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

Average Error: 12.4 → 12.7

Time: 10.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -2.50693626867780328 \cdot 10^{-276}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\\ \mathbf{elif}\;j \le 6.99274296360215801 \cdot 10^{-128}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1057970 = x;
        double r1057971 = y;
        double r1057972 = z;
        double r1057973 = r1057971 * r1057972;
        double r1057974 = t;
        double r1057975 = a;
        double r1057976 = r1057974 * r1057975;
        double r1057977 = r1057973 - r1057976;
        double r1057978 = r1057970 * r1057977;
        double r1057979 = b;
        double r1057980 = c;
        double r1057981 = r1057980 * r1057972;
        double r1057982 = i;
        double r1057983 = r1057974 * r1057982;
        double r1057984 = r1057981 - r1057983;
        double r1057985 = r1057979 * r1057984;
        double r1057986 = r1057978 - r1057985;
        double r1057987 = j;
        double r1057988 = r1057980 * r1057975;
        double r1057989 = r1057971 * r1057982;
        double r1057990 = r1057988 - r1057989;
        double r1057991 = r1057987 * r1057990;
        double r1057992 = r1057986 + r1057991;
        return r1057992;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1057993 = j;
        double r1057994 = -2.5069362686778033e-276;
        bool r1057995 = r1057993 <= r1057994;
        double r1057996 = x;
        double r1057997 = y;
        double r1057998 = z;
        double r1057999 = r1057997 * r1057998;
        double r1058000 = t;
        double r1058001 = a;
        double r1058002 = r1058000 * r1058001;
        double r1058003 = r1057999 - r1058002;
        double r1058004 = r1057996 * r1058003;
        double r1058005 = b;
        double r1058006 = c;
        double r1058007 = r1058006 * r1057998;
        double r1058008 = i;
        double r1058009 = r1058000 * r1058008;
        double r1058010 = r1058007 - r1058009;
        double r1058011 = r1058005 * r1058010;
        double r1058012 = r1058004 - r1058011;
        double r1058013 = r1058006 * r1058001;
        double r1058014 = r1057997 * r1058008;
        double r1058015 = r1058013 - r1058014;
        double r1058016 = r1057993 * r1058015;
        double r1058017 = cbrt(r1058016);
        double r1058018 = r1058017 * r1058017;
        double r1058019 = cbrt(r1057993);
        double r1058020 = cbrt(r1058015);
        double r1058021 = r1058019 * r1058020;
        double r1058022 = r1058018 * r1058021;
        double r1058023 = r1058012 + r1058022;
        double r1058024 = 6.992742963602158e-128;
        bool r1058025 = r1057993 <= r1058024;
        double r1058026 = 0.0;
        double r1058027 = r1058012 + r1058026;
        double r1058028 = cbrt(r1058005);
        double r1058029 = r1058028 * r1058028;
        double r1058030 = r1058028 * r1058010;
        double r1058031 = r1058029 * r1058030;
        double r1058032 = r1058004 - r1058031;
        double r1058033 = r1058032 + r1058016;
        double r1058034 = r1058025 ? r1058027 : r1058033;
        double r1058035 = r1057995 ? r1058023 : r1058034;
        return r1058035;
}

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	12.4
Target	20.0
Herbie	12.7

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

Derivation

1. Split input into 3 regimes

2. if j < -2.5069362686778033e-276

   1. Initial program 12.3

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

   3. Applied add-cube-cbrt 12.6

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

   5. Applied cbrt-prod 12.5

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

   if -2.5069362686778033e-276 < j < 6.992742963602158e-128

   1. Initial program 17.5

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

   2. Taylor expanded around 0 18.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{0}\]

   if 6.992742963602158e-128 < j

   1. Initial program 8.8

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

   3. Applied add-cube-cbrt 9.1

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

   4. Applied associate-*l* 9.1

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -2.50693626867780328 \cdot 10^{-276}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\\ \mathbf{elif}\;j \le 6.99274296360215801 \cdot 10^{-128}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))))