Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 9.2

Time: 27.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -329700211979175.0625:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-x\right) \cdot \left(a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;j \le 4.422735920963178095732739417621615994879 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(j \cdot y\right) \cdot \left(-i\right) + c \cdot \left(j \cdot t\right)\right) + \left(\mathsf{fma}\left(x \cdot a, -t, x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - \left(a \cdot \left(b \cdot \left(-i\right)\right) + b \cdot \left(c \cdot z\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r335736 = x;
        double r335737 = y;
        double r335738 = z;
        double r335739 = r335737 * r335738;
        double r335740 = t;
        double r335741 = a;
        double r335742 = r335740 * r335741;
        double r335743 = r335739 - r335742;
        double r335744 = r335736 * r335743;
        double r335745 = b;
        double r335746 = c;
        double r335747 = r335746 * r335738;
        double r335748 = i;
        double r335749 = r335748 * r335741;
        double r335750 = r335747 - r335749;
        double r335751 = r335745 * r335750;
        double r335752 = r335744 - r335751;
        double r335753 = j;
        double r335754 = r335746 * r335740;
        double r335755 = r335748 * r335737;
        double r335756 = r335754 - r335755;
        double r335757 = r335753 * r335756;
        double r335758 = r335752 + r335757;
        return r335758;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r335759 = j;
        double r335760 = -329700211979175.06;
        bool r335761 = r335759 <= r335760;
        double r335762 = z;
        double r335763 = x;
        double r335764 = r335762 * r335763;
        double r335765 = y;
        double r335766 = r335764 * r335765;
        double r335767 = -r335763;
        double r335768 = a;
        double r335769 = t;
        double r335770 = r335768 * r335769;
        double r335771 = r335767 * r335770;
        double r335772 = r335766 + r335771;
        double r335773 = b;
        double r335774 = c;
        double r335775 = r335774 * r335762;
        double r335776 = i;
        double r335777 = r335776 * r335768;
        double r335778 = r335775 - r335777;
        double r335779 = r335773 * r335778;
        double r335780 = r335772 - r335779;
        double r335781 = r335774 * r335769;
        double r335782 = r335765 * r335776;
        double r335783 = r335781 - r335782;
        double r335784 = r335759 * r335783;
        double r335785 = r335780 + r335784;
        double r335786 = 4.422735920963178e-44;
        bool r335787 = r335759 <= r335786;
        double r335788 = r335759 * r335765;
        double r335789 = -r335776;
        double r335790 = r335788 * r335789;
        double r335791 = r335759 * r335769;
        double r335792 = r335774 * r335791;
        double r335793 = r335790 + r335792;
        double r335794 = r335763 * r335768;
        double r335795 = -r335769;
        double r335796 = r335762 * r335765;
        double r335797 = r335763 * r335796;
        double r335798 = fma(r335794, r335795, r335797);
        double r335799 = r335798 - r335779;
        double r335800 = r335793 + r335799;
        double r335801 = r335796 - r335770;
        double r335802 = r335763 * r335801;
        double r335803 = r335773 * r335789;
        double r335804 = r335768 * r335803;
        double r335805 = r335773 * r335775;
        double r335806 = r335804 + r335805;
        double r335807 = r335802 - r335806;
        double r335808 = r335807 + r335784;
        double r335809 = r335787 ? r335800 : r335808;
        double r335810 = r335761 ? r335785 : r335809;
        return r335810;
}

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	12.2
Target	16.3
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if j < -329700211979175.06

   1. Initial program 7.5

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

   3. Applied sub-neg 7.5

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

   4. Applied distribute-lft-in 7.5

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

   5. Simplified 8.1

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

   6. Simplified 8.1

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

   if -329700211979175.06 < j < 4.422735920963178e-44

   1. Initial program 15.7

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

   3. Applied sub-neg 15.7

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

   4. Applied distribute-lft-in 15.7

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

   5. Simplified 13.2

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

   6. Simplified 10.5

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

   7. Taylor expanded around inf 11.0

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

   8. Simplified 10.1

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

   10. Applied associate-*l* 10.1

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

   11. Simplified 10.1

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

   if 4.422735920963178e-44 < j

   1. Initial program 7.4

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

   3. Applied sub-neg 7.4

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

   4. Applied distribute-lft-in 7.4

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

   5. Simplified 7.7

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -329700211979175.0625:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-x\right) \cdot \left(a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;j \le 4.422735920963178095732739417621615994879 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(j \cdot y\right) \cdot \left(-i\right) + c \cdot \left(j \cdot t\right)\right) + \left(\mathsf{fma}\left(x \cdot a, -t, x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - \left(a \cdot \left(b \cdot \left(-i\right)\right) + b \cdot \left(c \cdot z\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :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) 2.0) (pow (* i y) 2.0))) (+ (* 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) 2.0) (pow (* i y) 2.0))) (+ (* 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)))))