qlog (example 3.10)

Average Error: 61.4 → 0.5

Time: 12.4s

Precision: binary64

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

Math

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

↓

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

C

double code(double x) {
	return ((double) (((double) log(((double) (1.0 - x)))) / ((double) log(((double) (1.0 + x))))));
}

↓

double code(double x) {
	return ((double) (((double) cbrt(((double) (((double) (((double) log(1.0)) - ((double) (x * ((double) (1.0 + ((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))))))))) / ((double) (((double) log(1.0)) - ((double) (x * ((double) (((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))) - 1.0)))))))))) * ((double) (((double) cbrt(((double) (((double) (((double) log(1.0)) - ((double) (x * ((double) (1.0 + ((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))))))))) / ((double) (((double) log(1.0)) - ((double) (x * ((double) (((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))) - 1.0)))))))))) * ((double) cbrt(((double) (((double) (((double) log(1.0)) - ((double) (x * ((double) (1.0 + ((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))))))))) / ((double) (((double) log(1.0)) - ((double) (x * ((double) (((double) (((double) (0.5 / 1.0)) * ((double) (x / 1.0)))) - 1.0))))))))))))));
}

Error

Bits error versus x

Target

Original	61.4
Target	0.3
Herbie	0.5

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

1. Initial program 61.4

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

2. Taylor expanded around 0 60.5

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

3. Simplified 60.5

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

7. Applied add-cube-cbrt 0.5

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

8. Simplified 0.5

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

9. Simplified 0.5

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

10. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

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

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