qlog (example 3.10)

Average Error: 61.4 → 0.4

Time: 37.7s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r3892226 = 1.0;
        double r3892227 = x;
        double r3892228 = r3892226 - r3892227;
        double r3892229 = log(r3892228);
        double r3892230 = r3892226 + r3892227;
        double r3892231 = log(r3892230);
        double r3892232 = r3892229 / r3892231;
        return r3892232;
}

↓

double f(double x) {
        double r3892233 = 1.0;
        double r3892234 = log(r3892233);
        double r3892235 = 0.5;
        double r3892236 = x;
        double r3892237 = r3892236 / r3892233;
        double r3892238 = r3892237 * r3892237;
        double r3892239 = r3892236 * r3892233;
        double r3892240 = fma(r3892235, r3892238, r3892239);
        double r3892241 = r3892234 - r3892240;
        double r3892242 = -0.5;
        double r3892243 = fma(r3892233, r3892236, r3892234);
        double r3892244 = fma(r3892242, r3892238, r3892243);
        double r3892245 = r3892241 / r3892244;
        double r3892246 = cbrt(r3892245);
        double r3892247 = r3892246 * r3892246;
        double r3892248 = r3892247 * r3892246;
        return r3892248;
}

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

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

10. Applied associate-*r* 0.4

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

11. Final simplification 0.4

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

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))))