Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 9.7

Time: 33.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -0.01018236418344508308064799706471603712998:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;j \le 6.630197526137251571160185503510114222161 \cdot 10^{136}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) - i \cdot \left(y \cdot j\right)\right) + \left(\left(z \cdot y - t \cdot a\right) \cdot x - \left(c \cdot z - a \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y - t \cdot a\right) \cdot x - \sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b} \cdot \left(\sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b} \cdot \sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r12774762 = x;
        double r12774763 = y;
        double r12774764 = z;
        double r12774765 = r12774763 * r12774764;
        double r12774766 = t;
        double r12774767 = a;
        double r12774768 = r12774766 * r12774767;
        double r12774769 = r12774765 - r12774768;
        double r12774770 = r12774762 * r12774769;
        double r12774771 = b;
        double r12774772 = c;
        double r12774773 = r12774772 * r12774764;
        double r12774774 = i;
        double r12774775 = r12774774 * r12774767;
        double r12774776 = r12774773 - r12774775;
        double r12774777 = r12774771 * r12774776;
        double r12774778 = r12774770 - r12774777;
        double r12774779 = j;
        double r12774780 = r12774772 * r12774766;
        double r12774781 = r12774774 * r12774763;
        double r12774782 = r12774780 - r12774781;
        double r12774783 = r12774779 * r12774782;
        double r12774784 = r12774778 + r12774783;
        return r12774784;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r12774785 = j;
        double r12774786 = -0.010182364183445083;
        bool r12774787 = r12774785 <= r12774786;
        double r12774788 = c;
        double r12774789 = t;
        double r12774790 = r12774788 * r12774789;
        double r12774791 = i;
        double r12774792 = y;
        double r12774793 = r12774791 * r12774792;
        double r12774794 = r12774790 - r12774793;
        double r12774795 = r12774794 * r12774785;
        double r12774796 = x;
        double r12774797 = z;
        double r12774798 = r12774797 * r12774792;
        double r12774799 = r12774796 * r12774798;
        double r12774800 = a;
        double r12774801 = r12774796 * r12774789;
        double r12774802 = r12774800 * r12774801;
        double r12774803 = r12774799 - r12774802;
        double r12774804 = r12774788 * r12774797;
        double r12774805 = r12774800 * r12774791;
        double r12774806 = r12774804 - r12774805;
        double r12774807 = b;
        double r12774808 = r12774806 * r12774807;
        double r12774809 = r12774803 - r12774808;
        double r12774810 = r12774795 + r12774809;
        double r12774811 = 6.630197526137252e+136;
        bool r12774812 = r12774785 <= r12774811;
        double r12774813 = r12774785 * r12774788;
        double r12774814 = r12774789 * r12774813;
        double r12774815 = r12774792 * r12774785;
        double r12774816 = r12774791 * r12774815;
        double r12774817 = r12774814 - r12774816;
        double r12774818 = r12774789 * r12774800;
        double r12774819 = r12774798 - r12774818;
        double r12774820 = r12774819 * r12774796;
        double r12774821 = r12774820 - r12774808;
        double r12774822 = r12774817 + r12774821;
        double r12774823 = cbrt(r12774808);
        double r12774824 = r12774823 * r12774823;
        double r12774825 = r12774823 * r12774824;
        double r12774826 = r12774820 - r12774825;
        double r12774827 = r12774826 + r12774795;
        double r12774828 = r12774812 ? r12774822 : r12774827;
        double r12774829 = r12774787 ? r12774810 : r12774828;
        return r12774829;
}

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	12.3
Target	16.1
Herbie	9.7

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

Derivation

1. Split input into 3 regimes

2. if j < -0.010182364183445083

   1. Initial program 8.1

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

   2. Taylor expanded around inf 8.0

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

   if -0.010182364183445083 < j < 6.630197526137252e+136

   1. Initial program 14.1

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

   2. Taylor expanded around inf 10.5

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

   if 6.630197526137252e+136 < j

   1. Initial program 6.9

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

   3. Applied add-cube-cbrt 7.0

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -0.01018236418344508308064799706471603712998:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right)\\ \mathbf{elif}\;j \le 6.630197526137251571160185503510114222161 \cdot 10^{136}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) - i \cdot \left(y \cdot j\right)\right) + \left(\left(z \cdot y - t \cdot a\right) \cdot x - \left(c \cdot z - a \cdot i\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y - t \cdot a\right) \cdot x - \sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b} \cdot \left(\sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b} \cdot \sqrt[3]{\left(c \cdot z - a \cdot i\right) \cdot b}\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))))