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

Average Error: 12.5 → 12.9

Time: 8.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.53757378257533171 \cdot 10^{-120}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;x \le 4.4248614871445536 \cdot 10^{-179}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1000635 = x;
        double r1000636 = y;
        double r1000637 = z;
        double r1000638 = r1000636 * r1000637;
        double r1000639 = t;
        double r1000640 = a;
        double r1000641 = r1000639 * r1000640;
        double r1000642 = r1000638 - r1000641;
        double r1000643 = r1000635 * r1000642;
        double r1000644 = b;
        double r1000645 = c;
        double r1000646 = r1000645 * r1000637;
        double r1000647 = i;
        double r1000648 = r1000639 * r1000647;
        double r1000649 = r1000646 - r1000648;
        double r1000650 = r1000644 * r1000649;
        double r1000651 = r1000643 - r1000650;
        double r1000652 = j;
        double r1000653 = r1000645 * r1000640;
        double r1000654 = r1000636 * r1000647;
        double r1000655 = r1000653 - r1000654;
        double r1000656 = r1000652 * r1000655;
        double r1000657 = r1000651 + r1000656;
        return r1000657;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1000658 = x;
        double r1000659 = -5.537573782575332e-120;
        bool r1000660 = r1000658 <= r1000659;
        double r1000661 = y;
        double r1000662 = z;
        double r1000663 = r1000661 * r1000662;
        double r1000664 = t;
        double r1000665 = a;
        double r1000666 = r1000664 * r1000665;
        double r1000667 = r1000663 - r1000666;
        double r1000668 = r1000658 * r1000667;
        double r1000669 = b;
        double r1000670 = c;
        double r1000671 = r1000670 * r1000662;
        double r1000672 = i;
        double r1000673 = r1000664 * r1000672;
        double r1000674 = r1000671 - r1000673;
        double r1000675 = r1000669 * r1000674;
        double r1000676 = r1000668 - r1000675;
        double r1000677 = j;
        double r1000678 = r1000670 * r1000665;
        double r1000679 = r1000661 * r1000672;
        double r1000680 = r1000678 - r1000679;
        double r1000681 = r1000677 * r1000680;
        double r1000682 = cbrt(r1000681);
        double r1000683 = r1000682 * r1000682;
        double r1000684 = r1000683 * r1000682;
        double r1000685 = r1000676 + r1000684;
        double r1000686 = 4.4248614871445536e-179;
        bool r1000687 = r1000658 <= r1000686;
        double r1000688 = cbrt(r1000658);
        double r1000689 = r1000688 * r1000688;
        double r1000690 = 0.0;
        double r1000691 = r1000689 * r1000690;
        double r1000692 = r1000691 - r1000675;
        double r1000693 = r1000692 + r1000681;
        double r1000694 = sqrt(r1000658);
        double r1000695 = r1000694 * r1000667;
        double r1000696 = r1000694 * r1000695;
        double r1000697 = r1000696 - r1000675;
        double r1000698 = r1000697 + r1000681;
        double r1000699 = r1000687 ? r1000693 : r1000698;
        double r1000700 = r1000660 ? r1000685 : r1000699;
        return r1000700;
}

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	12.5
Target	20.2
Herbie	12.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -5.537573782575332e-120

   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

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

   if -5.537573782575332e-120 < x < 4.4248614871445536e-179

   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. Using strategy rm

   3. Applied add-cube-cbrt 17.7

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

   4. Applied associate-*l* 17.7

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

   5. Taylor expanded around 0 18.4

      \[\leadsto \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{0} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   if 4.4248614871445536e-179 < x

   1. Initial program 10.5

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

   3. Applied add-sqr-sqrt 10.6

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

   4. Applied associate-*l* 10.6

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.53757378257533171 \cdot 10^{-120}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;x \le 4.4248614871445536 \cdot 10^{-179}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))))