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

Average Error: 12.1 → 9.6

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.757291648762964449191530452281109108034 \cdot 10^{-98} \lor \neg \left(b \le 1.138173278485745872842410902436161460661 \cdot 10^{45}\right):\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(\left(z \cdot b\right) \cdot c + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r701993 = x;
        double r701994 = y;
        double r701995 = z;
        double r701996 = r701994 * r701995;
        double r701997 = t;
        double r701998 = a;
        double r701999 = r701997 * r701998;
        double r702000 = r701996 - r701999;
        double r702001 = r701993 * r702000;
        double r702002 = b;
        double r702003 = c;
        double r702004 = r702003 * r701995;
        double r702005 = i;
        double r702006 = r701997 * r702005;
        double r702007 = r702004 - r702006;
        double r702008 = r702002 * r702007;
        double r702009 = r702001 - r702008;
        double r702010 = j;
        double r702011 = r702003 * r701998;
        double r702012 = r701994 * r702005;
        double r702013 = r702011 - r702012;
        double r702014 = r702010 * r702013;
        double r702015 = r702009 + r702014;
        return r702015;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r702016 = b;
        double r702017 = -2.7572916487629644e-98;
        bool r702018 = r702016 <= r702017;
        double r702019 = 1.1381732784857459e+45;
        bool r702020 = r702016 <= r702019;
        double r702021 = !r702020;
        bool r702022 = r702018 || r702021;
        double r702023 = x;
        double r702024 = y;
        double r702025 = z;
        double r702026 = r702024 * r702025;
        double r702027 = r702023 * r702026;
        double r702028 = a;
        double r702029 = t;
        double r702030 = r702023 * r702029;
        double r702031 = r702028 * r702030;
        double r702032 = -r702031;
        double r702033 = r702027 + r702032;
        double r702034 = c;
        double r702035 = r702034 * r702025;
        double r702036 = i;
        double r702037 = r702029 * r702036;
        double r702038 = r702035 - r702037;
        double r702039 = r702016 * r702038;
        double r702040 = r702033 - r702039;
        double r702041 = j;
        double r702042 = r702041 * r702034;
        double r702043 = r702028 * r702042;
        double r702044 = r702024 * r702036;
        double r702045 = -r702044;
        double r702046 = r702041 * r702045;
        double r702047 = r702043 + r702046;
        double r702048 = r702040 + r702047;
        double r702049 = r702034 * r702028;
        double r702050 = r702049 - r702044;
        double r702051 = r702041 * r702050;
        double r702052 = r702023 * r702028;
        double r702053 = r702029 * r702052;
        double r702054 = -r702053;
        double r702055 = r702027 + r702054;
        double r702056 = r702025 * r702016;
        double r702057 = r702056 * r702034;
        double r702058 = r702036 * r702016;
        double r702059 = r702029 * r702058;
        double r702060 = -r702059;
        double r702061 = r702057 + r702060;
        double r702062 = r702055 - r702061;
        double r702063 = r702051 + r702062;
        double r702064 = r702022 ? r702048 : r702063;
        return r702064;
}

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	20.2
Herbie	9.6

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

Derivation

1. Split input into 2 regimes

2. if b < -2.7572916487629644e-98 or 1.1381732784857459e+45 < b

   1. Initial program 8.5

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

   3. Applied sub-neg 8.5

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

   4. Applied distribute-lft-in 8.5

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

   5. Simplified 9.2

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

   7. Applied sub-neg 9.2

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

   8. Applied distribute-lft-in 9.2

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

   9. Simplified 8.9

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

   if -2.7572916487629644e-98 < b < 1.1381732784857459e+45

   1. Initial program 15.0

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

   3. Applied sub-neg 15.0

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

   4. Applied distribute-lft-in 15.0

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

   5. Simplified 15.3

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

   7. Applied sub-neg 15.3

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

   8. Applied distribute-lft-in 15.3

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

   9. Simplified 12.8

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

   10. Simplified 10.5

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

   12. Applied *-un-lft-identity 10.5

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

   13. Applied associate-*l* 10.5

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

   14. Simplified 10.2

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

   16. Applied associate-*r* 10.1

       \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + \left(-1 \cdot \left(t \cdot \left(x \cdot a\right)\right)\right)\right) - \left(\color{blue}{\left(z \cdot b\right) \cdot c} + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.757291648762964449191530452281109108034 \cdot 10^{-98} \lor \neg \left(b \le 1.138173278485745872842410902436161460661 \cdot 10^{45}\right):\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(\left(z \cdot b\right) \cdot c + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))))