Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.3

Time: 1.9m

Precision: 64

• could not determine a ground truth for program body

  1. wj = -4.662649594744133e+254
  2. x = 1.0857985921417446e-197

Math

\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -4.1691209072271064 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right)\\ \mathbf{elif}\;wj \le 6.696494124738262 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj} \cdot \frac{1}{wj + 1}\right), \left(wj + -1\right), wj\right)\\ \end{array}\]

C

double f(double wj, double x) {
        double r45589275 = wj;
        double r45589276 = exp(r45589275);
        double r45589277 = r45589275 * r45589276;
        double r45589278 = x;
        double r45589279 = r45589277 - r45589278;
        double r45589280 = r45589276 + r45589277;
        double r45589281 = r45589279 / r45589280;
        double r45589282 = r45589275 - r45589281;
        return r45589282;
}

↓

double f(double wj, double x) {
        double r45589283 = wj;
        double r45589284 = -4.1691209072271064e-09;
        bool r45589285 = r45589283 <= r45589284;
        double r45589286 = x;
        double r45589287 = exp(r45589283);
        double r45589288 = r45589286 / r45589287;
        double r45589289 = r45589283 - r45589288;
        double r45589290 = 1.0;
        double r45589291 = r45589283 * r45589283;
        double r45589292 = r45589290 - r45589291;
        double r45589293 = r45589289 / r45589292;
        double r45589294 = -1.0;
        double r45589295 = r45589283 + r45589294;
        double r45589296 = fma(r45589293, r45589295, r45589283);
        double r45589297 = 6.696494124738262e-09;
        bool r45589298 = r45589283 <= r45589297;
        double r45589299 = -2.0;
        double r45589300 = fma(r45589286, r45589299, r45589283);
        double r45589301 = fma(r45589300, r45589283, r45589286);
        double r45589302 = r45589290 - r45589283;
        double r45589303 = r45589289 / r45589302;
        double r45589304 = r45589283 + r45589290;
        double r45589305 = r45589290 / r45589304;
        double r45589306 = r45589303 * r45589305;
        double r45589307 = fma(r45589306, r45589295, r45589283);
        double r45589308 = r45589298 ? r45589301 : r45589307;
        double r45589309 = r45589285 ? r45589296 : r45589308;
        return r45589309;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.8
Target	13.0
Herbie	0.3

\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

1. Split input into 3 regimes

2. if wj < -4.1691209072271064e-09

   1. Initial program 4.7

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 40.7

      \[\leadsto wj - \color{blue}{\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}}\]

   4. Applied add-cube-cbrt 40.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}} - \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\]

   5. Applied prod-diff 40.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right), \left(\sqrt[3]{wj}\right), \left(-\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right)\right) + \mathsf{fma}\left(\left(-\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right), \left(\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right), \left(\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right)\right)}\]

   6. Simplified 40.6

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

   7. Simplified 4.4

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{1 + wj}\right) + \color{blue}{0}\]
   8. Using strategy rm

   9. Applied flip-+ 4.8

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{\color{blue}{\frac{1 \cdot 1 - wj \cdot wj}{1 - wj}}}\right) + 0\]

   10. Applied associate-/r/ 4.9

       \[\leadsto \left(wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj} \cdot \left(1 - wj\right)}\right) + 0\]

   11. Applied add-cube-cbrt 5.7

       \[\leadsto \left(\color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}} - \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj} \cdot \left(1 - wj\right)\right) + 0\]

   12. Applied prod-diff 5.6

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right), \left(\sqrt[3]{wj}\right), \left(-\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right) + \mathsf{fma}\left(\left(-\left(1 - wj\right)\right), \left(\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right), \left(\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right)\right)} + 0\]

   13. Simplified 4.9

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right)} + \mathsf{fma}\left(\left(-\left(1 - wj\right)\right), \left(\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right), \left(\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right)\right) + 0\]

   14. Simplified 4.8

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right) + \color{blue}{0}\right) + 0\]

   if -4.1691209072271064e-09 < wj < 6.696494124738262e-09

   1. Initial program 13.5

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)}\]

   if 6.696494124738262e-09 < wj

   1. Initial program 29.5

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 37.6

      \[\leadsto wj - \color{blue}{\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}}\]

   4. Applied add-cube-cbrt 38.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}} - \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\]

   5. Applied prod-diff 38.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right), \left(\sqrt[3]{wj}\right), \left(-\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right)\right) + \mathsf{fma}\left(\left(-\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right), \left(\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right), \left(\sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \cdot \sqrt{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}}\right)\right)}\]

   6. Simplified 37.4

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

   7. Simplified 3.0

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{1 + wj}\right) + \color{blue}{0}\]
   8. Using strategy rm

   9. Applied flip-+ 3.1

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{\color{blue}{\frac{1 \cdot 1 - wj \cdot wj}{1 - wj}}}\right) + 0\]

   10. Applied associate-/r/ 3.0

       \[\leadsto \left(wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj} \cdot \left(1 - wj\right)}\right) + 0\]

   11. Applied add-cube-cbrt 4.2

       \[\leadsto \left(\color{blue}{\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right) \cdot \sqrt[3]{wj}} - \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj} \cdot \left(1 - wj\right)\right) + 0\]

   12. Applied prod-diff 4.2

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\sqrt[3]{wj} \cdot \sqrt[3]{wj}\right), \left(\sqrt[3]{wj}\right), \left(-\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right) + \mathsf{fma}\left(\left(-\left(1 - wj\right)\right), \left(\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right), \left(\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right)\right)} + 0\]

   13. Simplified 3.0

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right)} + \mathsf{fma}\left(\left(-\left(1 - wj\right)\right), \left(\frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right), \left(\left(1 - wj\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 \cdot 1 - wj \cdot wj}\right)\right)\right) + 0\]

   14. Simplified 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right) + \color{blue}{0}\right) + 0\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{\color{blue}{1 \cdot 1} - wj \cdot wj}\right), \left(wj + -1\right), wj\right) + 0\right) + 0\]

   17. Applied difference-of-squares 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{\color{blue}{\left(1 + wj\right) \cdot \left(1 - wj\right)}}\right), \left(wj + -1\right), wj\right) + 0\right) + 0\]

   18. Applied *-un-lft-identity 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{wj - \color{blue}{1 \cdot \frac{x}{e^{wj}}}}{\left(1 + wj\right) \cdot \left(1 - wj\right)}\right), \left(wj + -1\right), wj\right) + 0\right) + 0\]

   19. Applied *-un-lft-identity 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{\color{blue}{1 \cdot wj} - 1 \cdot \frac{x}{e^{wj}}}{\left(1 + wj\right) \cdot \left(1 - wj\right)}\right), \left(wj + -1\right), wj\right) + 0\right) + 0\]

   20. Applied distribute-lft-out-- 3.0

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{\color{blue}{1 \cdot \left(wj - \frac{x}{e^{wj}}\right)}}{\left(1 + wj\right) \cdot \left(1 - wj\right)}\right), \left(wj + -1\right), wj\right) + 0\right) + 0\]

   21. Applied times-frac 3.2

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1 + wj} \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - wj}\right)}, \left(wj + -1\right), wj\right) + 0\right) + 0\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -4.1691209072271064 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\right), \left(wj + -1\right), wj\right)\\ \mathbf{elif}\;wj \le 6.696494124738262 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{1 - wj} \cdot \frac{1}{wj + 1}\right), \left(wj + -1\right), wj\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))