Average Error: 0.0 → 0.4
Time: 14.7s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\mathsf{fma}\left(\frac{-1}{2}, x, -1\right) \cdot x - \log x\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\left(\mathsf{fma}\left(\frac{-1}{2}, x, -1\right) \cdot x - \log x\right)
double f(double x) {
        double r279413 = 1.0;
        double r279414 = x;
        double r279415 = r279413 / r279414;
        double r279416 = r279415 - r279413;
        double r279417 = log(r279416);
        double r279418 = -r279417;
        return r279418;
}
↓
double f(double x) {
        double r279419 = -0.5;
        double r279420 = x;
        double r279421 = -1.0;
        double r279422 = fma(r279419, r279420, r279421);
        double r279423 = r279422 * r279420;
        double r279424 = log(r279420);
        double r279425 = r279423 - r279424;
        double r279426 = -r279425;
        return r279426;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1 x) 1))))