neg log

Average Error: 0.0 → 0.0

Time: 14.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r38808 = 1.0;
        double r38809 = x;
        double r38810 = r38808 / r38809;
        double r38811 = r38810 - r38808;
        double r38812 = log(r38811);
        double r38813 = -r38812;
        return r38813;
}

↓

double f(double x) {
        double r38814 = 1.0;
        double r38815 = sqrt(r38814);
        double r38816 = x;
        double r38817 = sqrt(r38816);
        double r38818 = r38815 / r38817;
        double r38819 = sqrt(r38818);
        double r38820 = -r38815;
        double r38821 = fma(r38819, r38819, r38820);
        double r38822 = log(r38821);
        double r38823 = r38815 + r38818;
        double r38824 = log(r38823);
        double r38825 = r38822 + r38824;
        double r38826 = -r38825;
        return r38826;
}

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

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

4. Applied add-sqr-sqrt 0.0

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

5. Applied add-sqr-sqrt 0.0

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

6. Applied times-frac 0.1

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

7. Applied difference-of-squares 0.1

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

8. Applied log-prod 0.0

   \[\leadsto -\color{blue}{\left(\log \left(\frac{\sqrt{1}}{\sqrt{x}} + \sqrt{1}\right) + \log \left(\frac{\sqrt{1}}{\sqrt{x}} - \sqrt{1}\right)\right)}\]
9. Using strategy rm

10. Applied add-sqr-sqrt 0.0

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

11. Applied fma-neg 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1.0 x) 1.0))))