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

Average Error: 25.7 → 6.5

Time: 18.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.360527887622532879859975400806262673533 \cdot 10^{74}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 5.226961588256762913949303420447053290802 \cdot 10^{-171}:\\ \;\;\;\;\frac{x \cdot z}{\sqrt{{z}^{2} - a \cdot t}} \cdot y\\ \mathbf{elif}\;z \le 6.533659553749963211432099594569213138326 \cdot 10^{149}:\\ \;\;\;\;x \cdot \left(\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r256139 = x;
        double r256140 = y;
        double r256141 = r256139 * r256140;
        double r256142 = z;
        double r256143 = r256141 * r256142;
        double r256144 = r256142 * r256142;
        double r256145 = t;
        double r256146 = a;
        double r256147 = r256145 * r256146;
        double r256148 = r256144 - r256147;
        double r256149 = sqrt(r256148);
        double r256150 = r256143 / r256149;
        return r256150;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r256151 = z;
        double r256152 = -4.360527887622533e+74;
        bool r256153 = r256151 <= r256152;
        double r256154 = x;
        double r256155 = y;
        double r256156 = -r256155;
        double r256157 = r256154 * r256156;
        double r256158 = 5.226961588256763e-171;
        bool r256159 = r256151 <= r256158;
        double r256160 = r256154 * r256151;
        double r256161 = 2.0;
        double r256162 = pow(r256151, r256161);
        double r256163 = a;
        double r256164 = t;
        double r256165 = r256163 * r256164;
        double r256166 = r256162 - r256165;
        double r256167 = sqrt(r256166);
        double r256168 = r256160 / r256167;
        double r256169 = r256168 * r256155;
        double r256170 = 6.533659553749963e+149;
        bool r256171 = r256151 <= r256170;
        double r256172 = r256151 * r256151;
        double r256173 = r256172 - r256165;
        double r256174 = sqrt(r256173);
        double r256175 = r256151 / r256174;
        double r256176 = r256175 * r256155;
        double r256177 = r256154 * r256176;
        double r256178 = r256154 * r256155;
        double r256179 = r256171 ? r256177 : r256178;
        double r256180 = r256159 ? r256169 : r256179;
        double r256181 = r256153 ? r256157 : r256180;
        return r256181;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.7
Target	8.1
Herbie	6.5

\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -4.360527887622533e+74

   1. Initial program 40.2

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

   2. Simplified 39.4

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

   4. Applied div-inv 39.5

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

   5. Applied associate-*l* 37.4

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

   6. Simplified 37.4

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

   7. Taylor expanded around -inf 3.0

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

   if -4.360527887622533e+74 < z < 5.226961588256763e-171

   1. Initial program 13.4

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

   2. Simplified 13.4

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

   4. Applied add-sqr-sqrt 13.4

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

   5. Applied sqrt-prod 13.5

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

   6. Applied associate-/r* 13.5

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

   7. Simplified 13.5

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

   9. Applied *-un-lft-identity 13.5

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

   10. Applied sqrt-prod 13.5

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

   11. Applied sqrt-prod 13.5

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

   12. Applied *-un-lft-identity 13.5

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

   13. Applied sqrt-prod 13.5

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

   14. Applied sqrt-prod 13.5

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

   15. Applied times-frac 13.5

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

   16. Applied times-frac 13.9

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

   17. Applied associate-*l* 12.9

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

   18. Simplified 12.4

       \[\leadsto \frac{\frac{y}{\sqrt{\sqrt{1}}}}{\sqrt{\sqrt{1}}} \cdot \color{blue}{\frac{x \cdot z}{\sqrt{{z}^{2} - a \cdot t}}}\]

   if 5.226961588256763e-171 < z < 6.533659553749963e+149

   1. Initial program 9.3

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

   2. Simplified 9.1

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

   4. Applied div-inv 9.2

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

   5. Applied associate-*l* 5.7

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

   6. Simplified 5.6

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

   8. Applied associate-*l* 5.3

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

   9. Simplified 8.8

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

   11. Applied *-un-lft-identity 8.8

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

   12. Applied sqrt-prod 8.8

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

   13. Applied times-frac 5.3

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

   14. Simplified 5.3

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

   if 6.533659553749963e+149 < z

   1. Initial program 53.7

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

   2. Simplified 53.2

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

   4. Applied div-inv 53.2

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

   5. Applied associate-*l* 53.1

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

   6. Simplified 53.1

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

   8. Applied associate-*l* 53.1

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

   9. Simplified 54.0

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

   10. Taylor expanded around inf 1.5

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.360527887622532879859975400806262673533 \cdot 10^{74}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 5.226961588256762913949303420447053290802 \cdot 10^{-171}:\\ \;\;\;\;\frac{x \cdot z}{\sqrt{{z}^{2} - a \cdot t}} \cdot y\\ \mathbf{elif}\;z \le 6.533659553749963211432099594569213138326 \cdot 10^{149}:\\ \;\;\;\;x \cdot \left(\frac{z}{\sqrt{z \cdot z - a \cdot t}} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))