Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 11.8 → 10.7

Time: 21.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.327198726263224763088598532410072030111 \cdot 10^{221}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(c \cdot z\right) \cdot b + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;i \le -1.860012936356754256230117739562338157089 \cdot 10^{-50}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{elif}\;i \le -2.244134716973463593118119927522988633023 \cdot 10^{-308}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;i \le 8.41379591870247978891855327424447970948 \cdot 10^{-57}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;i \le 200414012594118643998003271192018944:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r588097 = x;
        double r588098 = y;
        double r588099 = z;
        double r588100 = r588098 * r588099;
        double r588101 = t;
        double r588102 = a;
        double r588103 = r588101 * r588102;
        double r588104 = r588100 - r588103;
        double r588105 = r588097 * r588104;
        double r588106 = b;
        double r588107 = c;
        double r588108 = r588107 * r588099;
        double r588109 = i;
        double r588110 = r588101 * r588109;
        double r588111 = r588108 - r588110;
        double r588112 = r588106 * r588111;
        double r588113 = r588105 - r588112;
        double r588114 = j;
        double r588115 = r588107 * r588102;
        double r588116 = r588098 * r588109;
        double r588117 = r588115 - r588116;
        double r588118 = r588114 * r588117;
        double r588119 = r588113 + r588118;
        return r588119;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r588120 = i;
        double r588121 = -2.3271987262632248e+221;
        bool r588122 = r588120 <= r588121;
        double r588123 = x;
        double r588124 = y;
        double r588125 = z;
        double r588126 = r588124 * r588125;
        double r588127 = t;
        double r588128 = a;
        double r588129 = r588127 * r588128;
        double r588130 = r588126 - r588129;
        double r588131 = r588123 * r588130;
        double r588132 = c;
        double r588133 = r588132 * r588125;
        double r588134 = b;
        double r588135 = r588133 * r588134;
        double r588136 = r588120 * r588134;
        double r588137 = r588127 * r588136;
        double r588138 = -r588137;
        double r588139 = r588135 + r588138;
        double r588140 = r588131 - r588139;
        double r588141 = j;
        double r588142 = r588132 * r588128;
        double r588143 = r588124 * r588120;
        double r588144 = r588142 - r588143;
        double r588145 = r588141 * r588144;
        double r588146 = r588140 + r588145;
        double r588147 = -1.8600129363567543e-50;
        bool r588148 = r588120 <= r588147;
        double r588149 = r588141 * r588132;
        double r588150 = r588128 * r588149;
        double r588151 = r588141 * r588124;
        double r588152 = r588120 * r588151;
        double r588153 = -r588152;
        double r588154 = r588150 + r588153;
        double r588155 = r588127 * r588120;
        double r588156 = r588133 - r588155;
        double r588157 = r588134 * r588156;
        double r588158 = r588131 - r588157;
        double r588159 = r588154 + r588158;
        double r588160 = -2.2441347169734636e-308;
        bool r588161 = r588120 <= r588160;
        double r588162 = r588128 * r588141;
        double r588163 = r588162 * r588132;
        double r588164 = -r588143;
        double r588165 = r588164 * r588141;
        double r588166 = r588163 + r588165;
        double r588167 = r588158 + r588166;
        double r588168 = 8.41379591870248e-57;
        bool r588169 = r588120 <= r588168;
        double r588170 = r588125 * r588124;
        double r588171 = r588123 * r588170;
        double r588172 = r588123 * r588127;
        double r588173 = r588128 * r588172;
        double r588174 = r588171 - r588173;
        double r588175 = r588174 - r588157;
        double r588176 = r588150 + r588165;
        double r588177 = r588175 + r588176;
        double r588178 = 2.0041401259411864e+35;
        bool r588179 = r588120 <= r588178;
        double r588180 = r588179 ? r588167 : r588159;
        double r588181 = r588169 ? r588177 : r588180;
        double r588182 = r588161 ? r588167 : r588181;
        double r588183 = r588148 ? r588159 : r588182;
        double r588184 = r588122 ? r588146 : r588183;
        return r588184;
}

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	11.8
Target	19.3
Herbie	10.7

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if i < -2.3271987262632248e+221

   1. Initial program 27.6

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

   3. Applied sub-neg 27.6

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

   4. Applied distribute-lft-in 27.6

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

   5. Simplified 27.6

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

   6. Simplified 28.2

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

   if -2.3271987262632248e+221 < i < -1.8600129363567543e-50 or 2.0041401259411864e+35 < i

   1. Initial program 14.5

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

   3. Applied sub-neg 14.5

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

   4. Applied distribute-lft-in 14.5

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

   5. Simplified 14.3

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

   6. Simplified 14.3

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

   8. Applied neg-mul-1 14.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \color{blue}{\left(-1 \cdot \left(y \cdot i\right)\right)} \cdot j\right)\]

   9. Applied associate-*l* 14.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \color{blue}{-1 \cdot \left(\left(y \cdot i\right) \cdot j\right)}\right)\]

   10. Simplified 10.9

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)}\right)\]

   if -1.8600129363567543e-50 < i < -2.2441347169734636e-308 or 8.41379591870248e-57 < i < 2.0041401259411864e+35

   1. Initial program 9.1

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

   3. Applied sub-neg 9.1

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

   4. Applied distribute-lft-in 9.1

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

   5. Simplified 9.6

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

   6. Simplified 9.6

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

   8. Applied associate-*r* 9.3

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

   if -2.2441347169734636e-308 < i < 8.41379591870248e-57

   1. Initial program 9.3

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

   3. Applied sub-neg 9.3

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

   4. Applied distribute-lft-in 9.3

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

   5. Simplified 9.9

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

   6. Simplified 9.9

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

   7. Taylor expanded around inf 9.8

      \[\leadsto \left(\color{blue}{\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.327198726263224763088598532410072030111 \cdot 10^{221}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(c \cdot z\right) \cdot b + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;i \le -1.860012936356754256230117739562338157089 \cdot 10^{-50}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{elif}\;i \le -2.244134716973463593118119927522988633023 \cdot 10^{-308}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;i \le 8.41379591870247978891855327424447970948 \cdot 10^{-57}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;i \le 200414012594118643998003271192018944:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.46969429677770502e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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