Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Average Error: 12.0 → 6.5

Time: 4.4s

Precision: 64

Math

\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.57127147346290313 \cdot 10^{197}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r499250 = x;
        double r499251 = y;
        double r499252 = 2.0;
        double r499253 = r499251 * r499252;
        double r499254 = z;
        double r499255 = r499253 * r499254;
        double r499256 = r499254 * r499252;
        double r499257 = r499256 * r499254;
        double r499258 = t;
        double r499259 = r499251 * r499258;
        double r499260 = r499257 - r499259;
        double r499261 = r499255 / r499260;
        double r499262 = r499250 - r499261;
        return r499262;
}

↓

double f(double x, double y, double z, double t) {
        double r499263 = z;
        double r499264 = -2.571271473462903e+197;
        bool r499265 = r499263 <= r499264;
        double r499266 = x;
        double r499267 = y;
        double r499268 = 2.0;
        double r499269 = r499267 * r499268;
        double r499270 = 0.0;
        double r499271 = r499269 * r499270;
        double r499272 = r499266 - r499271;
        double r499273 = cbrt(r499263);
        double r499274 = r499273 * r499273;
        double r499275 = 2.0;
        double r499276 = pow(r499263, r499275);
        double r499277 = r499268 * r499276;
        double r499278 = t;
        double r499279 = r499278 * r499267;
        double r499280 = r499277 - r499279;
        double r499281 = cbrt(r499280);
        double r499282 = r499281 * r499281;
        double r499283 = r499274 / r499282;
        double r499284 = r499269 * r499283;
        double r499285 = r499273 / r499281;
        double r499286 = r499284 * r499285;
        double r499287 = r499266 - r499286;
        double r499288 = r499265 ? r499272 : r499287;
        return r499288;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	12.0
Target	0.1
Herbie	6.5

\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.571271473462903e+197

   1. Initial program 27.9

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 27.9

      \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{1 \cdot \left(\left(z \cdot 2\right) \cdot z - y \cdot t\right)}}\]

   4. Applied times-frac 14.4

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{1} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}}\]

   5. Simplified 14.4

      \[\leadsto x - \color{blue}{\left(y \cdot 2\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

   6. Simplified 14.4

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\frac{z}{2 \cdot {z}^{2} - t \cdot y}}\]

   7. Taylor expanded around 0 12.0

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{0}\]

   if -2.571271473462903e+197 < z

   1. Initial program 10.5

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 10.5

      \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{1 \cdot \left(\left(z \cdot 2\right) \cdot z - y \cdot t\right)}}\]

   4. Applied times-frac 6.2

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{1} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}}\]

   5. Simplified 6.2

      \[\leadsto x - \color{blue}{\left(y \cdot 2\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

   6. Simplified 6.2

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\frac{z}{2 \cdot {z}^{2} - t \cdot y}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 6.4

      \[\leadsto x - \left(y \cdot 2\right) \cdot \frac{z}{\color{blue}{\left(\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}\right) \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}}\]

   9. Applied add-cube-cbrt 6.5

      \[\leadsto x - \left(y \cdot 2\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}\right) \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\]

   10. Applied times-frac 6.5

       \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right)}\]

   11. Applied associate-*r* 6.0

       \[\leadsto x - \color{blue}{\left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.57127147346290313 \cdot 10^{197}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))