Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.7 → 10.7

Time: 28.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -7.9425973173534991 \cdot 10^{59}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)}\right) \cdot \sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 4.66076012081488949 \cdot 10^{119}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r76122 = x;
        double r76123 = y;
        double r76124 = z;
        double r76125 = r76123 * r76124;
        double r76126 = t;
        double r76127 = a;
        double r76128 = r76126 * r76127;
        double r76129 = r76125 - r76128;
        double r76130 = r76122 * r76129;
        double r76131 = b;
        double r76132 = c;
        double r76133 = r76132 * r76124;
        double r76134 = i;
        double r76135 = r76134 * r76127;
        double r76136 = r76133 - r76135;
        double r76137 = r76131 * r76136;
        double r76138 = r76130 - r76137;
        double r76139 = j;
        double r76140 = r76132 * r76126;
        double r76141 = r76134 * r76123;
        double r76142 = r76140 - r76141;
        double r76143 = r76139 * r76142;
        double r76144 = r76138 + r76143;
        return r76144;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r76145 = j;
        double r76146 = -7.942597317353499e+59;
        bool r76147 = r76145 <= r76146;
        double r76148 = x;
        double r76149 = y;
        double r76150 = z;
        double r76151 = r76149 * r76150;
        double r76152 = t;
        double r76153 = a;
        double r76154 = r76152 * r76153;
        double r76155 = r76151 - r76154;
        double r76156 = r76148 * r76155;
        double r76157 = b;
        double r76158 = c;
        double r76159 = r76157 * r76158;
        double r76160 = r76150 * r76159;
        double r76161 = i;
        double r76162 = r76161 * r76157;
        double r76163 = -r76153;
        double r76164 = r76162 * r76163;
        double r76165 = cbrt(r76164);
        double r76166 = r76165 * r76165;
        double r76167 = r76166 * r76165;
        double r76168 = r76160 + r76167;
        double r76169 = r76156 - r76168;
        double r76170 = r76158 * r76152;
        double r76171 = r76161 * r76149;
        double r76172 = r76170 - r76171;
        double r76173 = r76145 * r76172;
        double r76174 = r76169 + r76173;
        double r76175 = 4.6607601208148895e+119;
        bool r76176 = r76145 <= r76175;
        double r76177 = r76160 + r76164;
        double r76178 = r76156 - r76177;
        double r76179 = r76145 * r76158;
        double r76180 = r76152 * r76179;
        double r76181 = r76145 * r76149;
        double r76182 = r76161 * r76181;
        double r76183 = r76180 - r76182;
        double r76184 = r76178 + r76183;
        double r76185 = r76150 * r76157;
        double r76186 = r76185 * r76158;
        double r76187 = r76186 + r76164;
        double r76188 = r76156 - r76187;
        double r76189 = r76188 + r76173;
        double r76190 = r76176 ? r76184 : r76189;
        double r76191 = r76147 ? r76174 : r76190;
        return r76191;
}

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 < -7.942597317353499e+59

   1. Initial program 8.2

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

   3. Applied sub-neg 8.2

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

   4. Applied distribute-lft-in 8.2

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

   5. Simplified 8.3

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

   7. Applied distribute-rgt-neg-in 8.3

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

   8. Applied associate-*r* 8.5

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

   9. Simplified 8.5

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

   11. Applied add-cube-cbrt 8.6

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

   if -7.942597317353499e+59 < j < 4.6607601208148895e+119

   1. Initial program 14.1

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

   3. Applied sub-neg 14.1

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

   4. Applied distribute-lft-in 14.1

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

   5. Simplified 14.6

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

   7. Applied distribute-rgt-neg-in 14.6

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

   8. Applied associate-*r* 14.4

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

   9. Simplified 14.4

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

   10. Taylor expanded around inf 11.3

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

   if 4.6607601208148895e+119 < j

   1. Initial program 8.1

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

   3. Applied sub-neg 8.1

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

   4. Applied distribute-lft-in 8.1

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

   5. Simplified 8.3

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

   7. Applied distribute-rgt-neg-in 8.3

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

   8. Applied associate-*r* 8.1

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

   9. Simplified 8.1

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

   11. Applied associate-*r* 9.5

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -7.9425973173534991 \cdot 10^{59}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)}\right) \cdot \sqrt[3]{\left(i \cdot b\right) \cdot \left(-a\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 4.66076012081488949 \cdot 10^{119}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(i \cdot b\right) \cdot \left(-a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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)))))