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

Average Error: 24.5 → 6.3

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.6721891600828564 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 5.32450332568295542 \cdot 10^{121}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r299860 = x;
        double r299861 = y;
        double r299862 = r299860 * r299861;
        double r299863 = z;
        double r299864 = r299862 * r299863;
        double r299865 = r299863 * r299863;
        double r299866 = t;
        double r299867 = a;
        double r299868 = r299866 * r299867;
        double r299869 = r299865 - r299868;
        double r299870 = sqrt(r299869);
        double r299871 = r299864 / r299870;
        return r299871;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r299872 = z;
        double r299873 = -3.6721891600828564e+151;
        bool r299874 = r299872 <= r299873;
        double r299875 = -1.0;
        double r299876 = x;
        double r299877 = y;
        double r299878 = r299876 * r299877;
        double r299879 = r299875 * r299878;
        double r299880 = 5.324503325682955e+121;
        bool r299881 = r299872 <= r299880;
        double r299882 = r299872 * r299872;
        double r299883 = t;
        double r299884 = a;
        double r299885 = r299883 * r299884;
        double r299886 = r299882 - r299885;
        double r299887 = sqrt(r299886);
        double r299888 = r299872 / r299887;
        double r299889 = r299877 * r299888;
        double r299890 = r299876 * r299889;
        double r299891 = 1.0;
        double r299892 = r299878 * r299891;
        double r299893 = r299881 ? r299890 : r299892;
        double r299894 = r299874 ? r299879 : r299893;
        return r299894;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	7.5
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 < -3.6721891600828564e+151

   1. Initial program 52.8

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

   3. Applied *-un-lft-identity 52.8

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

   4. Applied sqrt-prod 52.8

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

   5. Applied times-frac 52.3

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

   6. Simplified 52.3

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

   8. Applied associate-*l* 52.3

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

   9. Taylor expanded around -inf 1.1

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

   if -3.6721891600828564e+151 < z < 5.324503325682955e+121

   1. Initial program 10.9

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

   3. Applied *-un-lft-identity 10.9

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

   4. Applied sqrt-prod 10.9

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

   5. Applied times-frac 8.8

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

   6. Simplified 8.8

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

   8. Applied associate-*l* 8.9

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

   if 5.324503325682955e+121 < z

   1. Initial program 47.0

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

   3. Applied *-un-lft-identity 47.0

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

   4. Applied sqrt-prod 47.0

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

   5. Applied times-frac 44.7

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

   6. Simplified 44.7

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

   7. Taylor expanded around inf 1.5

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.6721891600828564 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 5.32450332568295542 \cdot 10^{121}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))))