Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.4 → 10.0

Time: 8.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.988825475455037498549084748882071237206 \cdot 10^{-159} \lor \neg \left(x \le 8.590299919376646895424917696537023407421 \cdot 10^{46}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r641712 = x;
        double r641713 = y;
        double r641714 = z;
        double r641715 = r641713 * r641714;
        double r641716 = t;
        double r641717 = a;
        double r641718 = r641716 * r641717;
        double r641719 = r641715 - r641718;
        double r641720 = r641712 * r641719;
        double r641721 = b;
        double r641722 = c;
        double r641723 = r641722 * r641714;
        double r641724 = i;
        double r641725 = r641724 * r641717;
        double r641726 = r641723 - r641725;
        double r641727 = r641721 * r641726;
        double r641728 = r641720 - r641727;
        double r641729 = j;
        double r641730 = r641722 * r641716;
        double r641731 = r641724 * r641713;
        double r641732 = r641730 - r641731;
        double r641733 = r641729 * r641732;
        double r641734 = r641728 + r641733;
        return r641734;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r641735 = x;
        double r641736 = -3.9888254754550375e-159;
        bool r641737 = r641735 <= r641736;
        double r641738 = 8.590299919376647e+46;
        bool r641739 = r641735 <= r641738;
        double r641740 = !r641739;
        bool r641741 = r641737 || r641740;
        double r641742 = cbrt(r641735);
        double r641743 = r641742 * r641742;
        double r641744 = y;
        double r641745 = z;
        double r641746 = r641744 * r641745;
        double r641747 = r641742 * r641746;
        double r641748 = r641743 * r641747;
        double r641749 = t;
        double r641750 = a;
        double r641751 = r641749 * r641750;
        double r641752 = -r641751;
        double r641753 = r641735 * r641752;
        double r641754 = r641748 + r641753;
        double r641755 = b;
        double r641756 = c;
        double r641757 = r641756 * r641745;
        double r641758 = i;
        double r641759 = r641758 * r641750;
        double r641760 = r641757 - r641759;
        double r641761 = r641755 * r641760;
        double r641762 = r641754 - r641761;
        double r641763 = j;
        double r641764 = r641756 * r641749;
        double r641765 = r641758 * r641744;
        double r641766 = r641764 - r641765;
        double r641767 = r641763 * r641766;
        double r641768 = r641762 + r641767;
        double r641769 = r641735 * r641744;
        double r641770 = r641769 * r641745;
        double r641771 = -1.0;
        double r641772 = r641735 * r641749;
        double r641773 = r641750 * r641772;
        double r641774 = r641771 * r641773;
        double r641775 = 1.0;
        double r641776 = pow(r641774, r641775);
        double r641777 = r641770 + r641776;
        double r641778 = r641777 - r641761;
        double r641779 = cbrt(r641767);
        double r641780 = r641779 * r641779;
        double r641781 = r641780 * r641779;
        double r641782 = r641778 + r641781;
        double r641783 = r641741 ? r641768 : r641782;
        return r641783;
}

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	12.4
Target	16.1
Herbie	10.0

\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -3.9888254754550375e-159 or 8.590299919376647e+46 < x

   1. Initial program 9.2

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

   3. Applied sub-neg 9.2

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

   4. Applied distribute-lft-in 9.2

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

   6. Applied add-cube-cbrt 9.4

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

   7. Applied associate-*l* 9.4

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

   if -3.9888254754550375e-159 < x < 8.590299919376647e+46

   1. Initial program 15.6

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

   3. Applied sub-neg 15.6

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

   4. Applied distribute-lft-in 15.6

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

   6. Applied associate-*r* 13.0

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

   8. Applied pow1 13.0

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

   9. Applied pow1 13.0

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

   10. Applied pow-prod-down 13.0

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

   11. Simplified 10.3

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

   13. Applied add-cube-cbrt 10.6

       \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.988825475455037498549084748882071237206 \cdot 10^{-159} \lor \neg \left(x \le 8.590299919376646895424917696537023407421 \cdot 10^{46}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

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