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

Average Error: 12.3 → 12.7

Time: 22.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.251443854831909976072487415727622261891 \cdot 10^{-212}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \left(\sqrt[3]{i} \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;x \le 2.327388989455034964452189310677033160838 \cdot 10^{-241}:\\ \;\;\;\;\left(x \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-t \cdot i\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r542556 = x;
        double r542557 = y;
        double r542558 = z;
        double r542559 = r542557 * r542558;
        double r542560 = t;
        double r542561 = a;
        double r542562 = r542560 * r542561;
        double r542563 = r542559 - r542562;
        double r542564 = r542556 * r542563;
        double r542565 = b;
        double r542566 = c;
        double r542567 = r542566 * r542558;
        double r542568 = i;
        double r542569 = r542560 * r542568;
        double r542570 = r542567 - r542569;
        double r542571 = r542565 * r542570;
        double r542572 = r542564 - r542571;
        double r542573 = j;
        double r542574 = r542566 * r542561;
        double r542575 = r542557 * r542568;
        double r542576 = r542574 - r542575;
        double r542577 = r542573 * r542576;
        double r542578 = r542572 + r542577;
        return r542578;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r542579 = x;
        double r542580 = -4.25144385483191e-212;
        bool r542581 = r542579 <= r542580;
        double r542582 = y;
        double r542583 = z;
        double r542584 = r542582 * r542583;
        double r542585 = t;
        double r542586 = a;
        double r542587 = r542585 * r542586;
        double r542588 = r542584 - r542587;
        double r542589 = r542579 * r542588;
        double r542590 = b;
        double r542591 = c;
        double r542592 = r542591 * r542583;
        double r542593 = i;
        double r542594 = r542585 * r542593;
        double r542595 = r542592 - r542594;
        double r542596 = r542590 * r542595;
        double r542597 = r542589 - r542596;
        double r542598 = j;
        double r542599 = r542586 * r542598;
        double r542600 = r542599 * r542591;
        double r542601 = cbrt(r542593);
        double r542602 = r542601 * r542601;
        double r542603 = r542598 * r542582;
        double r542604 = r542601 * r542603;
        double r542605 = r542602 * r542604;
        double r542606 = -r542605;
        double r542607 = r542600 + r542606;
        double r542608 = r542597 + r542607;
        double r542609 = 2.327388989455035e-241;
        bool r542610 = r542579 <= r542609;
        double r542611 = 0.0;
        double r542612 = r542579 * r542611;
        double r542613 = r542612 - r542596;
        double r542614 = r542591 * r542586;
        double r542615 = r542582 * r542593;
        double r542616 = r542614 - r542615;
        double r542617 = r542598 * r542616;
        double r542618 = r542613 + r542617;
        double r542619 = r542590 * r542591;
        double r542620 = r542583 * r542619;
        double r542621 = -r542594;
        double r542622 = r542590 * r542621;
        double r542623 = r542620 + r542622;
        double r542624 = r542589 - r542623;
        double r542625 = -r542615;
        double r542626 = r542598 * r542625;
        double r542627 = r542600 + r542626;
        double r542628 = r542624 + r542627;
        double r542629 = r542610 ? r542618 : r542628;
        double r542630 = r542581 ? r542608 : r542629;
        return r542630;
}

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	12.3
Target	19.5
Herbie	12.7

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

Derivation

1. Split input into 3 regimes

2. if x < -4.25144385483191e-212

   1. Initial program 10.8

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

   3. Applied sub-neg 10.8

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

   4. Applied distribute-lft-in 10.8

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

   5. Simplified 10.8

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

   7. Applied associate-*r* 10.9

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

   8. Taylor expanded around inf 11.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)}\right)\]

   9. Simplified 11.0

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

   11. Applied add-cube-cbrt 11.1

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

   12. Applied associate-*l* 11.1

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

   if -4.25144385483191e-212 < x < 2.327388989455035e-241

   1. Initial program 17.7

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

   2. Taylor expanded around 0 16.4

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

   if 2.327388989455035e-241 < x

   1. Initial program 11.6

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

   3. Applied sub-neg 11.6

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

   4. Applied distribute-lft-in 11.6

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

   5. Simplified 12.2

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

   7. Applied associate-*r* 12.2

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

   9. Applied sub-neg 12.2

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

   10. Applied distribute-lft-in 12.2

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

   11. Simplified 12.7

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.251443854831909976072487415727622261891 \cdot 10^{-212}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \left(\sqrt[3]{i} \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;x \le 2.327388989455034964452189310677033160838 \cdot 10^{-241}:\\ \;\;\;\;\left(x \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-t \cdot i\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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)))))