Average Error: 61.4 → 0.4
Time: 1.2m
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot x + \left(\frac{1}{2} \cdot \frac{x}{1}\right) \cdot \frac{x}{1}\right)}{\left(1 \cdot x + \log 1\right) + \left(\frac{x}{1} \cdot \frac{-1}{2}\right) \cdot \frac{x}{1}}\]
\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}{1}\right)}{\left(1 \cdot x + \log 1\right) + \left(\frac{x}{1} \cdot \frac{-1}{2}\right) \cdot \frac{x}{1}}
double f(double x) {
        double r4816485 = 1.0;
        double r4816486 = x;
        double r4816487 = r4816485 - r4816486;
        double r4816488 = log(r4816487);
        double r4816489 = r4816485 + r4816486;
        double r4816490 = log(r4816489);
        double r4816491 = r4816488 / r4816490;
        return r4816491;
}
↓
double f(double x) {
        double r4816492 = 1.0;
        double r4816493 = log(r4816492);
        double r4816494 = x;
        double r4816495 = r4816492 * r4816494;
        double r4816496 = 0.5;
        double r4816497 = r4816494 / r4816492;
        double r4816498 = r4816496 * r4816497;
        double r4816499 = r4816498 * r4816497;
        double r4816500 = r4816495 + r4816499;
        double r4816501 = r4816493 - r4816500;
        double r4816502 = r4816495 + r4816493;
        double r4816503 = -0.5;
        double r4816504 = r4816497 * r4816503;
        double r4816505 = r4816504 * r4816497;
        double r4816506 = r4816502 + r4816505;
        double r4816507 = r4816501 / r4816506;
        return r4816507;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
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 61.4

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

  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. Simplified60.5

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied associate-*r*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 
(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))))