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

Average Error: 25.2 → 6.3

Time: 53.1s

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 r479425 = x;
        double r479426 = y;
        double r479427 = r479425 * r479426;
        double r479428 = z;
        double r479429 = r479427 * r479428;
        double r479430 = r479428 * r479428;
        double r479431 = t;
        double r479432 = a;
        double r479433 = r479431 * r479432;
        double r479434 = r479430 - r479433;
        double r479435 = sqrt(r479434);
        double r479436 = r479429 / r479435;
        return r479436;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r479437 = z;
        double r479438 = -4.611098198946549e+108;
        bool r479439 = r479437 <= r479438;
        double r479440 = -1.0;
        double r479441 = x;
        double r479442 = y;
        double r479443 = r479441 * r479442;
        double r479444 = r479440 * r479443;
        double r479445 = 3.100709508274363e+95;
        bool r479446 = r479437 <= r479445;
        double r479447 = cbrt(r479442);
        double r479448 = r479437 * r479437;
        double r479449 = t;
        double r479450 = a;
        double r479451 = r479449 * r479450;
        double r479452 = r479448 - r479451;
        double r479453 = sqrt(r479452);
        double r479454 = cbrt(r479453);
        double r479455 = fabs(r479454);
        double r479456 = cbrt(r479437);
        double r479457 = r479455 / r479456;
        double r479458 = r479457 / r479456;
        double r479459 = r479458 / r479447;
        double r479460 = r479447 / r479459;
        double r479461 = sqrt(r479453);
        double r479462 = r479441 / r479461;
        double r479463 = r479460 * r479462;
        double r479464 = sqrt(r479454);
        double r479465 = r479464 / r479456;
        double r479466 = r479447 / r479465;
        double r479467 = r479463 * r479466;
        double r479468 = r479446 ? r479467 : r479443;
        double r479469 = r479439 ? r479444 : r479468;
        return r479469;
}

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 
(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)))))