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

Average Error: 11.6 → 11.8

Time: 29.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.5726372568800548 \cdot 10^{-194}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)} \cdot \left(\sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)}\right)\right)\\ \mathbf{elif}\;x \le 1.960703821127914 \cdot 10^{-183}:\\ \;\;\;\;\left(c \cdot z - t \cdot i\right) \cdot \left(-b\right) + \left(c \cdot a - y \cdot i\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32827436 = x;
        double r32827437 = y;
        double r32827438 = z;
        double r32827439 = r32827437 * r32827438;
        double r32827440 = t;
        double r32827441 = a;
        double r32827442 = r32827440 * r32827441;
        double r32827443 = r32827439 - r32827442;
        double r32827444 = r32827436 * r32827443;
        double r32827445 = b;
        double r32827446 = c;
        double r32827447 = r32827446 * r32827438;
        double r32827448 = i;
        double r32827449 = r32827440 * r32827448;
        double r32827450 = r32827447 - r32827449;
        double r32827451 = r32827445 * r32827450;
        double r32827452 = r32827444 - r32827451;
        double r32827453 = j;
        double r32827454 = r32827446 * r32827441;
        double r32827455 = r32827437 * r32827448;
        double r32827456 = r32827454 - r32827455;
        double r32827457 = r32827453 * r32827456;
        double r32827458 = r32827452 + r32827457;
        return r32827458;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32827459 = x;
        double r32827460 = -1.5726372568800548e-194;
        bool r32827461 = r32827459 <= r32827460;
        double r32827462 = c;
        double r32827463 = a;
        double r32827464 = r32827462 * r32827463;
        double r32827465 = y;
        double r32827466 = i;
        double r32827467 = r32827465 * r32827466;
        double r32827468 = r32827464 - r32827467;
        double r32827469 = j;
        double r32827470 = r32827468 * r32827469;
        double r32827471 = z;
        double r32827472 = r32827465 * r32827471;
        double r32827473 = t;
        double r32827474 = r32827463 * r32827473;
        double r32827475 = r32827472 - r32827474;
        double r32827476 = r32827475 * r32827459;
        double r32827477 = b;
        double r32827478 = cbrt(r32827477);
        double r32827479 = r32827478 * r32827478;
        double r32827480 = r32827462 * r32827471;
        double r32827481 = r32827473 * r32827466;
        double r32827482 = r32827480 - r32827481;
        double r32827483 = r32827482 * r32827478;
        double r32827484 = r32827479 * r32827483;
        double r32827485 = cbrt(r32827484);
        double r32827486 = r32827485 * r32827485;
        double r32827487 = r32827485 * r32827486;
        double r32827488 = r32827476 - r32827487;
        double r32827489 = r32827470 + r32827488;
        double r32827490 = 1.960703821127914e-183;
        bool r32827491 = r32827459 <= r32827490;
        double r32827492 = -r32827477;
        double r32827493 = r32827482 * r32827492;
        double r32827494 = r32827493 + r32827470;
        double r32827495 = r32827477 * r32827482;
        double r32827496 = r32827476 - r32827495;
        double r32827497 = cbrt(r32827469);
        double r32827498 = r32827497 * r32827497;
        double r32827499 = r32827497 * r32827468;
        double r32827500 = r32827498 * r32827499;
        double r32827501 = r32827496 + r32827500;
        double r32827502 = r32827491 ? r32827494 : r32827501;
        double r32827503 = r32827461 ? r32827489 : r32827502;
        return r32827503;
}

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	11.6
Target	19.2
Herbie	11.8

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.5726372568800548e-194

   1. Initial program 9.9

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

   3. Applied add-cube-cbrt 10.2

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

   4. Applied associate-*l* 10.2

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

   6. Applied add-cube-cbrt 10.3

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

   if -1.5726372568800548e-194 < x < 1.960703821127914e-183

   1. Initial program 17.4

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

   2. Taylor expanded around 0 17.2

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

   if 1.960703821127914e-183 < x

   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

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

   4. Applied associate-*l* 9.8

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5726372568800548 \cdot 10^{-194}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)} \cdot \left(\sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - t \cdot i\right) \cdot \sqrt[3]{b}\right)}\right)\right)\\ \mathbf{elif}\;x \le 1.960703821127914 \cdot 10^{-183}:\\ \;\;\;\;\left(c \cdot z - t \cdot i\right) \cdot \left(-b\right) + \left(c \cdot a - y \cdot i\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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) (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)))))