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

Average Error: 12.4 → 11.0

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le \frac{-260022612279875}{2.977131414714805823690030317109266572713 \cdot 10^{138}} \lor \neg \left(x \le \frac{7823478373404693}{6.864797660130609714981900799081393217269 \cdot 10^{156}}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot \left(c \cdot a - y \cdot i\right) + t \cdot \left(i \cdot b - a \cdot x\right)\right) - z \cdot \left(b \cdot c\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r598391 = x;
        double r598392 = y;
        double r598393 = z;
        double r598394 = r598392 * r598393;
        double r598395 = t;
        double r598396 = a;
        double r598397 = r598395 * r598396;
        double r598398 = r598394 - r598397;
        double r598399 = r598391 * r598398;
        double r598400 = b;
        double r598401 = c;
        double r598402 = r598401 * r598393;
        double r598403 = i;
        double r598404 = r598395 * r598403;
        double r598405 = r598402 - r598404;
        double r598406 = r598400 * r598405;
        double r598407 = r598399 - r598406;
        double r598408 = j;
        double r598409 = r598401 * r598396;
        double r598410 = r598392 * r598403;
        double r598411 = r598409 - r598410;
        double r598412 = r598408 * r598411;
        double r598413 = r598407 + r598412;
        return r598413;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r598414 = x;
        double r598415 = -260022612279875.0;
        double r598416 = 2.977131414714806e+138;
        double r598417 = r598415 / r598416;
        bool r598418 = r598414 <= r598417;
        double r598419 = 7823478373404693.0;
        double r598420 = 6.86479766013061e+156;
        double r598421 = r598419 / r598420;
        bool r598422 = r598414 <= r598421;
        double r598423 = !r598422;
        bool r598424 = r598418 || r598423;
        double r598425 = y;
        double r598426 = z;
        double r598427 = r598425 * r598426;
        double r598428 = t;
        double r598429 = a;
        double r598430 = r598428 * r598429;
        double r598431 = r598427 - r598430;
        double r598432 = r598414 * r598431;
        double r598433 = b;
        double r598434 = c;
        double r598435 = r598434 * r598426;
        double r598436 = i;
        double r598437 = r598428 * r598436;
        double r598438 = r598435 - r598437;
        double r598439 = r598433 * r598438;
        double r598440 = r598432 - r598439;
        double r598441 = j;
        double r598442 = r598434 * r598429;
        double r598443 = r598425 * r598436;
        double r598444 = r598442 - r598443;
        double r598445 = r598441 * r598444;
        double r598446 = r598440 + r598445;
        double r598447 = r598436 * r598433;
        double r598448 = r598429 * r598414;
        double r598449 = r598447 - r598448;
        double r598450 = r598428 * r598449;
        double r598451 = r598445 + r598450;
        double r598452 = r598433 * r598434;
        double r598453 = r598426 * r598452;
        double r598454 = r598451 - r598453;
        double r598455 = r598424 ? r598446 : r598454;
        return r598455;
}

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	19.4
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if x < -8.733998472310765e-125

   1. Initial program 9.5

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

   3. Applied add-cube-cbrt 9.8

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

   if -8.733998472310765e-125 < x < 1.139651707266175e-141

   1. Initial program 17.2

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

   2. Taylor expanded around inf 13.4

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

   if 1.139651707266175e-141 < x

   1. Initial program 9.7

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

   3. Applied add-cube-cbrt 9.9

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

   4. Applied associate-*l* 9.9

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-260022612279875}{2.977131414714805823690030317109266572713 \cdot 10^{138}} \lor \neg \left(x \le \frac{7823478373404693}{6.864797660130609714981900799081393217269 \cdot 10^{156}}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot \left(c \cdot a - y \cdot i\right) + t \cdot \left(i \cdot b - a \cdot x\right)\right) - z \cdot \left(b \cdot c\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))))