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

Average Error: 11.6 → 7.4

Time: 9.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) = -\infty \lor \neg \left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \le 4.999599852208662350331065524329530205172 \cdot 10^{305}\right):\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) + a \cdot \left(j \cdot c\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r875513 = x;
        double r875514 = y;
        double r875515 = z;
        double r875516 = r875514 * r875515;
        double r875517 = t;
        double r875518 = a;
        double r875519 = r875517 * r875518;
        double r875520 = r875516 - r875519;
        double r875521 = r875513 * r875520;
        double r875522 = b;
        double r875523 = c;
        double r875524 = r875523 * r875515;
        double r875525 = i;
        double r875526 = r875517 * r875525;
        double r875527 = r875524 - r875526;
        double r875528 = r875522 * r875527;
        double r875529 = r875521 - r875528;
        double r875530 = j;
        double r875531 = r875523 * r875518;
        double r875532 = r875514 * r875525;
        double r875533 = r875531 - r875532;
        double r875534 = r875530 * r875533;
        double r875535 = r875529 + r875534;
        return r875535;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r875536 = x;
        double r875537 = y;
        double r875538 = z;
        double r875539 = r875537 * r875538;
        double r875540 = t;
        double r875541 = a;
        double r875542 = r875540 * r875541;
        double r875543 = r875539 - r875542;
        double r875544 = r875536 * r875543;
        double r875545 = b;
        double r875546 = c;
        double r875547 = r875546 * r875538;
        double r875548 = i;
        double r875549 = r875540 * r875548;
        double r875550 = r875547 - r875549;
        double r875551 = r875545 * r875550;
        double r875552 = r875544 - r875551;
        double r875553 = j;
        double r875554 = r875546 * r875541;
        double r875555 = r875537 * r875548;
        double r875556 = r875554 - r875555;
        double r875557 = r875553 * r875556;
        double r875558 = r875552 + r875557;
        double r875559 = -inf.0;
        bool r875560 = r875558 <= r875559;
        double r875561 = 4.999599852208662e+305;
        bool r875562 = r875558 <= r875561;
        double r875563 = !r875562;
        bool r875564 = r875560 || r875563;
        double r875565 = r875548 * r875545;
        double r875566 = r875540 * r875565;
        double r875567 = r875553 * r875546;
        double r875568 = r875541 * r875567;
        double r875569 = r875566 + r875568;
        double r875570 = r875553 * r875537;
        double r875571 = r875548 * r875570;
        double r875572 = r875569 - r875571;
        double r875573 = r875545 * r875547;
        double r875574 = -r875549;
        double r875575 = r875545 * r875574;
        double r875576 = r875573 + r875575;
        double r875577 = r875544 - r875576;
        double r875578 = cbrt(r875553);
        double r875579 = r875578 * r875578;
        double r875580 = r875578 * r875556;
        double r875581 = r875579 * r875580;
        double r875582 = r875577 + r875581;
        double r875583 = r875564 ? r875572 : r875582;
        return r875583;
}

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	19.6
Herbie	7.4

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

Derivation

1. Split input into 2 regimes

2. if (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))) < -inf.0 or 4.999599852208662e+305 < (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i))))

   1. Initial program 62.6

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

   2. Taylor expanded around inf 37.0

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

   if -inf.0 < (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))) < 4.999599852208662e+305

   1. Initial program 0.7

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

   3. Applied sub-neg 0.7

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

   4. Applied distribute-lft-in 0.7

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

   6. Applied add-cube-cbrt 1.1

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

   7. Applied associate-*l* 1.1

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) = -\infty \lor \neg \left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \le 4.999599852208662350331065524329530205172 \cdot 10^{305}\right):\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) + a \cdot \left(j \cdot c\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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 -1.469694296777705e-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)))))