Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 11.8 → 8.4

Time: 12.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(j \cdot i\right)\\ t_2 := \mathsf{fma}\left(b, t \cdot i - c \cdot z, \mathsf{fma}\left(y, x \cdot z - j \cdot i, a \cdot \left(c \cdot j - x \cdot t\right)\right)\right)\\ t_3 := c \cdot a - y \cdot i\\ t_4 := t \cdot \left(b \cdot i - x \cdot a\right)\\ t_5 := i \cdot \left(t \cdot b\right)\\ t_6 := y \cdot z - t \cdot a\\ t_7 := a \cdot j - z \cdot b\\ t_8 := j \cdot t_3\\ \mathbf{if}\;c \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(x, t_6, c \cdot t_7 + \left(t_5 - t_1\right)\right)\\ \mathbf{elif}\;c \leq -6.894880890659607 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 2.4039080990269493 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - c \cdot b, \mathsf{fma}\left(t_3, j, t_4\right)\right)\\ \mathbf{elif}\;c \leq 2.8267541173163815 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 5.759062406266529 \cdot 10^{-235}:\\ \;\;\;\;\mathsf{fma}\left(x, t_6, t_5 + t_8\right)\\ \mathbf{elif}\;c \leq 2.28273651883575 \cdot 10^{-204}:\\ \;\;\;\;\left(t_4 + \left({\left(\sqrt[3]{y \cdot \left(x \cdot z\right)}\right)}^{3} - t_1\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;c \leq 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(x, t_6, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), t_8\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t_6, \mathsf{fma}\left(c, t_7, i \cdot \left(t \cdot b - y \cdot j\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

↓

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* y (* j i)))
        (t_2
         (fma
          b
          (- (* t i) (* c z))
          (fma y (- (* x z) (* j i)) (* a (- (* c j) (* x t))))))
        (t_3 (- (* c a) (* y i)))
        (t_4 (* t (- (* b i) (* x a))))
        (t_5 (* i (* t b)))
        (t_6 (- (* y z) (* t a)))
        (t_7 (- (* a j) (* z b)))
        (t_8 (* j t_3)))
   (if (<= c -1e+30)
     (fma x t_6 (+ (* c t_7) (- t_5 t_1)))
     (if (<= c -6.894880890659607e-187)
       t_2
       (if (<= c 2.4039080990269493e-287)
         (fma z (- (* x y) (* c b)) (fma t_3 j t_4))
         (if (<= c 2.8267541173163815e-252)
           t_2
           (if (<= c 5.759062406266529e-235)
             (fma x t_6 (+ t_5 t_8))
             (if (<= c 2.28273651883575e-204)
               (- (+ t_4 (- (pow (cbrt (* y (* x z))) 3.0) t_1)) (* c (* z b)))
               (if (<= c 1e-18)
                 (fma x t_6 (fma b (fma z (- c) (* t i)) t_8))
                 (fma x t_6 (fma c t_7 (* i (- (* t b) (* y j))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = y * (j * i);
	double t_2 = fma(b, ((t * i) - (c * z)), fma(y, ((x * z) - (j * i)), (a * ((c * j) - (x * t)))));
	double t_3 = (c * a) - (y * i);
	double t_4 = t * ((b * i) - (x * a));
	double t_5 = i * (t * b);
	double t_6 = (y * z) - (t * a);
	double t_7 = (a * j) - (z * b);
	double t_8 = j * t_3;
	double tmp;
	if (c <= -1e+30) {
		tmp = fma(x, t_6, ((c * t_7) + (t_5 - t_1)));
	} else if (c <= -6.894880890659607e-187) {
		tmp = t_2;
	} else if (c <= 2.4039080990269493e-287) {
		tmp = fma(z, ((x * y) - (c * b)), fma(t_3, j, t_4));
	} else if (c <= 2.8267541173163815e-252) {
		tmp = t_2;
	} else if (c <= 5.759062406266529e-235) {
		tmp = fma(x, t_6, (t_5 + t_8));
	} else if (c <= 2.28273651883575e-204) {
		tmp = (t_4 + (pow(cbrt((y * (x * z))), 3.0) - t_1)) - (c * (z * b));
	} else if (c <= 1e-18) {
		tmp = fma(x, t_6, fma(b, fma(z, -c, (t * i)), t_8));
	} else {
		tmp = fma(x, t_6, fma(c, t_7, (i * ((t * b) - (y * j)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

↓

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(y * Float64(j * i))
	t_2 = fma(b, Float64(Float64(t * i) - Float64(c * z)), fma(y, Float64(Float64(x * z) - Float64(j * i)), Float64(a * Float64(Float64(c * j) - Float64(x * t)))))
	t_3 = Float64(Float64(c * a) - Float64(y * i))
	t_4 = Float64(t * Float64(Float64(b * i) - Float64(x * a)))
	t_5 = Float64(i * Float64(t * b))
	t_6 = Float64(Float64(y * z) - Float64(t * a))
	t_7 = Float64(Float64(a * j) - Float64(z * b))
	t_8 = Float64(j * t_3)
	tmp = 0.0
	if (c <= -1e+30)
		tmp = fma(x, t_6, Float64(Float64(c * t_7) + Float64(t_5 - t_1)));
	elseif (c <= -6.894880890659607e-187)
		tmp = t_2;
	elseif (c <= 2.4039080990269493e-287)
		tmp = fma(z, Float64(Float64(x * y) - Float64(c * b)), fma(t_3, j, t_4));
	elseif (c <= 2.8267541173163815e-252)
		tmp = t_2;
	elseif (c <= 5.759062406266529e-235)
		tmp = fma(x, t_6, Float64(t_5 + t_8));
	elseif (c <= 2.28273651883575e-204)
		tmp = Float64(Float64(t_4 + Float64((cbrt(Float64(y * Float64(x * z))) ^ 3.0) - t_1)) - Float64(c * Float64(z * b)));
	elseif (c <= 1e-18)
		tmp = fma(x, t_6, fma(b, fma(z, Float64(-c), Float64(t * i)), t_8));
	else
		tmp = fma(x, t_6, fma(c, t_7, Float64(i * Float64(Float64(t * b) - Float64(y * j)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(y * N[(j * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(N[(t * i), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(x * z), $MachinePrecision] - N[(j * i), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(c * j), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(j * t$95$3), $MachinePrecision]}, If[LessEqual[c, -1e+30], N[(x * t$95$6 + N[(N[(c * t$95$7), $MachinePrecision] + N[(t$95$5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -6.894880890659607e-187], t$95$2, If[LessEqual[c, 2.4039080990269493e-287], N[(z * N[(N[(x * y), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * j + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.8267541173163815e-252], t$95$2, If[LessEqual[c, 5.759062406266529e-235], N[(x * t$95$6 + N[(t$95$5 + t$95$8), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.28273651883575e-204], N[(N[(t$95$4 + N[(N[Power[N[Power[N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e-18], N[(x * t$95$6 + N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision], N[(x * t$95$6 + N[(c * t$95$7 + N[(i * N[(N[(t * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Target

Original	11.8
Target	19.8
Herbie	8.4

\[\begin{array}{l} \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \]

Derivation

1. Split input into 7 regimes

2. if c < -1e30

   1. Initial program 17.7

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

   2. Simplified 17.7

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

   3. Taylor expanded in c around 0 7.4

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{c \cdot \left(a \cdot j + -1 \cdot \left(z \cdot b\right)\right) + \left(i \cdot \left(t \cdot b\right) + -1 \cdot \left(y \cdot \left(i \cdot j\right)\right)\right)}\right) \]

   if -1e30 < c < -6.8948808906596067e-187 or 2.40390809902694934e-287 < c < 2.82675411731638154e-252

   1. Initial program 8.5

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

   2. Simplified 8.5

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

   3. Applied egg-rr 8.8

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

   4. Taylor expanded in x around 0 8.5

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot i\right) + c \cdot a\right) \cdot j + \left(b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(y \cdot z - a \cdot t\right) \cdot x\right)} \]

   5. Simplified 8.6

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

   if -6.8948808906596067e-187 < c < 2.40390809902694934e-287

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   3. Applied egg-rr 10.7

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

   4. Taylor expanded in z around 0 8.9

      \[\leadsto \color{blue}{\left(y \cdot x + -1 \cdot \left(c \cdot b\right)\right) \cdot z + \left(\left(-1 \cdot \left(y \cdot i\right) + c \cdot a\right) \cdot j + \left(i \cdot \left(t \cdot b\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)\right)} \]

   5. Simplified 8.8

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

   if 2.82675411731638154e-252 < c < 5.75906240626652913e-235

   1. Initial program 11.2

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

   2. Simplified 11.2

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

   3. Applied egg-rr 11.5

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

   4. Taylor expanded in z around 0 14.2

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{i \cdot \left(t \cdot b\right) + \left(-1 \cdot \left(y \cdot i\right) + c \cdot a\right) \cdot j}\right) \]

   if 5.75906240626652913e-235 < c < 2.2827365188357501e-204

   1. Initial program 7.5

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

   2. Simplified 7.5

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

   3. Taylor expanded in t around 0 14.4

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(b \cdot z\right)\right) + \left(t \cdot \left(i \cdot b + -1 \cdot \left(a \cdot x\right)\right) + \left(y \cdot \left(z \cdot x\right) + j \cdot \left(c \cdot a - y \cdot i\right)\right)\right)} \]

   4. Applied egg-rr 14.5

      \[\leadsto -1 \cdot \left(c \cdot \left(b \cdot z\right)\right) + \left(t \cdot \left(i \cdot b + -1 \cdot \left(a \cdot x\right)\right) + \left(\color{blue}{{\left(\sqrt[3]{y \cdot \left(z \cdot x\right)}\right)}^{3}} + j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]

   5. Taylor expanded in y around inf 18.9

      \[\leadsto -1 \cdot \left(c \cdot \left(b \cdot z\right)\right) + \left(t \cdot \left(i \cdot b + -1 \cdot \left(a \cdot x\right)\right) + \left({\left(\sqrt[3]{y \cdot \left(z \cdot x\right)}\right)}^{3} + \color{blue}{-1 \cdot \left(y \cdot \left(i \cdot j\right)\right)}\right)\right) \]

   if 2.2827365188357501e-204 < c < 1.0000000000000001e-18

   1. Initial program 8.3

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

   2. Simplified 8.2

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

   3. Applied egg-rr 8.6

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

   4. Taylor expanded in j around 0 8.2

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), \color{blue}{\left(-1 \cdot \left(y \cdot i\right) + c \cdot a\right) \cdot j}\right)\right) \]

   5. Simplified 8.2

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

   if 1.0000000000000001e-18 < c

   1. Initial program 15.7

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

   2. Simplified 15.7

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

   3. Taylor expanded in b around 0 15.7

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(c \cdot a - i \cdot y\right) \cdot j}\right) \]

   4. Simplified 7.1

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{\mathsf{fma}\left(c, j \cdot a - b \cdot z, i \cdot \left(b \cdot t - j \cdot y\right)\right)}\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(a \cdot j - z \cdot b\right) + \left(i \cdot \left(t \cdot b\right) - y \cdot \left(j \cdot i\right)\right)\right)\\ \mathbf{elif}\;c \leq -6.894880890659607 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, \mathsf{fma}\left(y, x \cdot z - j \cdot i, a \cdot \left(c \cdot j - x \cdot t\right)\right)\right)\\ \mathbf{elif}\;c \leq 2.4039080990269493 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - c \cdot b, \mathsf{fma}\left(c \cdot a - y \cdot i, j, t \cdot \left(b \cdot i - x \cdot a\right)\right)\right)\\ \mathbf{elif}\;c \leq 2.8267541173163815 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, \mathsf{fma}\left(y, x \cdot z - j \cdot i, a \cdot \left(c \cdot j - x \cdot t\right)\right)\right)\\ \mathbf{elif}\;c \leq 5.759062406266529 \cdot 10^{-235}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \left(t \cdot b\right) + j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{elif}\;c \leq 2.28273651883575 \cdot 10^{-204}:\\ \;\;\;\;\left(t \cdot \left(b \cdot i - x \cdot a\right) + \left({\left(\sqrt[3]{y \cdot \left(x \cdot z\right)}\right)}^{3} - y \cdot \left(j \cdot i\right)\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;c \leq 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(c \cdot a - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(c, a \cdot j - z \cdot b, i \cdot \left(t \cdot b - y \cdot j\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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