Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 9.8

Time: 14.1s

Precision: 64

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

Math

\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r698864 = x;
        double r698865 = y;
        double r698866 = z;
        double r698867 = r698865 * r698866;
        double r698868 = t;
        double r698869 = a;
        double r698870 = r698868 * r698869;
        double r698871 = r698867 - r698870;
        double r698872 = r698864 * r698871;
        double r698873 = b;
        double r698874 = c;
        double r698875 = r698874 * r698866;
        double r698876 = i;
        double r698877 = r698876 * r698869;
        double r698878 = r698875 - r698877;
        double r698879 = r698873 * r698878;
        double r698880 = r698872 - r698879;
        double r698881 = j;
        double r698882 = r698874 * r698868;
        double r698883 = r698876 * r698865;
        double r698884 = r698882 - r698883;
        double r698885 = r698881 * r698884;
        double r698886 = r698880 + r698885;
        return r698886;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r698887 = b;
        double r698888 = -6.833466350363758e+180;
        bool r698889 = r698887 <= r698888;
        double r698890 = x;
        double r698891 = y;
        double r698892 = z;
        double r698893 = r698891 * r698892;
        double r698894 = t;
        double r698895 = a;
        double r698896 = r698894 * r698895;
        double r698897 = r698893 - r698896;
        double r698898 = r698890 * r698897;
        double r698899 = c;
        double r698900 = r698899 * r698892;
        double r698901 = i;
        double r698902 = r698901 * r698895;
        double r698903 = r698900 - r698902;
        double r698904 = r698887 * r698903;
        double r698905 = r698898 - r698904;
        double r698906 = j;
        double r698907 = cbrt(r698906);
        double r698908 = r698907 * r698907;
        double r698909 = r698899 * r698894;
        double r698910 = r698901 * r698891;
        double r698911 = r698909 - r698910;
        double r698912 = r698907 * r698911;
        double r698913 = r698908 * r698912;
        double r698914 = r698905 + r698913;
        double r698915 = 0.015564147372555343;
        bool r698916 = r698887 <= r698915;
        double r698917 = r698892 * r698887;
        double r698918 = r698917 * r698899;
        double r698919 = r698901 * r698887;
        double r698920 = -r698895;
        double r698921 = r698919 * r698920;
        double r698922 = r698918 + r698921;
        double r698923 = r698898 - r698922;
        double r698924 = r698909 * r698906;
        double r698925 = r698906 * r698891;
        double r698926 = r698901 * r698925;
        double r698927 = -r698926;
        double r698928 = r698924 + r698927;
        double r698929 = r698923 + r698928;
        double r698930 = r698901 * r698906;
        double r698931 = r698930 * r698891;
        double r698932 = -r698931;
        double r698933 = r698924 + r698932;
        double r698934 = r698905 + r698933;
        double r698935 = r698916 ? r698929 : r698934;
        double r698936 = r698889 ? r698914 : r698935;
        return r698936;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Target

Original	12.1
Target	16.0
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -6.833466350363758e+180

   1. Initial program 5.5

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 5.7

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \sqrt[3]{j}\right)} \cdot \left(c \cdot t - i \cdot y\right)\]

   4. Applied associate-*l* 5.7

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)}\]

   if -6.833466350363758e+180 < b < 0.015564147372555343

   1. Initial program 14.0

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(c \cdot t\right) \cdot j} + j \cdot \left(-i \cdot y\right)\right)\]

   6. Simplified 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \color{blue}{\left(-i \cdot \left(j \cdot y\right)\right)}\right)\]
   7. Using strategy rm

   8. Applied sub-neg 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z + \left(-i \cdot a\right)\right)}\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   9. Applied distribute-lft-in 14.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(b \cdot \left(c \cdot z\right) + b \cdot \left(-i \cdot a\right)\right)}\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   10. Simplified 12.9

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{z \cdot \left(b \cdot c\right)} + b \cdot \left(-i \cdot a\right)\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]
   11. Using strategy rm

   12. Applied distribute-rgt-neg-in 12.9

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \color{blue}{\left(i \cdot \left(-a\right)\right)}\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   13. Applied associate-*r* 11.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{\left(b \cdot i\right) \cdot \left(-a\right)}\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   14. Simplified 11.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{\left(i \cdot b\right)} \cdot \left(-a\right)\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]
   15. Using strategy rm

   16. Applied associate-*r* 10.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{\left(z \cdot b\right) \cdot c} + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   if 0.015564147372555343 < b

   1. Initial program 6.8

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 6.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 6.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 6.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(c \cdot t\right) \cdot j} + j \cdot \left(-i \cdot y\right)\right)\]

   6. Simplified 7.9

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \color{blue}{\left(-i \cdot \left(j \cdot y\right)\right)}\right)\]
   7. Using strategy rm

   8. Applied associate-*r* 7.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\color{blue}{\left(i \cdot j\right) \cdot y}\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))