Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.0 → 11.4

Time: 30.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.070193068145833190122055894254729817899 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(\left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 1.273716925755540258919533772715346247145 \cdot 10^{-203}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(i \cdot b\right) \cdot a - \left(\left(x \cdot t\right) \cdot a + \left(b \cdot c\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r24260742 = x;
        double r24260743 = y;
        double r24260744 = z;
        double r24260745 = r24260743 * r24260744;
        double r24260746 = t;
        double r24260747 = a;
        double r24260748 = r24260746 * r24260747;
        double r24260749 = r24260745 - r24260748;
        double r24260750 = r24260742 * r24260749;
        double r24260751 = b;
        double r24260752 = c;
        double r24260753 = r24260752 * r24260744;
        double r24260754 = i;
        double r24260755 = r24260754 * r24260747;
        double r24260756 = r24260753 - r24260755;
        double r24260757 = r24260751 * r24260756;
        double r24260758 = r24260750 - r24260757;
        double r24260759 = j;
        double r24260760 = r24260752 * r24260746;
        double r24260761 = r24260754 * r24260743;
        double r24260762 = r24260760 - r24260761;
        double r24260763 = r24260759 * r24260762;
        double r24260764 = r24260758 + r24260763;
        return r24260764;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r24260765 = x;
        double r24260766 = -3.070193068145833e-283;
        bool r24260767 = r24260765 <= r24260766;
        double r24260768 = y;
        double r24260769 = z;
        double r24260770 = r24260768 * r24260769;
        double r24260771 = a;
        double r24260772 = t;
        double r24260773 = r24260771 * r24260772;
        double r24260774 = r24260770 - r24260773;
        double r24260775 = r24260774 * r24260765;
        double r24260776 = b;
        double r24260777 = c;
        double r24260778 = r24260769 * r24260777;
        double r24260779 = i;
        double r24260780 = r24260779 * r24260771;
        double r24260781 = r24260778 - r24260780;
        double r24260782 = r24260776 * r24260781;
        double r24260783 = r24260775 - r24260782;
        double r24260784 = r24260777 * r24260772;
        double r24260785 = r24260779 * r24260768;
        double r24260786 = r24260784 - r24260785;
        double r24260787 = cbrt(r24260786);
        double r24260788 = r24260787 * r24260787;
        double r24260789 = j;
        double r24260790 = r24260788 * r24260789;
        double r24260791 = r24260790 * r24260787;
        double r24260792 = r24260783 + r24260791;
        double r24260793 = 1.2737169257555403e-203;
        bool r24260794 = r24260765 <= r24260793;
        double r24260795 = r24260789 * r24260786;
        double r24260796 = r24260779 * r24260776;
        double r24260797 = r24260796 * r24260771;
        double r24260798 = r24260765 * r24260772;
        double r24260799 = r24260798 * r24260771;
        double r24260800 = r24260776 * r24260777;
        double r24260801 = r24260800 * r24260769;
        double r24260802 = r24260799 + r24260801;
        double r24260803 = r24260797 - r24260802;
        double r24260804 = r24260795 + r24260803;
        double r24260805 = sqrt(r24260765);
        double r24260806 = r24260805 * r24260774;
        double r24260807 = r24260805 * r24260806;
        double r24260808 = r24260807 - r24260782;
        double r24260809 = r24260795 + r24260808;
        double r24260810 = r24260794 ? r24260804 : r24260809;
        double r24260811 = r24260767 ? r24260792 : r24260810;
        return r24260811;
}

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	12.0
Target	15.7
Herbie	11.4

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

Derivation

1. Split input into 3 regimes

2. if x < -3.070193068145833e-283

   1. Initial program 11.7

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

   3. Applied add-cube-cbrt 12.0

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

   4. Applied associate-*r* 12.0

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

   if -3.070193068145833e-283 < x < 1.2737169257555403e-203

   1. Initial program 16.9

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

   2. Taylor expanded around inf 11.2

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

   if 1.2737169257555403e-203 < x

   1. Initial program 10.6

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

   3. Applied add-sqr-sqrt 10.7

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

   4. Applied associate-*l* 10.7

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

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.070193068145833190122055894254729817899 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(\left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 1.273716925755540258919533772715346247145 \cdot 10^{-203}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(i \cdot b\right) \cdot a - \left(\left(x \cdot t\right) \cdot a + \left(b \cdot c\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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)))))