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

Average Error: 11.9 → 9.1

Time: 31.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.734842912893802954211285640305587494427 \cdot 10^{65}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\right)\\ \mathbf{elif}\;x \le 4.989813711427708520463647735774083823355 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-t\right) \cdot \left(a \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r39350307 = x;
        double r39350308 = y;
        double r39350309 = z;
        double r39350310 = r39350308 * r39350309;
        double r39350311 = t;
        double r39350312 = a;
        double r39350313 = r39350311 * r39350312;
        double r39350314 = r39350310 - r39350313;
        double r39350315 = r39350307 * r39350314;
        double r39350316 = b;
        double r39350317 = c;
        double r39350318 = r39350317 * r39350309;
        double r39350319 = i;
        double r39350320 = r39350311 * r39350319;
        double r39350321 = r39350318 - r39350320;
        double r39350322 = r39350316 * r39350321;
        double r39350323 = r39350315 - r39350322;
        double r39350324 = j;
        double r39350325 = r39350317 * r39350312;
        double r39350326 = r39350308 * r39350319;
        double r39350327 = r39350325 - r39350326;
        double r39350328 = r39350324 * r39350327;
        double r39350329 = r39350323 + r39350328;
        return r39350329;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r39350330 = x;
        double r39350331 = -8.734842912893803e+65;
        bool r39350332 = r39350330 <= r39350331;
        double r39350333 = y;
        double r39350334 = z;
        double r39350335 = r39350333 * r39350334;
        double r39350336 = a;
        double r39350337 = t;
        double r39350338 = r39350336 * r39350337;
        double r39350339 = r39350335 - r39350338;
        double r39350340 = r39350339 * r39350330;
        double r39350341 = b;
        double r39350342 = c;
        double r39350343 = r39350334 * r39350342;
        double r39350344 = i;
        double r39350345 = r39350344 * r39350337;
        double r39350346 = r39350343 - r39350345;
        double r39350347 = r39350341 * r39350346;
        double r39350348 = r39350340 - r39350347;
        double r39350349 = j;
        double r39350350 = r39350342 * r39350336;
        double r39350351 = r39350344 * r39350333;
        double r39350352 = r39350350 - r39350351;
        double r39350353 = r39350349 * r39350352;
        double r39350354 = cbrt(r39350353);
        double r39350355 = r39350354 * r39350354;
        double r39350356 = r39350354 * r39350355;
        double r39350357 = r39350348 + r39350356;
        double r39350358 = 4.9898137114277085e+50;
        bool r39350359 = r39350330 <= r39350358;
        double r39350360 = r39350334 * r39350330;
        double r39350361 = r39350360 * r39350333;
        double r39350362 = -r39350337;
        double r39350363 = r39350336 * r39350330;
        double r39350364 = r39350362 * r39350363;
        double r39350365 = r39350361 + r39350364;
        double r39350366 = r39350365 - r39350347;
        double r39350367 = r39350366 + r39350353;
        double r39350368 = cbrt(r39350347);
        double r39350369 = r39350368 * r39350368;
        double r39350370 = r39350369 * r39350368;
        double r39350371 = r39350340 - r39350370;
        double r39350372 = r39350371 + r39350353;
        double r39350373 = r39350359 ? r39350367 : r39350372;
        double r39350374 = r39350332 ? r39350357 : r39350373;
        return r39350374;
}

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	11.9
Target	19.7
Herbie	9.1

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

Derivation

1. Split input into 3 regimes

2. if x < -8.734842912893803e+65

   1. Initial program 6.2

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

   3. Applied add-cube-cbrt 6.4

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

   if -8.734842912893803e+65 < x < 4.9898137114277085e+50

   1. Initial program 14.0

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

   3. Applied add-cube-cbrt 14.2

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

   4. Applied associate-*l* 14.2

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

   6. Applied sub-neg 14.2

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

   7. Applied distribute-lft-in 14.2

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

   8. Applied distribute-lft-in 14.2

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

   9. Simplified 12.1

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

   10. Simplified 10.0

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

   if 4.9898137114277085e+50 < x

   1. Initial program 7.3

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

   3. Applied add-cube-cbrt 7.5

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.734842912893802954211285640305587494427 \cdot 10^{65}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\right)\\ \mathbf{elif}\;x \le 4.989813711427708520463647735774083823355 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-t\right) \cdot \left(a \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))))