neg log

Average Error: 0.0 → 0.0

Time: 4.3s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r29783 = 1.0;
        double r29784 = x;
        double r29785 = r29783 / r29784;
        double r29786 = r29785 - r29783;
        double r29787 = log(r29786);
        double r29788 = -r29787;
        return r29788;
}

↓

double f(double x) {
        double r29789 = 1.0;
        double r29790 = x;
        double r29791 = r29789 / r29790;
        double r29792 = sqrt(r29791);
        double r29793 = -r29789;
        double r29794 = fma(r29792, r29792, r29793);
        double r29795 = log(r29794);
        double r29796 = -r29795;
        return r29796;
}

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-sqrt 0.0

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

4. Applied fma-neg 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))