Average Error: 12.2 → 12.2
Time: 29.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\ \;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \end{array}\]
\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 -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\
\;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\

\mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\
\;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\

\mathbf{else}:\\
\;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r20706801 = x;
        double r20706802 = y;
        double r20706803 = z;
        double r20706804 = r20706802 * r20706803;
        double r20706805 = t;
        double r20706806 = a;
        double r20706807 = r20706805 * r20706806;
        double r20706808 = r20706804 - r20706807;
        double r20706809 = r20706801 * r20706808;
        double r20706810 = b;
        double r20706811 = c;
        double r20706812 = r20706811 * r20706803;
        double r20706813 = i;
        double r20706814 = r20706813 * r20706806;
        double r20706815 = r20706812 - r20706814;
        double r20706816 = r20706810 * r20706815;
        double r20706817 = r20706809 - r20706816;
        double r20706818 = j;
        double r20706819 = r20706811 * r20706805;
        double r20706820 = r20706813 * r20706802;
        double r20706821 = r20706819 - r20706820;
        double r20706822 = r20706818 * r20706821;
        double r20706823 = r20706817 + r20706822;
        return r20706823;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r20706824 = x;
        double r20706825 = -4.7326183276391225e-237;
        bool r20706826 = r20706824 <= r20706825;
        double r20706827 = a;
        double r20706828 = -r20706827;
        double r20706829 = t;
        double r20706830 = r20706827 * r20706829;
        double r20706831 = fma(r20706828, r20706829, r20706830);
        double r20706832 = r20706831 * r20706824;
        double r20706833 = y;
        double r20706834 = z;
        double r20706835 = r20706828 * r20706829;
        double r20706836 = fma(r20706833, r20706834, r20706835);
        double r20706837 = r20706836 * r20706824;
        double r20706838 = r20706832 + r20706837;
        double r20706839 = b;
        double r20706840 = c;
        double r20706841 = r20706840 * r20706834;
        double r20706842 = i;
        double r20706843 = r20706827 * r20706842;
        double r20706844 = r20706841 - r20706843;
        double r20706845 = r20706839 * r20706844;
        double r20706846 = r20706838 - r20706845;
        double r20706847 = j;
        double r20706848 = cbrt(r20706847);
        double r20706849 = r20706848 * r20706848;
        double r20706850 = r20706829 * r20706840;
        double r20706851 = r20706833 * r20706842;
        double r20706852 = r20706850 - r20706851;
        double r20706853 = r20706852 * r20706848;
        double r20706854 = r20706849 * r20706853;
        double r20706855 = r20706846 + r20706854;
        double r20706856 = 2.7147272054845436e-192;
        bool r20706857 = r20706824 <= r20706856;
        double r20706858 = -r20706839;
        double r20706859 = r20706844 * r20706858;
        double r20706860 = r20706847 * r20706852;
        double r20706861 = r20706859 + r20706860;
        double r20706862 = sqrt(r20706824);
        double r20706863 = r20706833 * r20706834;
        double r20706864 = r20706863 - r20706830;
        double r20706865 = r20706864 * r20706862;
        double r20706866 = r20706862 * r20706865;
        double r20706867 = r20706866 - r20706845;
        double r20706868 = r20706860 + r20706867;
        double r20706869 = r20706857 ? r20706861 : r20706868;
        double r20706870 = r20706826 ? r20706855 : r20706869;
        return r20706870;
}

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

Original12.2
Target16.3
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.7326183276391225e-237

    1. Initial program 11.1

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

    3. Applied prod-diff11.1

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

    4. Applied distribute-rgt-in11.1

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

    6. Applied add-cube-cbrt11.4

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

    7. Applied associate-*l*11.4

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

    if -4.7326183276391225e-237 < x < 2.7147272054845436e-192

    1. Initial program 17.4

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

    2. Taylor expanded around 0 16.8

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    if 2.7147272054845436e-192 < 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-sqrt10.7

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

    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 simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\ \;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \end{array}\]

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)))))