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

Average Error: 11.9 → 9.2

Time: 16.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -2.56366436408222289 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le -4.37991542778717814 \cdot 10^{-234}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{elif}\;j \le 3.78998645893523151 \cdot 10^{67}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{elif}\;j \le 1.69248794364116096 \cdot 10^{134}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r946142 = x;
        double r946143 = y;
        double r946144 = z;
        double r946145 = r946143 * r946144;
        double r946146 = t;
        double r946147 = a;
        double r946148 = r946146 * r946147;
        double r946149 = r946145 - r946148;
        double r946150 = r946142 * r946149;
        double r946151 = b;
        double r946152 = c;
        double r946153 = r946152 * r946144;
        double r946154 = i;
        double r946155 = r946146 * r946154;
        double r946156 = r946153 - r946155;
        double r946157 = r946151 * r946156;
        double r946158 = r946150 - r946157;
        double r946159 = j;
        double r946160 = r946152 * r946147;
        double r946161 = r946143 * r946154;
        double r946162 = r946160 - r946161;
        double r946163 = r946159 * r946162;
        double r946164 = r946158 + r946163;
        return r946164;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r946165 = j;
        double r946166 = -2.563664364082223e-07;
        bool r946167 = r946165 <= r946166;
        double r946168 = x;
        double r946169 = y;
        double r946170 = z;
        double r946171 = r946169 * r946170;
        double r946172 = t;
        double r946173 = a;
        double r946174 = r946172 * r946173;
        double r946175 = r946171 - r946174;
        double r946176 = r946168 * r946175;
        double r946177 = b;
        double r946178 = cbrt(r946177);
        double r946179 = r946178 * r946178;
        double r946180 = c;
        double r946181 = r946180 * r946170;
        double r946182 = i;
        double r946183 = r946172 * r946182;
        double r946184 = r946181 - r946183;
        double r946185 = r946178 * r946184;
        double r946186 = r946179 * r946185;
        double r946187 = r946176 - r946186;
        double r946188 = r946180 * r946173;
        double r946189 = r946169 * r946182;
        double r946190 = r946188 - r946189;
        double r946191 = r946165 * r946190;
        double r946192 = r946187 + r946191;
        double r946193 = -4.379915427787178e-234;
        bool r946194 = r946165 <= r946193;
        double r946195 = r946171 * r946168;
        double r946196 = r946168 * r946172;
        double r946197 = r946173 * r946196;
        double r946198 = -r946197;
        double r946199 = r946195 + r946198;
        double r946200 = r946177 * r946184;
        double r946201 = r946199 - r946200;
        double r946202 = r946165 * r946169;
        double r946203 = r946182 * r946202;
        double r946204 = -r946203;
        double r946205 = r946165 * r946180;
        double r946206 = r946173 * r946205;
        double r946207 = r946204 + r946206;
        double r946208 = r946201 + r946207;
        double r946209 = 3.7899864589352315e+67;
        bool r946210 = r946165 <= r946209;
        double r946211 = r946177 * r946180;
        double r946212 = r946170 * r946211;
        double r946213 = -r946183;
        double r946214 = r946213 * r946177;
        double r946215 = r946212 + r946214;
        double r946216 = r946176 - r946215;
        double r946217 = r946216 + r946207;
        double r946218 = 1.692487943641161e+134;
        bool r946219 = r946165 <= r946218;
        double r946220 = cbrt(r946175);
        double r946221 = r946220 * r946220;
        double r946222 = r946168 * r946221;
        double r946223 = r946222 * r946220;
        double r946224 = r946223 - r946200;
        double r946225 = r946224 + r946191;
        double r946226 = r946219 ? r946208 : r946225;
        double r946227 = r946210 ? r946217 : r946226;
        double r946228 = r946194 ? r946208 : r946227;
        double r946229 = r946167 ? r946192 : r946228;
        return r946229;
}

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	11.9
Target	19.8
Herbie	9.2

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

Derivation

1. Split input into 4 regimes

2. if j < -2.563664364082223e-07

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 7.2

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

   4. Applied associate-*l* 7.2

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

   if -2.563664364082223e-07 < j < -4.379915427787178e-234 or 3.7899864589352315e+67 < j < 1.692487943641161e+134

   1. Initial program 12.7

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

   3. Applied sub-neg 12.7

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

   4. Applied distribute-lft-in 12.7

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

   5. Simplified 11.5

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

   6. Simplified 11.5

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

   8. Applied neg-mul-1 11.5

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

   9. Applied associate-*l* 11.5

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

   10. Simplified 10.0

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

   12. Applied sub-neg 10.0

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

   13. Applied distribute-lft-in 10.0

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

   14. Simplified 10.0

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

   15. Simplified 10.1

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

   if -4.379915427787178e-234 < j < 3.7899864589352315e+67

   1. Initial program 14.6

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

   3. Applied sub-neg 14.6

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

   4. Applied distribute-lft-in 14.6

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

   5. Simplified 12.3

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

   6. Simplified 12.3

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

   8. Applied neg-mul-1 12.3

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

   9. Applied associate-*l* 12.3

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

   10. Simplified 9.6

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

   12. Applied sub-neg 9.6

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

   13. Applied distribute-lft-in 9.6

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

   14. Simplified 10.0

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

   15. Simplified 10.0

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

   if 1.692487943641161e+134 < j

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied associate-*r* 7.6

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -2.56366436408222289 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le -4.37991542778717814 \cdot 10^{-234}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{elif}\;j \le 3.78998645893523151 \cdot 10^{67}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{elif}\;j \le 1.69248794364116096 \cdot 10^{134}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(-i \cdot \left(j \cdot y\right)\right) + a \cdot \left(j \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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 -1.469694296777705e-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)))))