Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 25.2 → 6.3

Time: 51.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.611098198946549 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 3.1007095082743632 \cdot 10^{95}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{\frac{\frac{\frac{\left|\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right|}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\sqrt[3]{y}}} \cdot \frac{x}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r476247 = x;
        double r476248 = y;
        double r476249 = r476247 * r476248;
        double r476250 = z;
        double r476251 = r476249 * r476250;
        double r476252 = r476250 * r476250;
        double r476253 = t;
        double r476254 = a;
        double r476255 = r476253 * r476254;
        double r476256 = r476252 - r476255;
        double r476257 = sqrt(r476256);
        double r476258 = r476251 / r476257;
        return r476258;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r476259 = z;
        double r476260 = -4.611098198946549e+108;
        bool r476261 = r476259 <= r476260;
        double r476262 = -1.0;
        double r476263 = x;
        double r476264 = y;
        double r476265 = r476263 * r476264;
        double r476266 = r476262 * r476265;
        double r476267 = 3.100709508274363e+95;
        bool r476268 = r476259 <= r476267;
        double r476269 = cbrt(r476264);
        double r476270 = r476259 * r476259;
        double r476271 = t;
        double r476272 = a;
        double r476273 = r476271 * r476272;
        double r476274 = r476270 - r476273;
        double r476275 = sqrt(r476274);
        double r476276 = cbrt(r476275);
        double r476277 = fabs(r476276);
        double r476278 = cbrt(r476259);
        double r476279 = r476277 / r476278;
        double r476280 = r476279 / r476278;
        double r476281 = r476280 / r476269;
        double r476282 = r476269 / r476281;
        double r476283 = sqrt(r476275);
        double r476284 = r476263 / r476283;
        double r476285 = r476282 * r476284;
        double r476286 = sqrt(r476276);
        double r476287 = r476286 / r476278;
        double r476288 = r476269 / r476287;
        double r476289 = r476285 * r476288;
        double r476290 = r476268 ? r476289 : r476265;
        double r476291 = r476261 ? r476266 : r476290;
        return r476291;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.2
Target	7.9
Herbie	6.3

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -4.611098198946549e+108

   1. Initial program 44.9

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

   2. Taylor expanded around -inf 2.4

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

   if -4.611098198946549e+108 < z < 3.100709508274363e+95

   1. Initial program 11.9

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

   3. Applied associate-/l* 10.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 10.0

      \[\leadsto \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{\color{blue}{1 \cdot z}}}\]

   6. Applied add-sqr-sqrt 10.0

      \[\leadsto \frac{x \cdot y}{\frac{\sqrt{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}}}{1 \cdot z}}\]

   7. Applied sqrt-prod 10.3

      \[\leadsto \frac{x \cdot y}{\frac{\color{blue}{\sqrt{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt{z \cdot z - t \cdot a}}}}{1 \cdot z}}\]

   8. Applied times-frac 10.2

      \[\leadsto \frac{x \cdot y}{\color{blue}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{1} \cdot \frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{z}}}\]

   9. Applied times-frac 11.2

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{1}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{z}}}\]

   10. Simplified 11.2

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}} \cdot \frac{y}{\frac{\sqrt{\sqrt{z \cdot z - t \cdot a}}}{z}}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 11.7

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

   13. Applied add-cube-cbrt 11.5

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

   14. Applied sqrt-prod 11.6

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

   15. Applied times-frac 11.5

       \[\leadsto \frac{x}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{y}{\color{blue}{\frac{\sqrt{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt[3]{z}}}}\]

   16. Applied add-cube-cbrt 11.8

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

   17. Applied times-frac 11.3

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

   18. Applied associate-*r* 9.0

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

   19. Simplified 9.0

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

   if 3.100709508274363e+95 < z

   1. Initial program 42.8

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

   2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{x \cdot y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.611098198946549 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 3.1007095082743632 \cdot 10^{95}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{\frac{\frac{\frac{\left|\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}\right|}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\sqrt[3]{y}}} \cdot \frac{x}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))