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

Average Error: 12.1 → 9.4

Time: 29.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.127944646935574702339279358207920026995 \cdot 10^{-134}:\\ \;\;\;\;\left(a \cdot \left(c \cdot j\right) - \left(j \cdot y\right) \cdot i\right) + \left(\left(\left(y \cdot z\right) \cdot x + x \cdot \left(-t \cdot a\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;x \le 6.207332660334129063897749455315579858521 \cdot 10^{69}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(z \cdot x\right) \cdot y + t \cdot \left(a \cdot \left(-x\right)\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot a - y \cdot i} + \left(\left(\left(y \cdot z\right) \cdot x + x \cdot \left(-t \cdot a\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r22922635 = x;
        double r22922636 = y;
        double r22922637 = z;
        double r22922638 = r22922636 * r22922637;
        double r22922639 = t;
        double r22922640 = a;
        double r22922641 = r22922639 * r22922640;
        double r22922642 = r22922638 - r22922641;
        double r22922643 = r22922635 * r22922642;
        double r22922644 = b;
        double r22922645 = c;
        double r22922646 = r22922645 * r22922637;
        double r22922647 = i;
        double r22922648 = r22922639 * r22922647;
        double r22922649 = r22922646 - r22922648;
        double r22922650 = r22922644 * r22922649;
        double r22922651 = r22922643 - r22922650;
        double r22922652 = j;
        double r22922653 = r22922645 * r22922640;
        double r22922654 = r22922636 * r22922647;
        double r22922655 = r22922653 - r22922654;
        double r22922656 = r22922652 * r22922655;
        double r22922657 = r22922651 + r22922656;
        return r22922657;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r22922658 = x;
        double r22922659 = -4.1279446469355747e-134;
        bool r22922660 = r22922658 <= r22922659;
        double r22922661 = a;
        double r22922662 = c;
        double r22922663 = j;
        double r22922664 = r22922662 * r22922663;
        double r22922665 = r22922661 * r22922664;
        double r22922666 = y;
        double r22922667 = r22922663 * r22922666;
        double r22922668 = i;
        double r22922669 = r22922667 * r22922668;
        double r22922670 = r22922665 - r22922669;
        double r22922671 = z;
        double r22922672 = r22922666 * r22922671;
        double r22922673 = r22922672 * r22922658;
        double r22922674 = t;
        double r22922675 = r22922674 * r22922661;
        double r22922676 = -r22922675;
        double r22922677 = r22922658 * r22922676;
        double r22922678 = r22922673 + r22922677;
        double r22922679 = r22922662 * r22922671;
        double r22922680 = r22922674 * r22922668;
        double r22922681 = r22922679 - r22922680;
        double r22922682 = b;
        double r22922683 = r22922681 * r22922682;
        double r22922684 = r22922678 - r22922683;
        double r22922685 = r22922670 + r22922684;
        double r22922686 = 6.207332660334129e+69;
        bool r22922687 = r22922658 <= r22922686;
        double r22922688 = r22922662 * r22922661;
        double r22922689 = r22922666 * r22922668;
        double r22922690 = r22922688 - r22922689;
        double r22922691 = r22922690 * r22922663;
        double r22922692 = r22922671 * r22922658;
        double r22922693 = r22922692 * r22922666;
        double r22922694 = -r22922658;
        double r22922695 = r22922661 * r22922694;
        double r22922696 = r22922674 * r22922695;
        double r22922697 = r22922693 + r22922696;
        double r22922698 = r22922697 - r22922683;
        double r22922699 = r22922691 + r22922698;
        double r22922700 = cbrt(r22922690);
        double r22922701 = r22922700 * r22922700;
        double r22922702 = r22922701 * r22922663;
        double r22922703 = r22922702 * r22922700;
        double r22922704 = r22922703 + r22922684;
        double r22922705 = r22922687 ? r22922699 : r22922704;
        double r22922706 = r22922660 ? r22922685 : r22922705;
        return r22922706;
}

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	19.7
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if x < -4.1279446469355747e-134

   1. Initial program 9.0

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

   3. Applied sub-neg 9.0

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

   4. Applied distribute-rgt-in 9.0

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

   5. Taylor expanded around inf 9.3

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

   if -4.1279446469355747e-134 < x < 6.207332660334129e+69

   1. Initial program 15.1

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

   3. Applied sub-neg 15.1

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

   4. Applied distribute-rgt-in 15.1

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

   6. Applied distribute-rgt-neg-in 15.1

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

   7. Applied associate-*l* 12.5

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

   9. Applied associate-*l* 9.9

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

   if 6.207332660334129e+69 < x

   1. Initial program 7.1

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

   3. Applied sub-neg 7.1

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

   4. Applied distribute-rgt-in 7.1

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

   6. Applied add-cube-cbrt 7.3

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

   7. Applied associate-*r* 7.3

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.127944646935574702339279358207920026995 \cdot 10^{-134}:\\ \;\;\;\;\left(a \cdot \left(c \cdot j\right) - \left(j \cdot y\right) \cdot i\right) + \left(\left(\left(y \cdot z\right) \cdot x + x \cdot \left(-t \cdot a\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;x \le 6.207332660334129063897749455315579858521 \cdot 10^{69}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\left(z \cdot x\right) \cdot y + t \cdot \left(a \cdot \left(-x\right)\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot a - y \cdot i} + \left(\left(\left(y \cdot z\right) \cdot x + x \cdot \left(-t \cdot a\right)\right) - \left(c \cdot z - t \cdot i\right) \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))