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

Average Error: 12.1 → 9.5

Time: 28.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -286718132868874816:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(\left(-x\right) \cdot a\right) \cdot t\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \le 9.580447942447200128217912121628776656147 \cdot 10^{105}:\\ \;\;\;\;\left(y \cdot \left(-j \cdot i\right) + c \cdot \left(a \cdot j\right)\right) + \left(\left(x \cdot \left(z \cdot y\right) + \left(a \cdot t\right) \cdot \left(-x\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) + \left(a \cdot t\right) \cdot \left(-x\right)\right) - \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r43992055 = x;
        double r43992056 = y;
        double r43992057 = z;
        double r43992058 = r43992056 * r43992057;
        double r43992059 = t;
        double r43992060 = a;
        double r43992061 = r43992059 * r43992060;
        double r43992062 = r43992058 - r43992061;
        double r43992063 = r43992055 * r43992062;
        double r43992064 = b;
        double r43992065 = c;
        double r43992066 = r43992065 * r43992057;
        double r43992067 = i;
        double r43992068 = r43992059 * r43992067;
        double r43992069 = r43992066 - r43992068;
        double r43992070 = r43992064 * r43992069;
        double r43992071 = r43992063 - r43992070;
        double r43992072 = j;
        double r43992073 = r43992065 * r43992060;
        double r43992074 = r43992056 * r43992067;
        double r43992075 = r43992073 - r43992074;
        double r43992076 = r43992072 * r43992075;
        double r43992077 = r43992071 + r43992076;
        return r43992077;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r43992078 = j;
        double r43992079 = -2.867181328688748e+17;
        bool r43992080 = r43992078 <= r43992079;
        double r43992081 = x;
        double r43992082 = y;
        double r43992083 = r43992081 * r43992082;
        double r43992084 = z;
        double r43992085 = r43992083 * r43992084;
        double r43992086 = -r43992081;
        double r43992087 = a;
        double r43992088 = r43992086 * r43992087;
        double r43992089 = t;
        double r43992090 = r43992088 * r43992089;
        double r43992091 = r43992085 + r43992090;
        double r43992092 = b;
        double r43992093 = c;
        double r43992094 = r43992093 * r43992084;
        double r43992095 = i;
        double r43992096 = r43992089 * r43992095;
        double r43992097 = r43992094 - r43992096;
        double r43992098 = r43992092 * r43992097;
        double r43992099 = r43992091 - r43992098;
        double r43992100 = r43992087 * r43992093;
        double r43992101 = r43992082 * r43992095;
        double r43992102 = r43992100 - r43992101;
        double r43992103 = r43992078 * r43992102;
        double r43992104 = r43992099 + r43992103;
        double r43992105 = 9.5804479424472e+105;
        bool r43992106 = r43992078 <= r43992105;
        double r43992107 = r43992078 * r43992095;
        double r43992108 = -r43992107;
        double r43992109 = r43992082 * r43992108;
        double r43992110 = r43992087 * r43992078;
        double r43992111 = r43992093 * r43992110;
        double r43992112 = r43992109 + r43992111;
        double r43992113 = r43992084 * r43992082;
        double r43992114 = r43992081 * r43992113;
        double r43992115 = r43992087 * r43992089;
        double r43992116 = r43992115 * r43992086;
        double r43992117 = r43992114 + r43992116;
        double r43992118 = r43992117 - r43992098;
        double r43992119 = r43992112 + r43992118;
        double r43992120 = cbrt(r43992098);
        double r43992121 = r43992120 * r43992120;
        double r43992122 = r43992120 * r43992121;
        double r43992123 = r43992117 - r43992122;
        double r43992124 = r43992103 + r43992123;
        double r43992125 = r43992106 ? r43992119 : r43992124;
        double r43992126 = r43992080 ? r43992104 : r43992125;
        return r43992126;
}

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	12.1
Target	19.7
Herbie	9.5

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

Derivation

1. Split input into 3 regimes

2. if j < -2.867181328688748e+17

   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 sub-neg 7.4

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

   4. Applied distribute-lft-in 7.4

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

   5. Taylor expanded around inf 8.0

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

   6. Simplified 8.1

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

   8. Applied associate-*r* 7.6

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

   if -2.867181328688748e+17 < j < 9.5804479424472e+105

   1. Initial program 14.1

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

   3. Applied sub-neg 14.1

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

   4. Applied distribute-lft-in 14.1

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

   6. Applied add-cube-cbrt 14.3

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

   7. Applied associate-*l* 14.3

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

   9. Applied sub-neg 14.3

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

   10. Applied distribute-lft-in 14.3

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

   11. Applied distribute-lft-in 14.3

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

   12. Simplified 12.5

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

   13. Simplified 10.4

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

   if 9.5804479424472e+105 < j

   1. Initial program 6.2

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

   3. Applied sub-neg 6.2

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

   4. Applied distribute-lft-in 6.2

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

   6. Applied add-cube-cbrt 6.3

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -286718132868874816:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(\left(-x\right) \cdot a\right) \cdot t\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \le 9.580447942447200128217912121628776656147 \cdot 10^{105}:\\ \;\;\;\;\left(y \cdot \left(-j \cdot i\right) + c \cdot \left(a \cdot j\right)\right) + \left(\left(x \cdot \left(z \cdot y\right) + \left(a \cdot t\right) \cdot \left(-x\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) + \left(a \cdot t\right) \cdot \left(-x\right)\right) - \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))))