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

Average Error: 24.8 → 6.1

Time: 16.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.667193347780096 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le -1.0059935812111571 \cdot 10^{-240}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 4.2013669449125285 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{z \cdot \left(y \cdot x\right)}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{elif}\;z \le 9.659251240846705 \cdot 10^{+99}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r13550676 = x;
        double r13550677 = y;
        double r13550678 = r13550676 * r13550677;
        double r13550679 = z;
        double r13550680 = r13550678 * r13550679;
        double r13550681 = r13550679 * r13550679;
        double r13550682 = t;
        double r13550683 = a;
        double r13550684 = r13550682 * r13550683;
        double r13550685 = r13550681 - r13550684;
        double r13550686 = sqrt(r13550685);
        double r13550687 = r13550680 / r13550686;
        return r13550687;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r13550688 = z;
        double r13550689 = -5.667193347780096e+153;
        bool r13550690 = r13550688 <= r13550689;
        double r13550691 = y;
        double r13550692 = x;
        double r13550693 = -r13550692;
        double r13550694 = r13550691 * r13550693;
        double r13550695 = -1.0059935812111571e-240;
        bool r13550696 = r13550688 <= r13550695;
        double r13550697 = r13550688 * r13550688;
        double r13550698 = t;
        double r13550699 = a;
        double r13550700 = r13550698 * r13550699;
        double r13550701 = r13550697 - r13550700;
        double r13550702 = sqrt(r13550701);
        double r13550703 = r13550688 / r13550702;
        double r13550704 = r13550692 * r13550703;
        double r13550705 = r13550691 * r13550704;
        double r13550706 = 4.2013669449125285e-250;
        bool r13550707 = r13550688 <= r13550706;
        double r13550708 = r13550691 * r13550692;
        double r13550709 = r13550688 * r13550708;
        double r13550710 = sqrt(r13550702);
        double r13550711 = r13550709 / r13550710;
        double r13550712 = r13550711 / r13550710;
        double r13550713 = 9.659251240846705e+99;
        bool r13550714 = r13550688 <= r13550713;
        double r13550715 = r13550714 ? r13550705 : r13550708;
        double r13550716 = r13550707 ? r13550712 : r13550715;
        double r13550717 = r13550696 ? r13550705 : r13550716;
        double r13550718 = r13550690 ? r13550694 : r13550717;
        return r13550718;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	7.9
Herbie	6.1

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -5.667193347780096e+153

   1. Initial program 53.5

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

   3. Applied *-un-lft-identity 53.5

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

   4. Applied sqrt-prod 53.5

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

   5. Applied times-frac 53.5

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

   6. Simplified 53.5

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

   8. Applied associate-*l* 53.5

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

   9. Taylor expanded around -inf 1.0

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

   10. Simplified 1.0

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

   if -5.667193347780096e+153 < z < -1.0059935812111571e-240 or 4.2013669449125285e-250 < z < 9.659251240846705e+99

   1. Initial program 10.7

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

   3. Applied *-un-lft-identity 10.7

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

   4. Applied sqrt-prod 10.7

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

   5. Applied times-frac 8.2

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

   6. Simplified 8.2

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

   8. Applied associate-*l* 7.6

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

   if -1.0059935812111571e-240 < z < 4.2013669449125285e-250

   1. Initial program 15.8

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

   3. Applied add-sqr-sqrt 15.8

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

   4. Applied sqrt-prod 15.9

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

   5. Applied associate-/r* 15.9

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

   if 9.659251240846705e+99 < z

   1. Initial program 43.9

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

   3. Applied *-un-lft-identity 43.9

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

   4. Applied sqrt-prod 43.9

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

   5. Applied times-frac 42.1

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

   6. Simplified 42.1

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

   8. Applied add-cube-cbrt 42.4

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

   9. Applied add-cube-cbrt 42.2

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

   10. Applied times-frac 42.2

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

   11. Simplified 42.2

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

   12. Taylor expanded around inf 2.4

       \[\leadsto \color{blue}{x \cdot y}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.667193347780096 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \le -1.0059935812111571 \cdot 10^{-240}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 4.2013669449125285 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{z \cdot \left(y \cdot x\right)}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\\ \mathbf{elif}\;z \le 9.659251240846705 \cdot 10^{+99}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :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)))))