Average Error: 61.3 → 0.5
Time: 10.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}} \cdot \frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}} \cdot \frac{\sqrt[3]{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\sqrt[3]{\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}
double f(double x) {
        double r97462 = 1.0;
        double r97463 = x;
        double r97464 = r97462 - r97463;
        double r97465 = log(r97464);
        double r97466 = r97462 + r97463;
        double r97467 = log(r97466);
        double r97468 = r97465 / r97467;
        return r97468;
}
↓
double f(double x) {
        double r97469 = 1.0;
        double r97470 = log(r97469);
        double r97471 = x;
        double r97472 = r97469 * r97471;
        double r97473 = 0.5;
        double r97474 = 2.0;
        double r97475 = pow(r97471, r97474);
        double r97476 = pow(r97469, r97474);
        double r97477 = r97475 / r97476;
        double r97478 = r97473 * r97477;
        double r97479 = r97472 + r97478;
        double r97480 = r97470 - r97479;
        double r97481 = cbrt(r97480);
        double r97482 = r97481 * r97481;
        double r97483 = r97470 - r97478;
        double r97484 = fma(r97471, r97469, r97483);
        double r97485 = cbrt(r97484);
        double r97486 = r97485 * r97485;
        double r97487 = r97482 / r97486;
        double r97488 = r97481 / r97485;
        double r97489 = r97487 * r97488;
        return r97489;
}

Error

Bits error versus x

Target

Original61.3
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.3

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

  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. Simplified60.5

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

  4. Taylor expanded around 0 0.5

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

  6. Applied add-cube-cbrt1.8

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied times-frac0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020081 +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))))