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

Average Error: 12.0 → 10.2

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -1.512608529495292496256238020861527295075 \cdot 10^{146}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(z \cdot y - a \cdot t\right) \cdot x\\ \mathbf{elif}\;j \le -4.392384127110078940945257259549888818845 \cdot 10^{-291}:\\ \;\;\;\;\left(c \cdot \left(j \cdot a\right) + \left(y \cdot j\right) \cdot \left(-i\right)\right) + \left(\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;j \le 3.116586143558746623708399161663533727855 \cdot 10^{62}:\\ \;\;\;\;\left(\left(c \cdot j\right) \cdot a + \left(i \cdot \left(-j\right)\right) \cdot y\right) + \left(\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(\sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x} \cdot \sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x}\right) \cdot \sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x} - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r42813352 = x;
        double r42813353 = y;
        double r42813354 = z;
        double r42813355 = r42813353 * r42813354;
        double r42813356 = t;
        double r42813357 = a;
        double r42813358 = r42813356 * r42813357;
        double r42813359 = r42813355 - r42813358;
        double r42813360 = r42813352 * r42813359;
        double r42813361 = b;
        double r42813362 = c;
        double r42813363 = r42813362 * r42813354;
        double r42813364 = i;
        double r42813365 = r42813356 * r42813364;
        double r42813366 = r42813363 - r42813365;
        double r42813367 = r42813361 * r42813366;
        double r42813368 = r42813360 - r42813367;
        double r42813369 = j;
        double r42813370 = r42813362 * r42813357;
        double r42813371 = r42813353 * r42813364;
        double r42813372 = r42813370 - r42813371;
        double r42813373 = r42813369 * r42813372;
        double r42813374 = r42813368 + r42813373;
        return r42813374;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r42813375 = j;
        double r42813376 = -1.5126085294952925e+146;
        bool r42813377 = r42813375 <= r42813376;
        double r42813378 = c;
        double r42813379 = a;
        double r42813380 = r42813378 * r42813379;
        double r42813381 = y;
        double r42813382 = i;
        double r42813383 = r42813381 * r42813382;
        double r42813384 = r42813380 - r42813383;
        double r42813385 = r42813375 * r42813384;
        double r42813386 = z;
        double r42813387 = r42813386 * r42813381;
        double r42813388 = t;
        double r42813389 = r42813379 * r42813388;
        double r42813390 = r42813387 - r42813389;
        double r42813391 = x;
        double r42813392 = r42813390 * r42813391;
        double r42813393 = r42813385 + r42813392;
        double r42813394 = -4.392384127110079e-291;
        bool r42813395 = r42813375 <= r42813394;
        double r42813396 = r42813375 * r42813379;
        double r42813397 = r42813378 * r42813396;
        double r42813398 = r42813381 * r42813375;
        double r42813399 = -r42813382;
        double r42813400 = r42813398 * r42813399;
        double r42813401 = r42813397 + r42813400;
        double r42813402 = r42813378 * r42813386;
        double r42813403 = r42813388 * r42813382;
        double r42813404 = r42813402 - r42813403;
        double r42813405 = b;
        double r42813406 = r42813404 * r42813405;
        double r42813407 = r42813392 - r42813406;
        double r42813408 = r42813401 + r42813407;
        double r42813409 = 3.1165861435587466e+62;
        bool r42813410 = r42813375 <= r42813409;
        double r42813411 = r42813378 * r42813375;
        double r42813412 = r42813411 * r42813379;
        double r42813413 = -r42813375;
        double r42813414 = r42813382 * r42813413;
        double r42813415 = r42813414 * r42813381;
        double r42813416 = r42813412 + r42813415;
        double r42813417 = r42813416 + r42813407;
        double r42813418 = cbrt(r42813392);
        double r42813419 = r42813418 * r42813418;
        double r42813420 = r42813419 * r42813418;
        double r42813421 = r42813420 - r42813406;
        double r42813422 = r42813385 + r42813421;
        double r42813423 = r42813410 ? r42813417 : r42813422;
        double r42813424 = r42813395 ? r42813408 : r42813423;
        double r42813425 = r42813377 ? r42813393 : r42813424;
        return r42813425;
}

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	19.1
Herbie	10.2

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

Derivation

1. Split input into 4 regimes

2. if j < -1.5126085294952925e+146

   1. Initial program 7.8

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

   2. Taylor expanded around 0 16.7

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

   if -1.5126085294952925e+146 < j < -4.392384127110079e-291

   1. Initial program 12.1

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

   3. Applied add-cube-cbrt 12.3

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

   4. Applied associate-*l* 12.3

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

   6. Applied sub-neg 12.3

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

   7. Applied distribute-lft-in 12.3

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

   8. Applied distribute-lft-in 12.3

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

   9. Simplified 11.2

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

   10. Simplified 10.2

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

   12. Applied associate-*r* 10.2

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

   if -4.392384127110079e-291 < j < 3.1165861435587466e+62

   1. Initial program 14.8

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

   3. Applied add-cube-cbrt 15.0

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

   4. Applied associate-*l* 15.0

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

   6. Applied sub-neg 15.0

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

   7. Applied distribute-lft-in 15.0

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

   8. Applied distribute-lft-in 15.0

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

   9. Simplified 12.3

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

   10. Simplified 10.2

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

   12. Applied distribute-lft-neg-in 10.2

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

   13. Applied associate-*r* 10.1

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

   if 3.1165861435587466e+62 < j

   1. Initial program 6.4

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

   3. Applied add-cube-cbrt 6.6

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.512608529495292496256238020861527295075 \cdot 10^{146}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(z \cdot y - a \cdot t\right) \cdot x\\ \mathbf{elif}\;j \le -4.392384127110078940945257259549888818845 \cdot 10^{-291}:\\ \;\;\;\;\left(c \cdot \left(j \cdot a\right) + \left(y \cdot j\right) \cdot \left(-i\right)\right) + \left(\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;j \le 3.116586143558746623708399161663533727855 \cdot 10^{62}:\\ \;\;\;\;\left(\left(c \cdot j\right) \cdot a + \left(i \cdot \left(-j\right)\right) \cdot y\right) + \left(\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(\left(\sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x} \cdot \sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x}\right) \cdot \sqrt[3]{\left(z \cdot y - a \cdot t\right) \cdot x} - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))))