Average Error: 0.0 → 0.5
Time: 6.6s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[x \cdot \left(0.5 \cdot \frac{x}{{1}^{2}} + 1\right) + \left(\log x - \log 1\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
x \cdot \left(0.5 \cdot \frac{x}{{1}^{2}} + 1\right) + \left(\log x - \log 1\right)
double f(double x) {
        double r11313 = 1.0;
        double r11314 = x;
        double r11315 = r11313 / r11314;
        double r11316 = r11315 - r11313;
        double r11317 = log(r11316);
        double r11318 = -r11317;
        return r11318;
}
↓
double f(double x) {
        double r11319 = x;
        double r11320 = 0.5;
        double r11321 = 1.0;
        double r11322 = 2.0;
        double r11323 = pow(r11321, r11322);
        double r11324 = r11319 / r11323;
        double r11325 = r11320 * r11324;
        double r11326 = r11325 + r11321;
        double r11327 = r11319 * r11326;
        double r11328 = log(r11319);
        double r11329 = log(r11321);
        double r11330 = r11328 - r11329;
        double r11331 = r11327 + r11330;
        return r11331;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.5

    \[\leadsto -\color{blue}{\left(\log 1 - \left(\log x + \left(1 \cdot x + 0.5 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)\right)}\]

  3. Final simplification0.5

    \[\leadsto x \cdot \left(0.5 \cdot \frac{x}{{1}^{2}} + 1\right) + \left(\log x - \log 1\right)\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))