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

Average Error: 12.1 → 11.3

Time: 15.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.34696385450285961 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 2.9280986771833522 \cdot 10^{-207}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 2.00957151932431567 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1003965 = x;
        double r1003966 = y;
        double r1003967 = z;
        double r1003968 = r1003966 * r1003967;
        double r1003969 = t;
        double r1003970 = a;
        double r1003971 = r1003969 * r1003970;
        double r1003972 = r1003968 - r1003971;
        double r1003973 = r1003965 * r1003972;
        double r1003974 = b;
        double r1003975 = c;
        double r1003976 = r1003975 * r1003967;
        double r1003977 = i;
        double r1003978 = r1003969 * r1003977;
        double r1003979 = r1003976 - r1003978;
        double r1003980 = r1003974 * r1003979;
        double r1003981 = r1003973 - r1003980;
        double r1003982 = j;
        double r1003983 = r1003975 * r1003970;
        double r1003984 = r1003966 * r1003977;
        double r1003985 = r1003983 - r1003984;
        double r1003986 = r1003982 * r1003985;
        double r1003987 = r1003981 + r1003986;
        return r1003987;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1003988 = a;
        double r1003989 = -7.34696385450286e-162;
        bool r1003990 = r1003988 <= r1003989;
        double r1003991 = x;
        double r1003992 = y;
        double r1003993 = z;
        double r1003994 = r1003992 * r1003993;
        double r1003995 = r1003991 * r1003994;
        double r1003996 = t;
        double r1003997 = r1003991 * r1003996;
        double r1003998 = r1003988 * r1003997;
        double r1003999 = -r1003998;
        double r1004000 = r1003995 + r1003999;
        double r1004001 = b;
        double r1004002 = c;
        double r1004003 = r1004002 * r1003993;
        double r1004004 = i;
        double r1004005 = r1003996 * r1004004;
        double r1004006 = r1004003 - r1004005;
        double r1004007 = r1004001 * r1004006;
        double r1004008 = r1004000 - r1004007;
        double r1004009 = j;
        double r1004010 = r1004002 * r1003988;
        double r1004011 = r1003992 * r1004004;
        double r1004012 = r1004010 - r1004011;
        double r1004013 = r1004009 * r1004012;
        double r1004014 = r1004008 + r1004013;
        double r1004015 = 2.9280986771833522e-207;
        bool r1004016 = r1003988 <= r1004015;
        double r1004017 = r1003996 * r1003988;
        double r1004018 = r1003994 - r1004017;
        double r1004019 = r1003991 * r1004018;
        double r1004020 = r1004001 * r1004002;
        double r1004021 = r1003993 * r1004020;
        double r1004022 = -r1004005;
        double r1004023 = r1004022 * r1004001;
        double r1004024 = r1004021 + r1004023;
        double r1004025 = r1004019 - r1004024;
        double r1004026 = r1004025 + r1004013;
        double r1004027 = 2.0095715193243157e-45;
        bool r1004028 = r1003988 <= r1004027;
        double r1004029 = cbrt(r1004019);
        double r1004030 = r1004029 * r1004029;
        double r1004031 = r1004030 * r1004029;
        double r1004032 = r1004031 - r1004007;
        double r1004033 = r1004032 + r1004013;
        double r1004034 = r1004019 - r1004007;
        double r1004035 = r1004009 * r1004002;
        double r1004036 = r1003988 * r1004035;
        double r1004037 = -r1004011;
        double r1004038 = r1004009 * r1004037;
        double r1004039 = r1004036 + r1004038;
        double r1004040 = r1004034 + r1004039;
        double r1004041 = r1004028 ? r1004033 : r1004040;
        double r1004042 = r1004016 ? r1004026 : r1004041;
        double r1004043 = r1003990 ? r1004014 : r1004042;
        return r1004043;
}

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	20.0
Herbie	11.3

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

Derivation

1. Split input into 4 regimes

2. if a < -7.34696385450286e-162

   1. Initial program 13.0

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

   3. Applied sub-neg 13.0

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

   4. Applied distribute-lft-in 13.0

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

   5. Simplified 12.5

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

   if -7.34696385450286e-162 < a < 2.9280986771833522e-207

   1. Initial program 10.3

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

   3. Applied sub-neg 10.3

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

   4. Applied distribute-lft-in 10.3

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

   5. Simplified 10.9

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

   6. Simplified 10.9

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

   if 2.9280986771833522e-207 < a < 2.0095715193243157e-45

   1. Initial program 8.9

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

   3. Applied add-cube-cbrt 9.2

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

   if 2.0095715193243157e-45 < a

   1. Initial program 15.2

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

   3. Applied sub-neg 15.2

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

   4. Applied distribute-lft-in 15.2

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

   5. Simplified 11.4

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.34696385450285961 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 2.9280986771833522 \cdot 10^{-207}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 2.00957151932431567 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))