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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -4.1241793159774735 \cdot 10^{-236} \lor \neg \left(j \le 1.402416700840359 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \end{array}\]

\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}\;j \le -4.1241793159774735 \cdot 10^{-236} \lor \neg \left(j \le 1.402416700840359 \cdot 10^{-93}\right):\\
\;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\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 r1020806 = x;
        double r1020807 = y;
        double r1020808 = z;
        double r1020809 = r1020807 * r1020808;
        double r1020810 = t;
        double r1020811 = a;
        double r1020812 = r1020810 * r1020811;
        double r1020813 = r1020809 - r1020812;
        double r1020814 = r1020806 * r1020813;
        double r1020815 = b;
        double r1020816 = c;
        double r1020817 = r1020816 * r1020808;
        double r1020818 = i;
        double r1020819 = r1020810 * r1020818;
        double r1020820 = r1020817 - r1020819;
        double r1020821 = r1020815 * r1020820;
        double r1020822 = r1020814 - r1020821;
        double r1020823 = j;
        double r1020824 = r1020816 * r1020811;
        double r1020825 = r1020807 * r1020818;
        double r1020826 = r1020824 - r1020825;
        double r1020827 = r1020823 * r1020826;
        double r1020828 = r1020822 + r1020827;
        return r1020828;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1020829 = j;
        double r1020830 = -4.1241793159774735e-236;
        bool r1020831 = r1020829 <= r1020830;
        double r1020832 = 1.402416700840359e-93;
        bool r1020833 = r1020829 <= r1020832;
        double r1020834 = !r1020833;
        bool r1020835 = r1020831 || r1020834;
        double r1020836 = t;
        double r1020837 = i;
        double r1020838 = r1020836 * r1020837;
        double r1020839 = c;
        double r1020840 = z;
        double r1020841 = r1020839 * r1020840;
        double r1020842 = r1020838 - r1020841;
        double r1020843 = b;
        double r1020844 = a;
        double r1020845 = r1020839 * r1020844;
        double r1020846 = y;
        double r1020847 = r1020846 * r1020837;
        double r1020848 = r1020845 - r1020847;
        double r1020849 = x;
        double r1020850 = cbrt(r1020849);
        double r1020851 = r1020850 * r1020850;
        double r1020852 = r1020846 * r1020840;
        double r1020853 = r1020836 * r1020844;
        double r1020854 = r1020852 - r1020853;
        double r1020855 = r1020850 * r1020854;
        double r1020856 = r1020851 * r1020855;
        double r1020857 = fma(r1020829, r1020848, r1020856);
        double r1020858 = fma(r1020842, r1020843, r1020857);
        double r1020859 = r1020849 * r1020840;
        double r1020860 = r1020837 * r1020829;
        double r1020861 = r1020859 - r1020860;
        double r1020862 = r1020846 * r1020861;
        double r1020863 = r1020849 * r1020836;
        double r1020864 = r1020844 * r1020863;
        double r1020865 = r1020862 - r1020864;
        double r1020866 = fma(r1020842, r1020843, r1020865);
        double r1020867 = r1020835 ? r1020858 : r1020866;
        return r1020867;
}

Original	12.1
Target	20.0
Herbie	11.4

Derivation

Split input into 2 regimes
if j < -4.1241793159774735e-236 or 1.402416700840359e-93 < j
1. Initial program 10.1
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
2. Simplified10.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt10.3
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*10.3
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
if -4.1241793159774735e-236 < j < 1.402416700840359e-93
1. Initial program 16.9
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
2. Simplified16.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt17.3
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*17.3
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
6. Taylor expanded around inf 15.5
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \color{blue}{x \cdot \left(z \cdot y\right) - \left(t \cdot \left(x \cdot a\right) + i \cdot \left(y \cdot j\right)\right)}\right)\]
7. Simplified13.8
  \[\leadsto \mathsf{fma}\left(t \cdot i - c \cdot z, b, \color{blue}{y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)}\right)\]
Recombined 2 regimes into one program.
Final simplification11.4
\[\leadsto \begin{array}{l} \mathbf{if}\;j \le -4.1241793159774735 \cdot 10^{-236} \lor \neg \left(j \le 1.402416700840359 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if j < -4.1241793159774735e-236 or 1.402416700840359e-93 < j`

`if -4.1241793159774735e-236 < j < 1.402416700840359e-93`

Reproduce