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

Average Error: 12.3 → 12.4

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.98155654580347545345036736346737136654 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{z \cdot y - t \cdot a}\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{elif}\;x \le 4.340999817937329746967166771511970045509 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{z \cdot y - t \cdot a}\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26176268 = x;
        double r26176269 = y;
        double r26176270 = z;
        double r26176271 = r26176269 * r26176270;
        double r26176272 = t;
        double r26176273 = a;
        double r26176274 = r26176272 * r26176273;
        double r26176275 = r26176271 - r26176274;
        double r26176276 = r26176268 * r26176275;
        double r26176277 = b;
        double r26176278 = c;
        double r26176279 = r26176278 * r26176270;
        double r26176280 = i;
        double r26176281 = r26176272 * r26176280;
        double r26176282 = r26176279 - r26176281;
        double r26176283 = r26176277 * r26176282;
        double r26176284 = r26176276 - r26176283;
        double r26176285 = j;
        double r26176286 = r26176278 * r26176273;
        double r26176287 = r26176269 * r26176280;
        double r26176288 = r26176286 - r26176287;
        double r26176289 = r26176285 * r26176288;
        double r26176290 = r26176284 + r26176289;
        return r26176290;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26176291 = x;
        double r26176292 = -7.981556545803475e-228;
        bool r26176293 = r26176291 <= r26176292;
        double r26176294 = a;
        double r26176295 = c;
        double r26176296 = r26176294 * r26176295;
        double r26176297 = i;
        double r26176298 = y;
        double r26176299 = r26176297 * r26176298;
        double r26176300 = r26176296 - r26176299;
        double r26176301 = j;
        double r26176302 = b;
        double r26176303 = t;
        double r26176304 = r26176303 * r26176297;
        double r26176305 = z;
        double r26176306 = r26176305 * r26176295;
        double r26176307 = r26176304 - r26176306;
        double r26176308 = r26176305 * r26176298;
        double r26176309 = r26176303 * r26176294;
        double r26176310 = r26176308 - r26176309;
        double r26176311 = cbrt(r26176310);
        double r26176312 = r26176311 * r26176311;
        double r26176313 = cbrt(r26176291);
        double r26176314 = r26176313 * r26176313;
        double r26176315 = r26176314 * r26176311;
        double r26176316 = r26176312 * r26176315;
        double r26176317 = r26176316 * r26176313;
        double r26176318 = fma(r26176302, r26176307, r26176317);
        double r26176319 = fma(r26176300, r26176301, r26176318);
        double r26176320 = 4.34099981793733e-204;
        bool r26176321 = r26176291 <= r26176320;
        double r26176322 = 0.0;
        double r26176323 = fma(r26176302, r26176307, r26176322);
        double r26176324 = fma(r26176300, r26176301, r26176323);
        double r26176325 = r26176321 ? r26176324 : r26176319;
        double r26176326 = r26176293 ? r26176319 : r26176325;
        return r26176326;
}

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	12.3
Target	19.7
Herbie	12.4

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

Derivation

1. Split input into 2 regimes

2. if x < -7.981556545803475e-228 or 4.34099981793733e-204 < x

   1. Initial program 10.9

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

   2. Simplified 10.9

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

   4. Applied add-cube-cbrt 11.2

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

   5. Applied associate-*r* 11.2

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

   7. Applied add-cube-cbrt 11.3

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

   8. Applied associate-*l* 11.3

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

   if -7.981556545803475e-228 < x < 4.34099981793733e-204

   1. Initial program 18.3

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

   2. Simplified 18.3

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

   3. Taylor expanded around 0 17.0

      \[\leadsto \mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(b, i \cdot t - z \cdot c, \color{blue}{0}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.98155654580347545345036736346737136654 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{z \cdot y - t \cdot a}\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{elif}\;x \le 4.340999817937329746967166771511970045509 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - i \cdot y, j, \mathsf{fma}\left(b, t \cdot i - z \cdot c, \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{z \cdot y - t \cdot a}\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))