Average Error: 61.3 → 0.4
Time: 13.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot x + \left(\frac{1}{2} \cdot \frac{x}{1}\right) \cdot \frac{x}{{\left(\sqrt{1}\right)}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(1 \cdot x + \left(\frac{1}{2} \cdot \frac{x}{1}\right) \cdot \frac{x}{{\left(\sqrt{1}\right)}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}
double f(double x) {
        double r93421 = 1.0;
        double r93422 = x;
        double r93423 = r93421 - r93422;
        double r93424 = log(r93423);
        double r93425 = r93421 + r93422;
        double r93426 = log(r93425);
        double r93427 = r93424 / r93426;
        return r93427;
}
↓
double f(double x) {
        double r93428 = 1.0;
        double r93429 = log(r93428);
        double r93430 = x;
        double r93431 = r93428 * r93430;
        double r93432 = 0.5;
        double r93433 = r93430 / r93428;
        double r93434 = r93432 * r93433;
        double r93435 = sqrt(r93428);
        double r93436 = 2.0;
        double r93437 = pow(r93435, r93436);
        double r93438 = r93430 / r93437;
        double r93439 = r93434 * r93438;
        double r93440 = r93431 + r93439;
        double r93441 = r93429 - r93440;
        double r93442 = r93431 + r93429;
        double r93443 = pow(r93430, r93436);
        double r93444 = pow(r93428, r93436);
        double r93445 = r93443 / r93444;
        double r93446 = r93432 * r93445;
        double r93447 = r93442 - r93446;
        double r93448 = r93441 / r93447;
        return r93448;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
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 61.3

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

  2. Taylor expanded around 0 60.4

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

  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-sqr-sqrt0.4

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

  6. Applied unpow-prod-down0.4

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

  7. Applied unpow20.4

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

  8. Applied times-frac0.4

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

  9. Applied associate-*r*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019351 
(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))))