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

Average Error: 12.5 → 9.1

Time: 28.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -942100039185.614990234375 \lor \neg \left(x \le 1.377993057330842504113383725556413732388 \cdot 10^{120}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t\right) \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1269288 = x;
        double r1269289 = y;
        double r1269290 = z;
        double r1269291 = r1269289 * r1269290;
        double r1269292 = t;
        double r1269293 = a;
        double r1269294 = r1269292 * r1269293;
        double r1269295 = r1269291 - r1269294;
        double r1269296 = r1269288 * r1269295;
        double r1269297 = b;
        double r1269298 = c;
        double r1269299 = r1269298 * r1269290;
        double r1269300 = i;
        double r1269301 = r1269292 * r1269300;
        double r1269302 = r1269299 - r1269301;
        double r1269303 = r1269297 * r1269302;
        double r1269304 = r1269296 - r1269303;
        double r1269305 = j;
        double r1269306 = r1269298 * r1269293;
        double r1269307 = r1269289 * r1269300;
        double r1269308 = r1269306 - r1269307;
        double r1269309 = r1269305 * r1269308;
        double r1269310 = r1269304 + r1269309;
        return r1269310;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1269311 = x;
        double r1269312 = -942100039185.615;
        bool r1269313 = r1269311 <= r1269312;
        double r1269314 = 1.3779930573308425e+120;
        bool r1269315 = r1269311 <= r1269314;
        double r1269316 = !r1269315;
        bool r1269317 = r1269313 || r1269316;
        double r1269318 = y;
        double r1269319 = z;
        double r1269320 = r1269318 * r1269319;
        double r1269321 = t;
        double r1269322 = a;
        double r1269323 = r1269321 * r1269322;
        double r1269324 = r1269320 - r1269323;
        double r1269325 = r1269311 * r1269324;
        double r1269326 = b;
        double r1269327 = c;
        double r1269328 = r1269327 * r1269319;
        double r1269329 = i;
        double r1269330 = r1269321 * r1269329;
        double r1269331 = r1269328 - r1269330;
        double r1269332 = r1269326 * r1269331;
        double r1269333 = r1269325 - r1269332;
        double r1269334 = j;
        double r1269335 = cbrt(r1269334);
        double r1269336 = r1269335 * r1269335;
        double r1269337 = r1269327 * r1269322;
        double r1269338 = r1269318 * r1269329;
        double r1269339 = r1269337 - r1269338;
        double r1269340 = r1269335 * r1269339;
        double r1269341 = r1269336 * r1269340;
        double r1269342 = r1269333 + r1269341;
        double r1269343 = r1269311 * r1269319;
        double r1269344 = r1269343 * r1269318;
        double r1269345 = r1269311 * r1269321;
        double r1269346 = r1269322 * r1269345;
        double r1269347 = -r1269346;
        double r1269348 = r1269344 + r1269347;
        double r1269349 = r1269326 * r1269327;
        double r1269350 = r1269319 * r1269349;
        double r1269351 = -r1269321;
        double r1269352 = r1269329 * r1269326;
        double r1269353 = r1269351 * r1269352;
        double r1269354 = r1269350 + r1269353;
        double r1269355 = r1269348 - r1269354;
        double r1269356 = r1269334 * r1269339;
        double r1269357 = r1269355 + r1269356;
        double r1269358 = r1269317 ? r1269342 : r1269357;
        return r1269358;
}

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	12.5
Target	19.9
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 2 regimes

2. if x < -942100039185.615 or 1.3779930573308425e+120 < x

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied associate-*l* 7.6

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

   if -942100039185.615 < x < 1.3779930573308425e+120

   1. Initial program 14.5

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

   3. Applied sub-neg 14.5

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

   4. Applied distribute-lft-in 14.5

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

   5. Simplified 14.5

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

   6. Simplified 12.5

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

   8. Applied sub-neg 12.5

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

   9. Applied distribute-lft-in 12.5

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

   10. Simplified 12.4

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

   11. Simplified 12.4

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

   13. Applied distribute-lft-neg-in 12.4

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

   14. Applied associate-*l* 12.0

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

   16. Applied associate-*r* 9.7

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -942100039185.614990234375 \lor \neg \left(x \le 1.377993057330842504113383725556413732388 \cdot 10^{120}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t\right) \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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 -1.46969429677770502e-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)))))