Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.4 → 10.9

Time: 34.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.262228259703944418388054681956517640133 \cdot 10^{72}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{x} \cdot \left(\left(y \cdot z - t \cdot a\right) \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x} - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \mathbf{elif}\;x \le 7.708714260711597468016611731654685252532 \cdot 10^{86}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(z \cdot \left(y \cdot x\right) + \left(x \cdot a\right) \cdot \left(-t\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r25596806 = x;
        double r25596807 = y;
        double r25596808 = z;
        double r25596809 = r25596807 * r25596808;
        double r25596810 = t;
        double r25596811 = a;
        double r25596812 = r25596810 * r25596811;
        double r25596813 = r25596809 - r25596812;
        double r25596814 = r25596806 * r25596813;
        double r25596815 = b;
        double r25596816 = c;
        double r25596817 = r25596816 * r25596808;
        double r25596818 = i;
        double r25596819 = r25596818 * r25596811;
        double r25596820 = r25596817 - r25596819;
        double r25596821 = r25596815 * r25596820;
        double r25596822 = r25596814 - r25596821;
        double r25596823 = j;
        double r25596824 = r25596816 * r25596810;
        double r25596825 = r25596818 * r25596807;
        double r25596826 = r25596824 - r25596825;
        double r25596827 = r25596823 * r25596826;
        double r25596828 = r25596822 + r25596827;
        return r25596828;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r25596829 = x;
        double r25596830 = -6.2622282597039444e+72;
        bool r25596831 = r25596829 <= r25596830;
        double r25596832 = c;
        double r25596833 = t;
        double r25596834 = r25596832 * r25596833;
        double r25596835 = y;
        double r25596836 = i;
        double r25596837 = r25596835 * r25596836;
        double r25596838 = r25596834 - r25596837;
        double r25596839 = j;
        double r25596840 = r25596838 * r25596839;
        double r25596841 = cbrt(r25596829);
        double r25596842 = z;
        double r25596843 = r25596835 * r25596842;
        double r25596844 = a;
        double r25596845 = r25596833 * r25596844;
        double r25596846 = r25596843 - r25596845;
        double r25596847 = r25596846 * r25596841;
        double r25596848 = r25596841 * r25596847;
        double r25596849 = r25596848 * r25596841;
        double r25596850 = b;
        double r25596851 = r25596832 * r25596842;
        double r25596852 = r25596844 * r25596836;
        double r25596853 = r25596851 - r25596852;
        double r25596854 = r25596850 * r25596853;
        double r25596855 = r25596849 - r25596854;
        double r25596856 = r25596840 + r25596855;
        double r25596857 = 7.708714260711597e+86;
        bool r25596858 = r25596829 <= r25596857;
        double r25596859 = r25596835 * r25596829;
        double r25596860 = r25596842 * r25596859;
        double r25596861 = r25596829 * r25596844;
        double r25596862 = -r25596833;
        double r25596863 = r25596861 * r25596862;
        double r25596864 = r25596860 + r25596863;
        double r25596865 = r25596864 - r25596854;
        double r25596866 = r25596840 + r25596865;
        double r25596867 = r25596829 * r25596846;
        double r25596868 = r25596840 + r25596867;
        double r25596869 = r25596858 ? r25596866 : r25596868;
        double r25596870 = r25596831 ? r25596856 : r25596869;
        return r25596870;
}

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	16.0
Herbie	10.9

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

Derivation

1. Split input into 3 regimes

2. if x < -6.2622282597039444e+72

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.8

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

   4. Applied associate-*l* 7.8

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

   6. Applied associate-*l* 7.8

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

   if -6.2622282597039444e+72 < x < 7.708714260711597e+86

   1. Initial program 14.3

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

   3. Applied add-cube-cbrt 14.5

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

   4. Applied associate-*l* 14.5

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

   6. Applied sub-neg 14.5

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

   7. Applied distribute-lft-in 14.5

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

   8. Applied distribute-lft-in 14.5

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

   9. Simplified 12.6

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

   10. Simplified 12.5

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

   12. Applied associate-*r* 10.5

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

   if 7.708714260711597e+86 < x

   1. Initial program 7.1

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

   2. Taylor expanded around 0 16.6

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.262228259703944418388054681956517640133 \cdot 10^{72}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{x} \cdot \left(\left(y \cdot z - t \cdot a\right) \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x} - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \mathbf{elif}\;x \le 7.708714260711597468016611731654685252532 \cdot 10^{86}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(z \cdot \left(y \cdot x\right) + \left(x \cdot a\right) \cdot \left(-t\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))))