Average Error: 61.6 → 0.4
Time: 19.6s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(x \cdot 1 + \frac{x}{1} \cdot \left(\frac{x}{1} \cdot \frac{1}{2}\right)\right)}{\frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + \left(x \cdot 1 + \log 1\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(x \cdot 1 + \frac{x}{1} \cdot \left(\frac{x}{1} \cdot \frac{1}{2}\right)\right)}{\frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + \left(x \cdot 1 + \log 1\right)}
double f(double x) {
        double r4207989 = 1.0;
        double r4207990 = x;
        double r4207991 = r4207989 - r4207990;
        double r4207992 = log(r4207991);
        double r4207993 = r4207989 + r4207990;
        double r4207994 = log(r4207993);
        double r4207995 = r4207992 / r4207994;
        return r4207995;
}
↓
double f(double x) {
        double r4207996 = 1.0;
        double r4207997 = log(r4207996);
        double r4207998 = x;
        double r4207999 = r4207998 * r4207996;
        double r4208000 = r4207998 / r4207996;
        double r4208001 = 0.5;
        double r4208002 = r4208000 * r4208001;
        double r4208003 = r4208000 * r4208002;
        double r4208004 = r4207999 + r4208003;
        double r4208005 = r4207997 - r4208004;
        double r4208006 = -0.5;
        double r4208007 = r4208000 * r4208000;
        double r4208008 = r4208006 * r4208007;
        double r4208009 = r4207999 + r4207997;
        double r4208010 = r4208008 + r4208009;
        double r4208011 = r4208005 / r4208010;
        return r4208011;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.6
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 61.6

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

  2. Taylor expanded around 0 60.5

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

  3. Simplified60.5

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 
(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))))