qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 20.6s

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 r68175 = 1.0;
        double r68176 = x;
        double r68177 = r68175 - r68176;
        double r68178 = log(r68177);
        double r68179 = r68175 + r68176;
        double r68180 = log(r68179);
        double r68181 = r68178 / r68180;
        return r68181;
}

↓

double f(double x) {
        double r68182 = 1.0;
        double r68183 = log(r68182);
        double r68184 = x;
        double r68185 = 0.5;
        double r68186 = 2.0;
        double r68187 = pow(r68184, r68186);
        double r68188 = pow(r68182, r68186);
        double r68189 = r68187 / r68188;
        double r68190 = r68185 * r68189;
        double r68191 = fma(r68182, r68184, r68190);
        double r68192 = r68183 - r68191;
        double r68193 = -0.5;
        double r68194 = fma(r68182, r68184, r68183);
        double r68195 = fma(r68193, r68189, r68194);
        double r68196 = r68192 / r68195;
        double r68197 = 3.0;
        double r68198 = pow(r68196, r68197);
        double r68199 = cbrt(r68198);
        return r68199;
}

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))))