neg log

Average Error: 0.0 → 0.0

Time: 4.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r33482 = 1.0;
        double r33483 = x;
        double r33484 = r33482 / r33483;
        double r33485 = r33484 - r33482;
        double r33486 = log(r33485);
        double r33487 = -r33486;
        return r33487;
}

↓

double f(double x) {
        double r33488 = 1.0;
        double r33489 = sqrt(r33488);
        double r33490 = x;
        double r33491 = sqrt(r33490);
        double r33492 = r33489 / r33491;
        double r33493 = r33492 + r33489;
        double r33494 = sqrt(r33493);
        double r33495 = r33492 - r33489;
        double r33496 = sqrt(r33495);
        double r33497 = r33494 * r33496;
        double r33498 = log(r33497);
        double r33499 = r33488 / r33490;
        double r33500 = r33499 - r33488;
        double r33501 = sqrt(r33500);
        double r33502 = log(r33501);
        double r33503 = r33498 + r33502;
        double r33504 = -r33503;
        return r33504;
}

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

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

4. Applied log-prod 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied add-sqr-sqrt 0.0

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

8. Applied add-sqr-sqrt 0.0

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

9. Applied times-frac 0.0

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

10. Applied difference-of-squares 0.0

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

11. Applied sqrt-prod 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))