qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 1.2m

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r4111269 = 1.0;
        double r4111270 = x;
        double r4111271 = r4111269 - r4111270;
        double r4111272 = log(r4111271);
        double r4111273 = r4111269 + r4111270;
        double r4111274 = log(r4111273);
        double r4111275 = r4111272 / r4111274;
        return r4111275;
}

↓

double f(double x) {
        double r4111276 = 1.0;
        double r4111277 = log(r4111276);
        double r4111278 = 0.5;
        double r4111279 = x;
        double r4111280 = r4111279 / r4111276;
        double r4111281 = r4111280 * r4111280;
        double r4111282 = r4111279 * r4111276;
        double r4111283 = fma(r4111278, r4111281, r4111282);
        double r4111284 = r4111277 - r4111283;
        double r4111285 = -0.5;
        double r4111286 = fma(r4111276, r4111279, r4111277);
        double r4111287 = fma(r4111285, r4111281, r4111286);
        double r4111288 = r4111284 / r4111287;
        double r4111289 = cbrt(r4111288);
        double r4111290 = r4111289 * r4111289;
        double r4111291 = r4111289 * r4111290;
        return r4111291;
}

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

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

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

7. Applied add-cube-cbrt 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))