Average Error: 0.0 → 0.0
Time: 17.2s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}}, \frac{\sqrt{1}}{\sqrt{x}}, -1\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}}, \frac{\sqrt{1}}{\sqrt{x}}, -1\right)\right)
double f(double x) {
        double r396233 = 1.0;
        double r396234 = x;
        double r396235 = r396233 / r396234;
        double r396236 = r396235 - r396233;
        double r396237 = log(r396236);
        double r396238 = -r396237;
        return r396238;
}
↓
double f(double x) {
        double r396239 = 1.0;
        double r396240 = sqrt(r396239);
        double r396241 = x;
        double r396242 = sqrt(r396241);
        double r396243 = r396240 / r396242;
        double r396244 = -r396239;
        double r396245 = fma(r396243, r396243, r396244);
        double r396246 = log(r396245);
        double r396247 = -r396246;
        return r396247;
}

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-sqr-sqrt0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied fma-neg0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1.0 x) 1.0))))