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

Average Error: 12.2 → 5.4

Time: 49.8s

Precision: binary64

Cost: 19080

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_2 := c \cdot \left(a \cdot j\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_4 := t_3 + \left(t_1 + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(\left(t_1 + t_2\right) + i \cdot \left(t \cdot b - y \cdot j\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(t_1 - \mathsf{fma}\left(b, z \cdot c - t \cdot i, \left(b + b\right) \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(t_2 - y \cdot \left(i \cdot j\right)\right)\\ \end{array} \]

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 (* x (- (* y z) (* t a))))
        (t_2 (* c (* a j)))
        (t_3 (* j (- (* a c) (* y i))))
        (t_4 (+ t_3 (+ t_1 (* b (- (* t i) (* z c)))))))
   (if (<= t_4 (- INFINITY))
     (- (+ (+ t_1 t_2) (* i (- (* t b) (* y j)))) (* c (* z b)))
     (if (<= t_4 2e+306)
       (+
        (- t_1 (fma b (- (* z c) (* t i)) (* (+ b b) (fma (- i) t (* t i)))))
        t_3)
       (+ (* t (- (* b i) (* x a))) (- t_2 (* y (* i 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 = x * ((y * z) - (t * a));
	double t_2 = c * (a * j);
	double t_3 = j * ((a * c) - (y * i));
	double t_4 = t_3 + (t_1 + (b * ((t * i) - (z * c))));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((t_1 + t_2) + (i * ((t * b) - (y * j)))) - (c * (z * b));
	} else if (t_4 <= 2e+306) {
		tmp = (t_1 - fma(b, ((z * c) - (t * i)), ((b + b) * fma(-i, t, (t * i))))) + t_3;
	} else {
		tmp = (t * ((b * i) - (x * a))) + (t_2 - (y * (i * 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(x * Float64(Float64(y * z) - Float64(t * a)))
	t_2 = Float64(c * Float64(a * j))
	t_3 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	t_4 = Float64(t_3 + Float64(t_1 + Float64(b * Float64(Float64(t * i) - Float64(z * c)))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 + t_2) + Float64(i * Float64(Float64(t * b) - Float64(y * j)))) - Float64(c * Float64(z * b)));
	elseif (t_4 <= 2e+306)
		tmp = Float64(Float64(t_1 - fma(b, Float64(Float64(z * c) - Float64(t * i)), Float64(Float64(b + b) * fma(Float64(-i), t, Float64(t * i))))) + t_3);
	else
		tmp = Float64(Float64(t * Float64(Float64(b * i) - Float64(x * a))) + Float64(t_2 - Float64(y * Float64(i * 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[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(t$95$1 + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(t$95$1 + t$95$2), $MachinePrecision] + N[(i * N[(N[(t * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+306], N[(N[(t$95$1 - N[(b * N[(N[(z * c), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision] + N[(N[(b + b), $MachinePrecision] * N[((-i) * t + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(t * N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	12.2
Target	20.0
Herbie	5.4

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      +-commutative [=>] 64.0	\[ \color{blue}{j \cdot \left(c \cdot a - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} \]
      fma-def [=>] 64.0	\[ \color{blue}{\mathsf{fma}\left(j, c \cdot a - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} \]
      *-commutative [=>] 64.0	\[ \mathsf{fma}\left(j, \color{blue}{a \cdot c} - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) \]
      *-commutative [=>] 64.0	\[ \mathsf{fma}\left(j, a \cdot c - y \cdot i, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{z \cdot c} - t \cdot i\right)\right) \]

   3. Taylor expanded in i around -inf 22.8

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

   if -inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < 2.00000000000000003e306

   1. Initial program 1.0

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

   2. Simplified 1.0

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

      [Start] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      sub-neg [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \color{blue}{\left(c \cdot a + \left(-y \cdot i\right)\right)} \]
      distribute-rgt-in [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\left(\left(c \cdot a\right) \cdot j + \left(-y \cdot i\right) \cdot j\right)} \]
      associate-+r+ [=>] 1.0	\[ \color{blue}{\left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(c \cdot a\right) \cdot j\right) + \left(-y \cdot i\right) \cdot j} \]
      *-commutative [=>] 1.0	\[ \left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(c \cdot a\right) \cdot j\right) + \color{blue}{j \cdot \left(-y \cdot i\right)} \]
      cancel-sign-sub [<=] 1.0	\[ \color{blue}{\left(\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(c \cdot a\right) \cdot j\right) - \left(-j\right) \cdot \left(-y \cdot i\right)} \]
      associate-+r- [<=] 1.0	\[ \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(c \cdot a\right) \cdot j - \left(-j\right) \cdot \left(-y \cdot i\right)\right)} \]
      *-commutative [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{z \cdot c} - t \cdot i\right)\right) + \left(\left(c \cdot a\right) \cdot j - \left(-j\right) \cdot \left(-y \cdot i\right)\right) \]
      cancel-sign-sub [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + \color{blue}{\left(\left(c \cdot a\right) \cdot j + j \cdot \left(-y \cdot i\right)\right)} \]
      *-commutative [<=] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + \left(\left(c \cdot a\right) \cdot j + \color{blue}{\left(-y \cdot i\right) \cdot j}\right) \]
      distribute-rgt-in [<=] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + \color{blue}{j \cdot \left(c \cdot a + \left(-y \cdot i\right)\right)} \]
      sub-neg [<=] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \color{blue}{\left(c \cdot a - y \cdot i\right)} \]
      *-commutative [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \left(\color{blue}{a \cdot c} - y \cdot i\right) \]

   3. Applied egg-rr 1.0

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

   4. Simplified 1.0

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

      [Start] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(z \cdot c - t \cdot i\right) + \left(b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]
      fma-def [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\mathsf{fma}\left(b, z \cdot c - t \cdot i, b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)}\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]
      *-commutative [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \mathsf{fma}\left(b, \color{blue}{c \cdot z} - t \cdot i, b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]
      *-commutative [<=] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \mathsf{fma}\left(b, c \cdot z - \color{blue}{i \cdot t}, b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]
      distribute-rgt-out [=>] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \mathsf{fma}\left(b, c \cdot z - i \cdot t, \color{blue}{\mathsf{fma}\left(-i, t, t \cdot i\right) \cdot \left(b + b\right)}\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]
      *-commutative [<=] 1.0	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - \mathsf{fma}\left(b, c \cdot z - i \cdot t, \mathsf{fma}\left(-i, t, \color{blue}{i \cdot t}\right) \cdot \left(b + b\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \]

   if 2.00000000000000003e306 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 62.5

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

   2. Simplified 62.5

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

      [Start] 62.5	\[ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      associate-+l- [=>] 62.5	\[ \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - t \cdot i\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)} \]
      fma-neg [=>] 62.5	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, -\left(b \cdot \left(c \cdot z - t \cdot i\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\right)} \]
      neg-sub0 [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{0 - \left(b \cdot \left(c \cdot z - t \cdot i\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)}\right) \]
      associate-+l- [<=] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)}\right) \]
      neg-sub0 [<=] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{\left(-b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right)\right) \]
      distribute-rgt-neg-in [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{b \cdot \left(-\left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right)\right) \]
      fma-def [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{\mathsf{fma}\left(b, -\left(c \cdot z - t \cdot i\right), j \cdot \left(c \cdot a - y \cdot i\right)\right)}\right) \]
      sub-neg [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, -\color{blue}{\left(c \cdot z + \left(-t \cdot i\right)\right)}, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      distribute-neg-in [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \color{blue}{\left(-c \cdot z\right) + \left(-\left(-t \cdot i\right)\right)}, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      +-commutative [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \color{blue}{\left(-\left(-t \cdot i\right)\right) + \left(-c \cdot z\right)}, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      remove-double-neg [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \color{blue}{t \cdot i} + \left(-c \cdot z\right), j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      sub-neg [<=] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \color{blue}{t \cdot i - c \cdot z}, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      *-commutative [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, t \cdot i - \color{blue}{z \cdot c}, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right) \]
      *-commutative [=>] 62.5	\[ \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, t \cdot i - z \cdot c, j \cdot \left(\color{blue}{a \cdot c} - y \cdot i\right)\right)\right) \]

   3. Taylor expanded in z around 0 44.8

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

   4. Simplified 41.7

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

      [Start] 44.8	\[ i \cdot \left(t \cdot b\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\right) \]
      associate-+r+ [=>] 44.8	\[ \color{blue}{\left(i \cdot \left(t \cdot b\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)} \]
      mul-1-neg [=>] 44.8	\[ \left(i \cdot \left(t \cdot b\right) + \color{blue}{\left(-a \cdot \left(t \cdot x\right)\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      unsub-neg [=>] 44.8	\[ \color{blue}{\left(i \cdot \left(t \cdot b\right) - a \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
      *-commutative [=>] 44.8	\[ \left(i \cdot \left(t \cdot b\right) - a \cdot \color{blue}{\left(x \cdot t\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      associate-*r* [=>] 43.3	\[ \left(i \cdot \left(t \cdot b\right) - \color{blue}{\left(a \cdot x\right) \cdot t}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      *-commutative [=>] 43.3	\[ \left(i \cdot \color{blue}{\left(b \cdot t\right)} - \left(a \cdot x\right) \cdot t\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      associate-*r* [=>] 41.7	\[ \left(\color{blue}{\left(i \cdot b\right) \cdot t} - \left(a \cdot x\right) \cdot t\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
      distribute-rgt-out-- [=>] 41.7	\[ \color{blue}{t \cdot \left(i \cdot b - a \cdot x\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
      *-commutative [<=] 41.7	\[ t \cdot \left(i \cdot b - a \cdot x\right) + j \cdot \left(c \cdot a - \color{blue}{i \cdot y}\right) \]

   5. Taylor expanded in c around 0 27.7

      \[\leadsto t \cdot \left(i \cdot b - a \cdot x\right) + \color{blue}{\left(-1 \cdot \left(y \cdot \left(i \cdot j\right)\right) + c \cdot \left(a \cdot j\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) \leq -\infty:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z - t \cdot a\right) + c \cdot \left(a \cdot j\right)\right) + i \cdot \left(t \cdot b - y \cdot j\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \mathsf{fma}\left(b, z \cdot c - t \cdot i, \left(b + b\right) \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(c \cdot \left(a \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.4
Cost	12680

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_2 := t_1 + b \cdot \left(t \cdot i - z \cdot c\right)\\ t_3 := c \cdot \left(a \cdot j\right)\\ t_4 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_5 := t_4 + t_2\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;\left(\left(t_1 + t_3\right) + i \cdot \left(t \cdot b - y \cdot j\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(j \cdot \mathsf{fma}\left(-i, y, y \cdot i\right) + t_4\right) + t_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(t_3 - y \cdot \left(i \cdot j\right)\right)\\ \end{array} \]

Alternative 2
Error	5.4
Cost	12680

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_2 := c \cdot \left(a \cdot j\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_4 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_5 := t_3 + \left(t_1 + t_4\right)\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;\left(\left(t_1 + t_2\right) + i \cdot \left(t \cdot b - y \cdot j\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(t_1 + \left(t_4 - b \cdot \mathsf{fma}\left(-i, t, t \cdot i\right)\right)\right) + t_3\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(t_2 - y \cdot \left(i \cdot j\right)\right)\\ \end{array} \]

Alternative 3
Error	5.9
Cost	5833

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_2 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_3 := t_2 + \left(t_1 + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{if}\;t_3 \leq -\infty \lor \neg \left(t_3 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(c \cdot \left(a \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(t_1 + \left(b \cdot \left(t \cdot i\right) - b \cdot \left(z \cdot c\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	5.4
Cost	5832

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_2 := c \cdot \left(a \cdot j\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_4 := t_3 + \left(t_1 + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\left(\left(t_1 + t_2\right) + i \cdot \left(t \cdot b - y \cdot j\right)\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t_3 + \left(t_1 + \left(b \cdot \left(t \cdot i\right) - b \cdot \left(z \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(t_2 - y \cdot \left(i \cdot j\right)\right)\\ \end{array} \]

Alternative 5
Error	5.9
Cost	5705

\[\begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right) + \left(c \cdot \left(a \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	36.8
Cost	2820

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i\right)\\ t_2 := z \cdot \left(x \cdot y - b \cdot c\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right) + t_1\\ t_4 := x \cdot z - i \cdot j\\ t_5 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_6 := y \cdot t_4\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.85 \cdot 10^{-188}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-223}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-285}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-304}:\\ \;\;\;\;t_1 - i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-294}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-251}:\\ \;\;\;\;\frac{y}{\frac{1}{t_4}}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-211}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-172}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;b \leq 2.9:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+120}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+159}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+166}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 7
Error	35.6
Cost	2808

\[\begin{array}{l} t_1 := y \cdot \left(x \cdot z\right)\\ t_2 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_3 := x \cdot z - i \cdot j\\ t_4 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_5 := c \cdot \left(a \cdot j\right) + i \cdot \left(t \cdot b - y \cdot j\right)\\ t_6 := z \cdot \left(x \cdot y - b \cdot c\right)\\ \mathbf{if}\;i \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;i \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\frac{1}{t_3}}\\ \mathbf{elif}\;i \leq -4.4 \cdot 10^{-122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -2.4 \cdot 10^{-165}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-189}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq -1.3 \cdot 10^{-215}:\\ \;\;\;\;t_1 - a \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;i \leq -4.8 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{-285}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{-190}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-182}:\\ \;\;\;\;y \cdot t_3\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-147}:\\ \;\;\;\;t_2 + t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 0.000112:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 8
Error	20.3
Cost	2784

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i - x \cdot a\right) + \left(c \cdot \left(a \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ t_2 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_3 := x \cdot \left(y \cdot z - t \cdot a\right) + t_2\\ t_4 := j \cdot \left(a \cdot c - y \cdot i\right) + \left(z \cdot \left(x \cdot y\right) + \left(b \cdot \left(t \cdot i\right) - b \cdot \left(z \cdot c\right)\right)\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-141}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-214}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-254}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-161}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \left(x \cdot z - i \cdot j\right) + t_2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 9
Error	28.4
Cost	2668

\[\begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_2 := x \cdot \left(y \cdot z - t \cdot a\right) + c \cdot \left(a \cdot j - z \cdot b\right)\\ t_3 := x \cdot z - i \cdot j\\ t_4 := t \cdot \left(b \cdot i - x \cdot a\right) + t_1\\ t_5 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{+22}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-188}:\\ \;\;\;\;y \cdot t_3\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-223}:\\ \;\;\;\;t_1 + t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-286}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-292}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-274}:\\ \;\;\;\;\frac{y}{\frac{1}{t_3}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.85:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 1.44 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+159}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 10
Error	24.3
Cost	2404

\[\begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_2 := x \cdot z - i \cdot j\\ t_3 := x \cdot \left(y \cdot z - t \cdot a\right) + c \cdot \left(a \cdot j - z \cdot b\right)\\ t_4 := y \cdot t_2 + b \cdot \left(t \cdot i - z \cdot c\right)\\ t_5 := t \cdot \left(b \cdot i - x \cdot a\right) + t_1\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -1.65 \cdot 10^{-102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -4.3 \cdot 10^{-188}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-223}:\\ \;\;\;\;t_1 + t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-285}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-295}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{y}{\frac{1}{t_2}}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-172}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-17}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 11
Error	23.5
Cost	2404

\[\begin{array}{l} t_1 := y \cdot \left(x \cdot z - i \cdot j\right)\\ t_2 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_3 := t_1 + t_2\\ t_4 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_5 := t \cdot \left(b \cdot i - x \cdot a\right) + t_4\\ t_6 := t_4 + t_2\\ t_7 := x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-80}:\\ \;\;\;\;t_7 + t_2\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-252}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-177}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-92}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-58}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+68}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+193}:\\ \;\;\;\;t_7 + c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+252}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	42.5
Cost	2292

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i - x \cdot a\right)\\ t_2 := a \cdot \left(c \cdot j - x \cdot t\right)\\ t_3 := c \cdot \left(a \cdot j - z \cdot b\right)\\ t_4 := y \cdot \left(x \cdot z - i \cdot j\right)\\ t_5 := i \cdot \left(t \cdot b - y \cdot j\right)\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+29}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-137}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-219}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{-300}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-167}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-56}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-47}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 13
Error	23.7
Cost	2272

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i - x \cdot a\right)\\ t_2 := b \cdot \left(t \cdot i - z \cdot c\right)\\ t_3 := c \cdot \left(a \cdot j\right)\\ t_4 := t_1 + \left(t_3 - y \cdot \left(i \cdot j\right)\right)\\ t_5 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_6 := x \cdot \left(y \cdot z - t \cdot a\right)\\ t_7 := y \cdot \left(x \cdot z - i \cdot j\right) + t_2\\ \mathbf{if}\;b \leq -1.96 \cdot 10^{-59}:\\ \;\;\;\;t_6 + t_2\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-121}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-187}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-223}:\\ \;\;\;\;t_5 + t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-286}:\\ \;\;\;\;t_6 + c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-301}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;\left(t_6 + t_3\right) - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-17}:\\ \;\;\;\;t_1 + t_5\\ \mathbf{else}:\\ \;\;\;\;t_7\\ \end{array} \]

Alternative 14
Error	39.0
Cost	2161

\[\begin{array}{l} t_1 := i \cdot \left(t \cdot b - y \cdot j\right)\\ t_2 := y \cdot \left(x \cdot z - i \cdot j\right)\\ t_3 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-52}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-91}:\\ \;\;\;\;z \cdot \left(x \cdot y - b \cdot c\right)\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 10^{-25}:\\ \;\;\;\;a \cdot \left(c \cdot j - x \cdot t\right)\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+43} \lor \neg \left(b \leq 1.05 \cdot 10^{+96}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	45.9
Cost	2029

\[\begin{array}{l} t_1 := c \cdot \left(a \cdot j - z \cdot b\right)\\ t_2 := a \cdot \left(c \cdot j - x \cdot t\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-114}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-177}:\\ \;\;\;\;\left(y \cdot i\right) \cdot \left(-j\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-231}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-265}:\\ \;\;\;\;i \cdot \left(y \cdot \left(-j\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+117} \lor \neg \left(x \leq 4 \cdot 10^{+201}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 16
Error	23.2
Cost	2008

\[\begin{array}{l} t_1 := i \cdot \left(t \cdot b - y \cdot j\right)\\ t_2 := c \cdot \left(a \cdot j\right) + t_1\\ t_3 := x \cdot \left(y \cdot z - t \cdot a\right) + c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{if}\;i \leq -1.28 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -1 \cdot 10^{+55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq -2.9 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 0.000105:\\ \;\;\;\;t_3\\ \mathbf{elif}\;i \leq 3 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	38.1
Cost	1896

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i - x \cdot a\right)\\ t_2 := i \cdot \left(t \cdot b - y \cdot j\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_4 := y \cdot \left(x \cdot z - i \cdot j\right)\\ \mathbf{if}\;j \leq -5.2 \cdot 10^{+65}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;j \leq -4.7 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq -4.5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;j \leq -4.3 \cdot 10^{-103}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;j \leq -5.6 \cdot 10^{-120}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \leq -1.35 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq -1 \cdot 10^{-266}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;j \leq 1.92 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;j \leq 3.4 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;j \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 18
Error	39.1
Cost	1765

\[\begin{array}{l} t_1 := i \cdot \left(t \cdot b - y \cdot j\right)\\ t_2 := y \cdot \left(x \cdot z - i \cdot j\right)\\ t_3 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-172}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;b \leq 1.62 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \left(c \cdot j - x \cdot t\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+44} \lor \neg \left(b \leq 2.3 \cdot 10^{+95}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	51.7
Cost	1245

\[\begin{array}{l} t_1 := y \cdot \left(x \cdot z\right)\\ t_2 := i \cdot \left(t \cdot b\right)\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.3 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-251}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-171}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{elif}\;b \leq 1.25 \lor \neg \left(b \leq 1.9 \cdot 10^{+96}\right):\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	49.5
Cost	1244

\[\begin{array}{l} t_1 := i \cdot \left(y \cdot \left(-j\right)\right)\\ t_2 := x \cdot \left(y \cdot z\right)\\ t_3 := t \cdot \left(b \cdot i\right)\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-134}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 235000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 21
Error	49.5
Cost	1244

\[\begin{array}{l} t_1 := y \cdot \left(i \cdot \left(-j\right)\right)\\ t_2 := x \cdot \left(y \cdot z\right)\\ t_3 := t \cdot \left(b \cdot i\right)\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9.4 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;i \cdot \left(y \cdot \left(-j\right)\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 22
Error	42.2
Cost	1236

\[\begin{array}{l} t_1 := a \cdot \left(c \cdot j - x \cdot t\right)\\ \mathbf{if}\;a \leq -9 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-256}:\\ \;\;\;\;\left(y \cdot i\right) \cdot \left(-j\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-272}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	37.4
Cost	1236

\[\begin{array}{l} t_1 := c \cdot \left(a \cdot j - z \cdot b\right)\\ t_2 := i \cdot \left(t \cdot b - y \cdot j\right)\\ \mathbf{if}\;i \leq -2.3 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -4.3 \cdot 10^{-284}:\\ \;\;\;\;a \cdot \left(c \cdot j - x \cdot t\right)\\ \mathbf{elif}\;i \leq 5.7 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 1.22 \cdot 10^{-102}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;i \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 24
Error	38.1
Cost	1236

\[\begin{array}{l} t_1 := c \cdot \left(a \cdot j - z \cdot b\right)\\ t_2 := i \cdot \left(t \cdot b - y \cdot j\right)\\ \mathbf{if}\;i \leq -1.35 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq -3 \cdot 10^{-215}:\\ \;\;\;\;t \cdot \left(b \cdot i - x \cdot a\right)\\ \mathbf{elif}\;i \leq 6 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;i \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 25
Error	49.4
Cost	716

\[\begin{array}{l} t_1 := t \cdot \left(b \cdot i\right)\\ \mathbf{if}\;t \leq -1.76 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-209}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;t \leq 29500000000:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 26
Error	49.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-24} \lor \neg \left(a \leq 7.6 \cdot 10^{+31}\right):\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \end{array} \]

Alternative 27
Error	53.4
Cost	320

\[a \cdot \left(c \cdot j\right) \]

Reproduce

herbie shell --seed 2022364 
(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)))))