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

Average Error: 12.8 → 10.1

Time: 24.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.7381012850626549 \cdot 10^{-82}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;t \le -5.6003152044590445 \cdot 10^{-248}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;t \le 6.1699886005730711 \cdot 10^{-238}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;t \le 4.47171486710020656 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r702065 = x;
        double r702066 = y;
        double r702067 = z;
        double r702068 = r702066 * r702067;
        double r702069 = t;
        double r702070 = a;
        double r702071 = r702069 * r702070;
        double r702072 = r702068 - r702071;
        double r702073 = r702065 * r702072;
        double r702074 = b;
        double r702075 = c;
        double r702076 = r702075 * r702067;
        double r702077 = i;
        double r702078 = r702069 * r702077;
        double r702079 = r702076 - r702078;
        double r702080 = r702074 * r702079;
        double r702081 = r702073 - r702080;
        double r702082 = j;
        double r702083 = r702075 * r702070;
        double r702084 = r702066 * r702077;
        double r702085 = r702083 - r702084;
        double r702086 = r702082 * r702085;
        double r702087 = r702081 + r702086;
        return r702087;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r702088 = t;
        double r702089 = -1.738101285062655e-82;
        bool r702090 = r702088 <= r702089;
        double r702091 = j;
        double r702092 = c;
        double r702093 = a;
        double r702094 = r702092 * r702093;
        double r702095 = y;
        double r702096 = i;
        double r702097 = r702095 * r702096;
        double r702098 = r702094 - r702097;
        double r702099 = r702091 * r702098;
        double r702100 = x;
        double r702101 = z;
        double r702102 = r702095 * r702101;
        double r702103 = r702100 * r702102;
        double r702104 = r702100 * r702093;
        double r702105 = r702088 * r702104;
        double r702106 = -r702105;
        double r702107 = r702103 + r702106;
        double r702108 = b;
        double r702109 = r702108 * r702092;
        double r702110 = r702101 * r702109;
        double r702111 = r702096 * r702108;
        double r702112 = r702088 * r702111;
        double r702113 = -r702112;
        double r702114 = r702110 + r702113;
        double r702115 = r702107 - r702114;
        double r702116 = r702099 + r702115;
        double r702117 = -5.6003152044590445e-248;
        bool r702118 = r702088 <= r702117;
        double r702119 = r702100 * r702088;
        double r702120 = r702093 * r702119;
        double r702121 = -r702120;
        double r702122 = r702103 + r702121;
        double r702123 = r702092 * r702101;
        double r702124 = r702088 * r702096;
        double r702125 = r702123 - r702124;
        double r702126 = r702108 * r702125;
        double r702127 = r702122 - r702126;
        double r702128 = r702091 * r702092;
        double r702129 = r702093 * r702128;
        double r702130 = -r702097;
        double r702131 = r702091 * r702130;
        double r702132 = r702129 + r702131;
        double r702133 = r702127 + r702132;
        double r702134 = 6.169988600573071e-238;
        bool r702135 = r702088 <= r702134;
        double r702136 = r702100 * r702095;
        double r702137 = r702136 * r702101;
        double r702138 = r702137 + r702121;
        double r702139 = cbrt(r702112);
        double r702140 = r702139 * r702139;
        double r702141 = r702140 * r702139;
        double r702142 = -r702141;
        double r702143 = r702110 + r702142;
        double r702144 = r702138 - r702143;
        double r702145 = r702144 + r702099;
        double r702146 = 4.4717148671002066e-37;
        bool r702147 = r702088 <= r702146;
        double r702148 = r702138 - r702126;
        double r702149 = r702148 + r702099;
        double r702150 = r702147 ? r702149 : r702116;
        double r702151 = r702135 ? r702145 : r702150;
        double r702152 = r702118 ? r702133 : r702151;
        double r702153 = r702090 ? r702116 : r702152;
        return r702153;
}

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	12.8
Target	20.1
Herbie	10.1

\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if t < -1.738101285062655e-82 or 4.4717148671002066e-37 < t

   1. Initial program 15.5

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

   3. Applied sub-neg 15.5

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

   4. Applied distribute-lft-in 15.5

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

   5. Simplified 15.5

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

   7. Applied sub-neg 15.5

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

   8. Applied distribute-lft-in 15.5

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

   9. Simplified 15.4

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

   10. Simplified 12.9

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

   12. Applied pow1 12.9

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

   13. Applied pow1 12.9

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

   14. Applied pow-prod-down 12.9

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

   15. Applied pow1 12.9

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

   16. Applied pow-prod-down 12.9

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

   17. Simplified 9.1

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

   if -1.738101285062655e-82 < t < -5.6003152044590445e-248

   1. Initial program 10.4

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

   3. Applied sub-neg 10.4

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

   4. Applied distribute-lft-in 10.4

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

   5. Simplified 10.3

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

   7. Applied sub-neg 10.3

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

   8. Applied distribute-lft-in 10.3

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

   9. Simplified 9.3

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

   if -5.6003152044590445e-248 < t < 6.169988600573071e-238

   1. Initial program 10.5

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

   3. Applied sub-neg 10.5

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

   4. Applied distribute-lft-in 10.5

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

   5. Simplified 10.6

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

   7. Applied sub-neg 10.6

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

   8. Applied distribute-lft-in 10.6

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

   9. Simplified 10.7

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

   10. Simplified 14.4

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

   12. Applied associate-*r* 13.9

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

   14. Applied add-cube-cbrt 13.9

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

   if 6.169988600573071e-238 < t < 4.4717148671002066e-37

   1. Initial program 10.1

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

   3. Applied sub-neg 10.1

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

   4. Applied distribute-lft-in 10.1

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

   5. Simplified 10.1

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

   7. Applied associate-*r* 10.8

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.7381012850626549 \cdot 10^{-82}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;t \le -5.6003152044590445 \cdot 10^{-248}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;t \le 6.1699886005730711 \cdot 10^{-238}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;t \le 4.47171486710020656 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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