Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.9 → 9.9

Time: 26.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -1.2880696392092357 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-a\right) \cdot \left(b \cdot i\right) + b \cdot \left(z \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - i \cdot y\right)\\ \mathbf{elif}\;j \le 3.0298788442355306 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(z \cdot c\right) + \left(a \cdot i\right) \cdot \left(-b\right)\right)\right) + \left(\left(-y\right) \cdot \left(j \cdot i\right) + \left(t \cdot j\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(a \cdot i\right) \cdot \left(-b\right) + \left(c \cdot b\right) \cdot z\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r6069994 = x;
        double r6069995 = y;
        double r6069996 = z;
        double r6069997 = r6069995 * r6069996;
        double r6069998 = t;
        double r6069999 = a;
        double r6070000 = r6069998 * r6069999;
        double r6070001 = r6069997 - r6070000;
        double r6070002 = r6069994 * r6070001;
        double r6070003 = b;
        double r6070004 = c;
        double r6070005 = r6070004 * r6069996;
        double r6070006 = i;
        double r6070007 = r6070006 * r6069999;
        double r6070008 = r6070005 - r6070007;
        double r6070009 = r6070003 * r6070008;
        double r6070010 = r6070002 - r6070009;
        double r6070011 = j;
        double r6070012 = r6070004 * r6069998;
        double r6070013 = r6070006 * r6069995;
        double r6070014 = r6070012 - r6070013;
        double r6070015 = r6070011 * r6070014;
        double r6070016 = r6070010 + r6070015;
        return r6070016;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r6070017 = j;
        double r6070018 = -1.2880696392092357e-15;
        bool r6070019 = r6070017 <= r6070018;
        double r6070020 = y;
        double r6070021 = z;
        double r6070022 = r6070020 * r6070021;
        double r6070023 = t;
        double r6070024 = a;
        double r6070025 = r6070023 * r6070024;
        double r6070026 = r6070022 - r6070025;
        double r6070027 = x;
        double r6070028 = r6070026 * r6070027;
        double r6070029 = -r6070024;
        double r6070030 = b;
        double r6070031 = i;
        double r6070032 = r6070030 * r6070031;
        double r6070033 = r6070029 * r6070032;
        double r6070034 = c;
        double r6070035 = r6070021 * r6070034;
        double r6070036 = r6070030 * r6070035;
        double r6070037 = r6070033 + r6070036;
        double r6070038 = r6070028 - r6070037;
        double r6070039 = r6070023 * r6070034;
        double r6070040 = r6070031 * r6070020;
        double r6070041 = r6070039 - r6070040;
        double r6070042 = r6070017 * r6070041;
        double r6070043 = r6070038 + r6070042;
        double r6070044 = 3.0298788442355306e-224;
        bool r6070045 = r6070017 <= r6070044;
        double r6070046 = r6070024 * r6070031;
        double r6070047 = -r6070030;
        double r6070048 = r6070046 * r6070047;
        double r6070049 = r6070036 + r6070048;
        double r6070050 = r6070028 - r6070049;
        double r6070051 = -r6070020;
        double r6070052 = r6070017 * r6070031;
        double r6070053 = r6070051 * r6070052;
        double r6070054 = r6070023 * r6070017;
        double r6070055 = r6070054 * r6070034;
        double r6070056 = r6070053 + r6070055;
        double r6070057 = r6070050 + r6070056;
        double r6070058 = r6070034 * r6070030;
        double r6070059 = r6070058 * r6070021;
        double r6070060 = r6070048 + r6070059;
        double r6070061 = r6070028 - r6070060;
        double r6070062 = r6070042 + r6070061;
        double r6070063 = r6070045 ? r6070057 : r6070062;
        double r6070064 = r6070019 ? r6070043 : r6070063;
        return r6070064;
}

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

Derivation

1. Split input into 3 regimes

2. if j < -1.2880696392092357e-15

   1. Initial program 7.2

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

   3. Applied sub-neg 7.2

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

   4. Applied distribute-lft-in 7.2

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

   6. Applied distribute-rgt-neg-in 7.2

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

   7. Applied associate-*r* 7.3

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

   if -1.2880696392092357e-15 < j < 3.0298788442355306e-224

   1. Initial program 15.9

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

   3. Applied sub-neg 15.9

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

   4. Applied distribute-lft-in 15.9

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

   6. Applied add-cube-cbrt 16.0

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

   7. Applied associate-*l* 16.0

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

   9. Applied sub-neg 16.0

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

   10. Applied distribute-lft-in 16.0

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

   11. Applied distribute-lft-in 16.0

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

   12. Simplified 13.1

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

   13. Simplified 10.4

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

   if 3.0298788442355306e-224 < j

   1. Initial program 10.6

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

   3. Applied sub-neg 10.6

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

   4. Applied distribute-lft-in 10.6

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

   6. Applied add-cube-cbrt 10.7

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

   7. Applied associate-*l* 10.7

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

   8. Taylor expanded around inf 10.8

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.2880696392092357 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-a\right) \cdot \left(b \cdot i\right) + b \cdot \left(z \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - i \cdot y\right)\\ \mathbf{elif}\;j \le 3.0298788442355306 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(z \cdot c\right) + \left(a \cdot i\right) \cdot \left(-b\right)\right)\right) + \left(\left(-y\right) \cdot \left(j \cdot i\right) + \left(t \cdot j\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(a \cdot i\right) \cdot \left(-b\right) + \left(c \cdot b\right) \cdot z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))