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

Average Error: 24.2 → 6.2

Time: 8.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.6183547901220424 \cdot 10^{153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 6.9260668882011419 \cdot 10^{147}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r293891 = x;
        double r293892 = y;
        double r293893 = r293891 * r293892;
        double r293894 = z;
        double r293895 = r293893 * r293894;
        double r293896 = r293894 * r293894;
        double r293897 = t;
        double r293898 = a;
        double r293899 = r293897 * r293898;
        double r293900 = r293896 - r293899;
        double r293901 = sqrt(r293900);
        double r293902 = r293895 / r293901;
        return r293902;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r293903 = z;
        double r293904 = -8.618354790122042e+153;
        bool r293905 = r293903 <= r293904;
        double r293906 = x;
        double r293907 = y;
        double r293908 = -r293907;
        double r293909 = r293906 * r293908;
        double r293910 = 6.926066888201142e+147;
        bool r293911 = r293903 <= r293910;
        double r293912 = r293903 * r293903;
        double r293913 = t;
        double r293914 = a;
        double r293915 = r293913 * r293914;
        double r293916 = r293912 - r293915;
        double r293917 = sqrt(r293916);
        double r293918 = r293917 / r293903;
        double r293919 = r293907 / r293918;
        double r293920 = r293906 * r293919;
        double r293921 = r293906 * r293907;
        double r293922 = r293911 ? r293920 : r293921;
        double r293923 = r293905 ? r293909 : r293922;
        return r293923;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	7.7
Herbie	6.2

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

Derivation

1. Split input into 3 regimes

2. if z < -8.618354790122042e+153

   1. Initial program 54.0

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

   3. Applied associate-/l* 53.6

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

   5. Applied *-un-lft-identity 53.6

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

   6. Applied *-un-lft-identity 53.6

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

   7. Applied sqrt-prod 53.6

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

   8. Applied times-frac 53.6

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

   9. Applied times-frac 53.6

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

   10. Simplified 53.6

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

   11. Taylor expanded around -inf 1.1

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

   12. Simplified 1.1

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

   if -8.618354790122042e+153 < z < 6.926066888201142e+147

   1. Initial program 11.2

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

   3. Applied associate-/l* 8.9

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

   5. Applied *-un-lft-identity 8.9

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

   6. Applied *-un-lft-identity 8.9

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

   7. Applied sqrt-prod 8.9

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

   8. Applied times-frac 8.9

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

   9. Applied times-frac 8.6

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

   10. Simplified 8.6

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

   if 6.926066888201142e+147 < z

   1. Initial program 51.6

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

   2. Taylor expanded around inf 1.2

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.6183547901220424 \cdot 10^{153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 6.9260668882011419 \cdot 10^{147}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))