Average Error: 0.0 → 0.0
Time: 8.8s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\log \left((e^{-\log x} - 1)^*\right)\]
double f(double x) {
        double r75253 = 1.0;
        double r75254 = x;
        double r75255 = r75253 / r75254;
        double r75256 = r75255 - r75253;
        double r75257 = log(r75256);
        double r75258 = -r75257;
        return r75258;
}
↓
double f(double x) {
        double r75259 = x;
        double r75260 = log(r75259);
        double r75261 = -r75260;
        double r75262 = expm1(r75261);
        double r75263 = log(r75262);
        double r75264 = -r75263;
        return r75264;
}


-\log \left(\frac{1}{x} - 1\right)
↓
-\log \left((e^{-\log x} - 1)^*\right)

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[-\log \left(\frac{1}{x} - 1\right)\]
  2. Using strategy rm

  3. Applied add-exp-log0.0

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

  4. Applied rec-exp0.0

    \[\leadsto -\log \left(\color{blue}{e^{-\log x}} - 1\right)\]

  5. Applied expm1-def0.0

    \[\leadsto -\log \color{blue}{\left((e^{-\log x} - 1)^*\right)}\]

  6. Final simplification0.0

    \[\leadsto -\log \left((e^{-\log x} - 1)^*\right)\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1 x) 1))))