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

Average Error: 12.1 → 12.0

Time: 11.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1011394834188850593395638272:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;x \le 5.532957602419463810363964349188881471165 \cdot 10^{-223}:\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(x \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j \cdot \left(c \cdot a\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a}\right)\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a\right)} + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r763632 = x;
        double r763633 = y;
        double r763634 = z;
        double r763635 = r763633 * r763634;
        double r763636 = t;
        double r763637 = a;
        double r763638 = r763636 * r763637;
        double r763639 = r763635 - r763638;
        double r763640 = r763632 * r763639;
        double r763641 = b;
        double r763642 = c;
        double r763643 = r763642 * r763634;
        double r763644 = i;
        double r763645 = r763636 * r763644;
        double r763646 = r763643 - r763645;
        double r763647 = r763641 * r763646;
        double r763648 = r763640 - r763647;
        double r763649 = j;
        double r763650 = r763642 * r763637;
        double r763651 = r763633 * r763644;
        double r763652 = r763650 - r763651;
        double r763653 = r763649 * r763652;
        double r763654 = r763648 + r763653;
        return r763654;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r763655 = x;
        double r763656 = -1.0113948341888506e+27;
        bool r763657 = r763655 <= r763656;
        double r763658 = y;
        double r763659 = z;
        double r763660 = r763658 * r763659;
        double r763661 = t;
        double r763662 = a;
        double r763663 = r763661 * r763662;
        double r763664 = r763660 - r763663;
        double r763665 = r763655 * r763664;
        double r763666 = b;
        double r763667 = c;
        double r763668 = r763667 * r763659;
        double r763669 = i;
        double r763670 = r763661 * r763669;
        double r763671 = r763668 - r763670;
        double r763672 = r763666 * r763671;
        double r763673 = cbrt(r763672);
        double r763674 = r763673 * r763673;
        double r763675 = r763674 * r763673;
        double r763676 = r763665 - r763675;
        double r763677 = j;
        double r763678 = r763667 * r763662;
        double r763679 = r763677 * r763678;
        double r763680 = r763658 * r763669;
        double r763681 = -r763680;
        double r763682 = r763677 * r763681;
        double r763683 = r763679 + r763682;
        double r763684 = r763676 + r763683;
        double r763685 = 5.532957602419464e-223;
        bool r763686 = r763655 <= r763685;
        double r763687 = r763669 * r763666;
        double r763688 = r763661 * r763687;
        double r763689 = r763666 * r763667;
        double r763690 = r763659 * r763689;
        double r763691 = r763655 * r763661;
        double r763692 = r763662 * r763691;
        double r763693 = r763690 + r763692;
        double r763694 = r763688 - r763693;
        double r763695 = r763678 - r763680;
        double r763696 = r763677 * r763695;
        double r763697 = r763694 + r763696;
        double r763698 = r763665 - r763672;
        double r763699 = cbrt(r763679);
        double r763700 = cbrt(r763677);
        double r763701 = cbrt(r763678);
        double r763702 = r763700 * r763701;
        double r763703 = r763699 * r763702;
        double r763704 = r763703 * r763699;
        double r763705 = r763704 + r763682;
        double r763706 = r763698 + r763705;
        double r763707 = r763686 ? r763697 : r763706;
        double r763708 = r763657 ? r763684 : r763707;
        return r763708;
}

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	12.1
Target	19.8
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0113948341888506e+27

   1. Initial program 7.2

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

   3. Applied sub-neg 7.2

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

   4. Applied distribute-lft-in 7.2

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

   6. Applied add-cube-cbrt 7.4

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

   if -1.0113948341888506e+27 < x < 5.532957602419464e-223

   1. Initial program 15.5

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

   2. Taylor expanded around inf 14.9

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

   if 5.532957602419464e-223 < x

   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. Using strategy rm

   6. Applied add-cube-cbrt 10.9

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

   8. Applied cbrt-prod 10.9

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1011394834188850593395638272:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;x \le 5.532957602419463810363964349188881471165 \cdot 10^{-223}:\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(x \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j \cdot \left(c \cdot a\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a}\right)\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a\right)} + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))))