Average Error: 0.0 → 0.0
Time: 22.8s
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 r39965 = 1.0;
        double r39966 = x;
        double r39967 = r39965 / r39966;
        double r39968 = r39967 - r39965;
        double r39969 = log(r39968);
        double r39970 = -r39969;
        return r39970;
}
↓
double f(double x) {
        double r39971 = 1.0;
        double r39972 = sqrt(r39971);
        double r39973 = x;
        double r39974 = sqrt(r39973);
        double r39975 = r39972 / r39974;
        double r39976 = -r39971;
        double r39977 = fma(r39975, r39975, r39976);
        double r39978 = log(r39977);
        double r39979 = -r39978;
        return r39979;
}

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 2019322 +o rules:numerics
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))