 = r67919 / r67922;
        double r67924 = 3.0;
        double r67925 = pow(r67923, r67924);
        double r67926 = cbrt(r67925);
        return r67926;
}

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

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

3. Simplified 60.5

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

7. Applied add-cbrt-cube 42.3

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

8. Applied add-cbrt-cube 41.7

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

9. Applied cbrt-undiv 41.7

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

10. Simplified 0.4

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

11. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019323 +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))))