Average Error: 61.5 → 0.4
Time: 12.5s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \mathsf{fma}\left(1, x, \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\mathsf{fma}\left(1, x, \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}
double f(double x) {
        double r77384 = 1.0;
        double r77385 = x;
        double r77386 = r77384 - r77385;
        double r77387 = log(r77386);
        double r77388 = r77384 + r77385;
        double r77389 = log(r77388);
        double r77390 = r77387 / r77389;
        return r77390;
}
↓
double f(double x) {
        double r77391 = 1.0;
        double r77392 = log(r77391);
        double r77393 = x;
        double r77394 = 0.5;
        double r77395 = 2.0;
        double r77396 = pow(r77393, r77395);
        double r77397 = pow(r77391, r77395);
        double r77398 = r77396 / r77397;
        double r77399 = r77394 * r77398;
        double r77400 = fma(r77391, r77393, r77399);
        double r77401 = r77392 - r77400;
        double r77402 = fma(r77391, r77393, r77392);
        double r77403 = r77402 - r77399;
        double r77404 = r77401 / r77403;
        return r77404;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 61.5

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

  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

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +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))))