Average Error: 61.7 → 0.4
Time: 14.3s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)
double f(double x) {
        double r105597 = 1.0;
        double r105598 = x;
        double r105599 = r105597 - r105598;
        double r105600 = log(r105599);
        double r105601 = r105597 + r105598;
        double r105602 = log(r105601);
        double r105603 = r105600 / r105602;
        return r105603;
}
↓
double f(double x) {
        double r105604 = 1.0;
        double r105605 = log(r105604);
        double r105606 = x;
        double r105607 = r105604 * r105606;
        double r105608 = 0.5;
        double r105609 = 2.0;
        double r105610 = pow(r105606, r105609);
        double r105611 = pow(r105604, r105609);
        double r105612 = r105610 / r105611;
        double r105613 = r105608 * r105612;
        double r105614 = r105607 + r105613;
        double r105615 = r105605 - r105614;
        double r105616 = r105607 + r105605;
        double r105617 = r105616 - r105613;
        double r105618 = r105615 / r105617;
        double r105619 = exp(r105618);
        double r105620 = log(r105619);
        return r105620;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.7
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.7

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

  2. Taylor expanded around 0 60.7

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

  3. Taylor expanded around 0 0.4

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

  5. Applied add-log-exp0.4

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

  6. Final simplification0.4

    \[\leadsto \log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)\]

Reproduce

herbie shell --seed 2019350 
(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))))