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

Average Error: 12.0 → 7.2

Time: 9.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) = -\infty \lor \neg \left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \le 3.1081462895994273 \cdot 10^{307}\right):\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) + a \cdot \left(j \cdot c\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}} \cdot \sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}}\right) \cdot \sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1055999 = x;
        double r1056000 = y;
        double r1056001 = z;
        double r1056002 = r1056000 * r1056001;
        double r1056003 = t;
        double r1056004 = a;
        double r1056005 = r1056003 * r1056004;
        double r1056006 = r1056002 - r1056005;
        double r1056007 = r1055999 * r1056006;
        double r1056008 = b;
        double r1056009 = c;
        double r1056010 = r1056009 * r1056001;
        double r1056011 = i;
        double r1056012 = r1056003 * r1056011;
        double r1056013 = r1056010 - r1056012;
        double r1056014 = r1056008 * r1056013;
        double r1056015 = r1056007 - r1056014;
        double r1056016 = j;
        double r1056017 = r1056009 * r1056004;
        double r1056018 = r1056000 * r1056011;
        double r1056019 = r1056017 - r1056018;
        double r1056020 = r1056016 * r1056019;
        double r1056021 = r1056015 + r1056020;
        return r1056021;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1056022 = x;
        double r1056023 = y;
        double r1056024 = z;
        double r1056025 = r1056023 * r1056024;
        double r1056026 = t;
        double r1056027 = a;
        double r1056028 = r1056026 * r1056027;
        double r1056029 = r1056025 - r1056028;
        double r1056030 = r1056022 * r1056029;
        double r1056031 = b;
        double r1056032 = c;
        double r1056033 = r1056032 * r1056024;
        double r1056034 = i;
        double r1056035 = r1056026 * r1056034;
        double r1056036 = r1056033 - r1056035;
        double r1056037 = r1056031 * r1056036;
        double r1056038 = r1056030 - r1056037;
        double r1056039 = j;
        double r1056040 = r1056032 * r1056027;
        double r1056041 = r1056023 * r1056034;
        double r1056042 = r1056040 - r1056041;
        double r1056043 = r1056039 * r1056042;
        double r1056044 = r1056038 + r1056043;
        double r1056045 = -inf.0;
        bool r1056046 = r1056044 <= r1056045;
        double r1056047 = 3.1081462895994273e+307;
        bool r1056048 = r1056044 <= r1056047;
        double r1056049 = !r1056048;
        bool r1056050 = r1056046 || r1056049;
        double r1056051 = r1056034 * r1056031;
        double r1056052 = r1056026 * r1056051;
        double r1056053 = r1056039 * r1056032;
        double r1056054 = r1056027 * r1056053;
        double r1056055 = r1056052 + r1056054;
        double r1056056 = r1056039 * r1056023;
        double r1056057 = r1056034 * r1056056;
        double r1056058 = r1056055 - r1056057;
        double r1056059 = cbrt(r1056042);
        double r1056060 = r1056059 * r1056059;
        double r1056061 = r1056039 * r1056060;
        double r1056062 = cbrt(r1056059);
        double r1056063 = r1056062 * r1056062;
        double r1056064 = r1056063 * r1056062;
        double r1056065 = r1056061 * r1056064;
        double r1056066 = r1056038 + r1056065;
        double r1056067 = r1056050 ? r1056058 : r1056066;
        return r1056067;
}

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	12.0
Target	20.1
Herbie	7.2

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

Derivation

1. Split input into 2 regimes

2. if (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))) < -inf.0 or 3.1081462895994273e+307 < (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i))))

   1. Initial program 63.8

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

   2. Taylor expanded around inf 34.4

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

   if -inf.0 < (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))) < 3.1081462895994273e+307

   1. Initial program 0.8

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

   3. Applied add-cube-cbrt 1.2

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

   4. Applied associate-*r* 1.2

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

   6. Applied add-cube-cbrt 1.3

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) = -\infty \lor \neg \left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \le 3.1081462895994273 \cdot 10^{307}\right):\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) + a \cdot \left(j \cdot c\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}} \cdot \sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}}\right) \cdot \sqrt[3]{\sqrt[3]{c \cdot a - y \cdot i}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))))