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

Average Error: 12.6 → 11.6

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.87029124636687325 \cdot 10^{-46}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \le 2.83754174328183624 \cdot 10^{-162}:\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(x \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r851634 = x;
        double r851635 = y;
        double r851636 = z;
        double r851637 = r851635 * r851636;
        double r851638 = t;
        double r851639 = a;
        double r851640 = r851638 * r851639;
        double r851641 = r851637 - r851640;
        double r851642 = r851634 * r851641;
        double r851643 = b;
        double r851644 = c;
        double r851645 = r851644 * r851636;
        double r851646 = i;
        double r851647 = r851638 * r851646;
        double r851648 = r851645 - r851647;
        double r851649 = r851643 * r851648;
        double r851650 = r851642 - r851649;
        double r851651 = j;
        double r851652 = r851644 * r851639;
        double r851653 = r851635 * r851646;
        double r851654 = r851652 - r851653;
        double r851655 = r851651 * r851654;
        double r851656 = r851650 + r851655;
        return r851656;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r851657 = x;
        double r851658 = -5.870291246366873e-46;
        bool r851659 = r851657 <= r851658;
        double r851660 = y;
        double r851661 = z;
        double r851662 = r851660 * r851661;
        double r851663 = t;
        double r851664 = a;
        double r851665 = r851663 * r851664;
        double r851666 = r851662 - r851665;
        double r851667 = r851657 * r851666;
        double r851668 = b;
        double r851669 = cbrt(r851668);
        double r851670 = r851669 * r851669;
        double r851671 = c;
        double r851672 = r851671 * r851661;
        double r851673 = i;
        double r851674 = r851663 * r851673;
        double r851675 = r851672 - r851674;
        double r851676 = r851669 * r851675;
        double r851677 = cbrt(r851676);
        double r851678 = r851677 * r851677;
        double r851679 = r851678 * r851677;
        double r851680 = r851670 * r851679;
        double r851681 = r851667 - r851680;
        double r851682 = j;
        double r851683 = r851671 * r851664;
        double r851684 = r851660 * r851673;
        double r851685 = r851683 - r851684;
        double r851686 = r851682 * r851685;
        double r851687 = r851681 + r851686;
        double r851688 = 2.837541743281836e-162;
        bool r851689 = r851657 <= r851688;
        double r851690 = r851673 * r851668;
        double r851691 = r851663 * r851690;
        double r851692 = r851668 * r851671;
        double r851693 = r851661 * r851692;
        double r851694 = r851657 * r851663;
        double r851695 = r851664 * r851694;
        double r851696 = r851693 + r851695;
        double r851697 = r851691 - r851696;
        double r851698 = r851697 + r851686;
        double r851699 = r851668 * r851675;
        double r851700 = r851667 - r851699;
        double r851701 = cbrt(r851686);
        double r851702 = r851701 * r851701;
        double r851703 = r851702 * r851701;
        double r851704 = r851700 + r851703;
        double r851705 = r851689 ? r851698 : r851704;
        double r851706 = r851659 ? r851687 : r851705;
        return r851706;
}

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	12.6
Target	20.2
Herbie	11.6

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

Derivation

1. Split input into 3 regimes

2. if x < -5.870291246366873e-46

   1. Initial program 8.2

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

   3. Applied add-cube-cbrt 8.4

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

   4. Applied associate-*l* 8.4

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

   6. Applied add-cube-cbrt 8.5

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

   if -5.870291246366873e-46 < x < 2.837541743281836e-162

   1. Initial program 17.2

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

   2. Taylor expanded around inf 14.2

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

   if 2.837541743281836e-162 < x

   1. Initial program 10.3

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

   3. Applied add-cube-cbrt 10.5

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.87029124636687325 \cdot 10^{-46}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \le 2.83754174328183624 \cdot 10^{-162}:\\ \;\;\;\;\left(t \cdot \left(i \cdot b\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(x \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))))