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

Average Error: 11.6 → 9.3

Time: 19.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.2224158461403074 \cdot 10^{-121}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot z\right) \cdot x + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(b \cdot \left(c \cdot z\right) + \left(-b \cdot \left(i \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;x \le 1.9015513961511429 \cdot 10^{-34}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot x\right) \cdot z + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(b \cdot \left(c \cdot z\right) + \left(i \cdot \left(-b\right)\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{x} \cdot \left(\left(-a\right) \cdot t\right)\right) \cdot \sqrt{x} + \left(y \cdot z\right) \cdot x\right) - \left(c \cdot z - i \cdot t\right) \cdot b\right) + \left(c \cdot a - y \cdot i\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r16113961 = x;
        double r16113962 = y;
        double r16113963 = z;
        double r16113964 = r16113962 * r16113963;
        double r16113965 = t;
        double r16113966 = a;
        double r16113967 = r16113965 * r16113966;
        double r16113968 = r16113964 - r16113967;
        double r16113969 = r16113961 * r16113968;
        double r16113970 = b;
        double r16113971 = c;
        double r16113972 = r16113971 * r16113963;
        double r16113973 = i;
        double r16113974 = r16113965 * r16113973;
        double r16113975 = r16113972 - r16113974;
        double r16113976 = r16113970 * r16113975;
        double r16113977 = r16113969 - r16113976;
        double r16113978 = j;
        double r16113979 = r16113971 * r16113966;
        double r16113980 = r16113962 * r16113973;
        double r16113981 = r16113979 - r16113980;
        double r16113982 = r16113978 * r16113981;
        double r16113983 = r16113977 + r16113982;
        return r16113983;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r16113984 = x;
        double r16113985 = -2.2224158461403074e-121;
        bool r16113986 = r16113984 <= r16113985;
        double r16113987 = c;
        double r16113988 = a;
        double r16113989 = r16113987 * r16113988;
        double r16113990 = y;
        double r16113991 = i;
        double r16113992 = r16113990 * r16113991;
        double r16113993 = r16113989 - r16113992;
        double r16113994 = j;
        double r16113995 = r16113993 * r16113994;
        double r16113996 = z;
        double r16113997 = r16113990 * r16113996;
        double r16113998 = r16113997 * r16113984;
        double r16113999 = t;
        double r16114000 = r16113999 * r16113984;
        double r16114001 = -r16113988;
        double r16114002 = r16114000 * r16114001;
        double r16114003 = r16113998 + r16114002;
        double r16114004 = b;
        double r16114005 = r16113987 * r16113996;
        double r16114006 = r16114004 * r16114005;
        double r16114007 = r16113991 * r16113999;
        double r16114008 = r16114004 * r16114007;
        double r16114009 = -r16114008;
        double r16114010 = r16114006 + r16114009;
        double r16114011 = r16114003 - r16114010;
        double r16114012 = r16113995 + r16114011;
        double r16114013 = 1.9015513961511429e-34;
        bool r16114014 = r16113984 <= r16114013;
        double r16114015 = r16113990 * r16113984;
        double r16114016 = r16114015 * r16113996;
        double r16114017 = r16114016 + r16114002;
        double r16114018 = -r16114004;
        double r16114019 = r16113991 * r16114018;
        double r16114020 = r16114019 * r16113999;
        double r16114021 = r16114006 + r16114020;
        double r16114022 = r16114017 - r16114021;
        double r16114023 = r16113995 + r16114022;
        double r16114024 = sqrt(r16113984);
        double r16114025 = r16114001 * r16113999;
        double r16114026 = r16114024 * r16114025;
        double r16114027 = r16114026 * r16114024;
        double r16114028 = r16114027 + r16113998;
        double r16114029 = r16114005 - r16114007;
        double r16114030 = r16114029 * r16114004;
        double r16114031 = r16114028 - r16114030;
        double r16114032 = r16114031 + r16113995;
        double r16114033 = r16114014 ? r16114023 : r16114032;
        double r16114034 = r16113986 ? r16114012 : r16114033;
        return r16114034;
}

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	11.6
Target	18.6
Herbie	9.3

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -2.2224158461403074e-121

   1. Initial program 8.3

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

   3. Applied sub-neg 8.3

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

   4. Applied distribute-lft-in 8.3

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

   6. Applied distribute-lft-neg-in 8.3

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

   7. Applied associate-*r* 10.2

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

   9. Applied sub-neg 10.2

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

   10. Applied distribute-lft-in 10.2

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

   11. Simplified 11.0

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

   13. Applied associate-*l* 10.2

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

   if -2.2224158461403074e-121 < x < 1.9015513961511429e-34

   1. Initial program 15.4

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

   3. Applied sub-neg 15.4

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

   4. Applied distribute-lft-in 15.4

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

   6. Applied distribute-lft-neg-in 15.4

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

   7. Applied associate-*r* 12.5

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

   9. Applied sub-neg 12.5

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

   10. Applied distribute-lft-in 12.5

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

   11. Simplified 12.2

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

   13. Applied associate-*r* 9.4

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

   if 1.9015513961511429e-34 < x

   1. Initial program 7.9

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

   3. Applied sub-neg 7.9

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

   4. Applied distribute-lft-in 7.9

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

   6. Applied add-sqr-sqrt 7.9

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

   7. Applied associate-*l* 7.9

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.2224158461403074 \cdot 10^{-121}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot z\right) \cdot x + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(b \cdot \left(c \cdot z\right) + \left(-b \cdot \left(i \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;x \le 1.9015513961511429 \cdot 10^{-34}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot x\right) \cdot z + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(b \cdot \left(c \cdot z\right) + \left(i \cdot \left(-b\right)\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{x} \cdot \left(\left(-a\right) \cdot t\right)\right) \cdot \sqrt{x} + \left(y \cdot z\right) \cdot x\right) - \left(c \cdot z - i \cdot t\right) \cdot b\right) + \left(c \cdot a - y \cdot i\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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) (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)))))