Average Error: 12.5 → 12.9
Time: 8.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.86574732953854097 \cdot 10^{-108} \lor \neg \left(b \le 1.68260521098177512 \cdot 10^{-166}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.86574732953854097 \cdot 10^{-108} \lor \neg \left(b \le 1.68260521098177512 \cdot 10^{-166}\right):\\
\;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r836807 = x;
        double r836808 = y;
        double r836809 = z;
        double r836810 = r836808 * r836809;
        double r836811 = t;
        double r836812 = a;
        double r836813 = r836811 * r836812;
        double r836814 = r836810 - r836813;
        double r836815 = r836807 * r836814;
        double r836816 = b;
        double r836817 = c;
        double r836818 = r836817 * r836809;
        double r836819 = i;
        double r836820 = r836811 * r836819;
        double r836821 = r836818 - r836820;
        double r836822 = r836816 * r836821;
        double r836823 = r836815 - r836822;
        double r836824 = j;
        double r836825 = r836817 * r836812;
        double r836826 = r836808 * r836819;
        double r836827 = r836825 - r836826;
        double r836828 = r836824 * r836827;
        double r836829 = r836823 + r836828;
        return r836829;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r836830 = b;
        double r836831 = -7.865747329538541e-108;
        bool r836832 = r836830 <= r836831;
        double r836833 = 1.682605210981775e-166;
        bool r836834 = r836830 <= r836833;
        double r836835 = !r836834;
        bool r836836 = r836832 || r836835;
        double r836837 = x;
        double r836838 = y;
        double r836839 = z;
        double r836840 = r836838 * r836839;
        double r836841 = t;
        double r836842 = a;
        double r836843 = r836841 * r836842;
        double r836844 = r836840 - r836843;
        double r836845 = r836837 * r836844;
        double r836846 = cbrt(r836845);
        double r836847 = r836846 * r836846;
        double r836848 = r836847 * r836846;
        double r836849 = c;
        double r836850 = r836849 * r836839;
        double r836851 = i;
        double r836852 = r836841 * r836851;
        double r836853 = r836850 - r836852;
        double r836854 = r836830 * r836853;
        double r836855 = r836848 - r836854;
        double r836856 = j;
        double r836857 = r836849 * r836842;
        double r836858 = r836838 * r836851;
        double r836859 = r836857 - r836858;
        double r836860 = r836856 * r836859;
        double r836861 = r836855 + r836860;
        double r836862 = 0.0;
        double r836863 = r836845 - r836862;
        double r836864 = r836863 + r836860;
        double r836865 = r836836 ? r836861 : r836864;
        return r836865;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.5
Target20.3
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -7.865747329538541e-108 or 1.682605210981775e-166 < b

    1. Initial program 9.9

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

    3. Applied add-cube-cbrt10.2

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

    if -7.865747329538541e-108 < b < 1.682605210981775e-166

    1. Initial program 17.5

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

    2. Taylor expanded around 0 18.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.86574732953854097 \cdot 10^{-108} \lor \neg \left(b \le 1.68260521098177512 \cdot 10^{-166}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))