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

Average Error: 16.9 → 19.1

Time: 15.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := c \cdot \left(a \cdot j - b \cdot z\right)\\ t_2 := i \cdot b - a \cdot x\\ \mathbf{if}\;y \leq -3.4146347982611155 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(t, t_2, t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := y \cdot \left(x \cdot z\right)\\ t_4 := y \cdot \left(i \cdot j\right)\\ \mathbf{if}\;y \leq -5.583327439550542 \cdot 10^{+166}:\\ \;\;\;\;\left(i \cdot \left(t \cdot b\right) + t_3\right) - \left(t_4 + a \cdot \left(t \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := c \cdot \left(b \cdot z\right)\\ \mathbf{if}\;y \leq -6.587183720127357 \cdot 10^{+113}:\\ \;\;\;\;\left(t_3 + c \cdot \left(a \cdot j\right)\right) - \left(t_4 + t_5\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_6 := y \cdot z - t \cdot a\\ \mathbf{if}\;y \leq -746741518026.9556:\\ \;\;\;\;x \cdot t_6 + i \cdot \left(t \cdot b - y \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_7 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;y \leq -5.418483375313162 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(x, t_6, t_7\right) - t_5\\ \mathbf{elif}\;y \leq 3.5212638255044147 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \left(t \cdot i\right) + \left(a \cdot \left(c \cdot j - t \cdot x\right) - b \cdot \left(c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_8 := t \cdot i - c \cdot z\\ \mathbf{if}\;y \leq 1.0736222073065391 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(b, t_8, \mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -t \cdot a\right), t_7\right)\right)\\ \mathbf{elif}\;y \leq 1.4276677929317117 \cdot 10^{-52}:\\ \;\;\;\;t_1 + t \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, t_8, \mathsf{fma}\left(x, t_6, -t_4\right)\right)\\ \end{array}\\ \end{array}\\ \end{array}\\ \end{array}\\ \end{array}\\ \end{array} \]

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 (* c (- (* a j) (* b z)))) (t_2 (- (* i b) (* a x))))
   (if (<= y -3.4146347982611155e+236)
     (log1p (expm1 (fma t t_2 t_1)))
     (let* ((t_3 (* y (* x z))) (t_4 (* y (* i j))))
       (if (<= y -5.583327439550542e+166)
         (- (+ (* i (* t b)) t_3) (+ t_4 (* a (* t x))))
         (let* ((t_5 (* c (* b z))))
           (if (<= y -6.587183720127357e+113)
             (- (+ t_3 (* c (* a j))) (+ t_4 t_5))
             (let* ((t_6 (- (* y z) (* t a))))
               (if (<= y -746741518026.9556)
                 (+ (* x t_6) (* i (- (* t b) (* y j))))
                 (let* ((t_7 (* j (- (* a c) (* y i)))))
                   (if (<= y -5.418483375313162e-135)
                     (- (fma x t_6 t_7) t_5)
                     (if (<= y 3.5212638255044147e-245)
                       (+
                        (* b (* t i))
                        (- (* a (- (* c j) (* t x))) (* b (* c z))))
                       (let* ((t_8 (- (* t i) (* c z))))
                         (if (<= y 1.0736222073065391e-79)
                           (fma b t_8 (fma x (fma y z (- (* t a))) t_7))
                           (if (<= y 1.4276677929317117e-52)
                             (+ t_1 (* t (log1p (expm1 t_2))))
                             (fma b t_8 (fma x t_6 (- t_4))))))))))))))))))

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 = c * ((a * j) - (b * z));
	double t_2 = (i * b) - (a * x);
	double tmp;
	if (y <= -3.4146347982611155e+236) {
		tmp = log1p(expm1(fma(t, t_2, t_1)));
	} else {
		double t_3 = y * (x * z);
		double t_4 = y * (i * j);
		double tmp_1;
		if (y <= -5.583327439550542e+166) {
			tmp_1 = ((i * (t * b)) + t_3) - (t_4 + (a * (t * x)));
		} else {
			double t_5 = c * (b * z);
			double tmp_2;
			if (y <= -6.587183720127357e+113) {
				tmp_2 = (t_3 + (c * (a * j))) - (t_4 + t_5);
			} else {
				double t_6 = (y * z) - (t * a);
				double tmp_3;
				if (y <= -746741518026.9556) {
					tmp_3 = (x * t_6) + (i * ((t * b) - (y * j)));
				} else {
					double t_7 = j * ((a * c) - (y * i));
					double tmp_4;
					if (y <= -5.418483375313162e-135) {
						tmp_4 = fma(x, t_6, t_7) - t_5;
					} else if (y <= 3.5212638255044147e-245) {
						tmp_4 = (b * (t * i)) + ((a * ((c * j) - (t * x))) - (b * (c * z)));
					} else {
						double t_8 = (t * i) - (c * z);
						double tmp_5;
						if (y <= 1.0736222073065391e-79) {
							tmp_5 = fma(b, t_8, fma(x, fma(y, z, -(t * a)), t_7));
						} else if (y <= 1.4276677929317117e-52) {
							tmp_5 = t_1 + (t * log1p(expm1(t_2)));
						} else {
							tmp_5 = fma(b, t_8, fma(x, t_6, -t_4));
						}
						tmp_4 = tmp_5;
					}
					tmp_3 = tmp_4;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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

Original	16.9
Target	25.9
Herbie	19.1

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

Derivation

1. Split input into 9 regimes

2. if y < -3.4146347982611155e236

   1. Initial program 27.3

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

   2. Simplified 24.4

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

   3. Taylor expanded in y around 0 42.5

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

   4. Simplified 39.2

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

   5. Applied log1p-expm1-u_binary64 36.3

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(c \cdot \left(a \cdot j - z \cdot b\right) + t \cdot \left(i \cdot b - a \cdot x\right)\right)\right)} \]

   6. Simplified 35.5

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(t, i \cdot b - a \cdot x, c \cdot \left(a \cdot j - b \cdot z\right)\right)\right)}\right) \]

   if -3.4146347982611155e236 < y < -5.5833274395505421e166

   1. Initial program 23.7

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

   2. Simplified 20.9

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

   3. Taylor expanded in c around 0 22.2

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

   if -5.5833274395505421e166 < y < -6.58718372012735694e113

   1. Initial program 19.1

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

   2. Simplified 16.2

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

   3. Taylor expanded in t around 0 19.7

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

   if -6.58718372012735694e113 < y < -746741518026.95557

   1. Initial program 15.3

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

   2. Simplified 13.7

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

   3. Taylor expanded in c around 0 23.6

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

   4. Simplified 21.4

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

   if -746741518026.95557 < y < -5.418483375313162e-135

   1. Initial program 11.8

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

   2. Simplified 10.7

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

   3. Applied fma-udef_binary64 11.0

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

   4. Taylor expanded in z around inf 18.2

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

   if -5.418483375313162e-135 < y < 3.52126382550441473e-245

   1. Initial program 11.9

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

   2. Simplified 11.0

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

   3. Applied fma-udef_binary64 11.6

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

   4. Applied sub-neg_binary64 11.6

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

   5. Applied distribute-rgt-in_binary64 12.3

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

   6. Applied associate-+l+_binary64 12.3

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

   7. Simplified 12.3

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

   8. Taylor expanded in y around 0 16.8

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

   9. Simplified 15.5

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

   if 3.52126382550441473e-245 < y < 1.07362220730653913e-79

   1. Initial program 14.0

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

   2. Simplified 13.1

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

   3. Applied fma-neg_binary64 13.1

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

   4. Simplified 13.1

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

   if 1.07362220730653913e-79 < y < 1.4276677929317117e-52

   1. Initial program 14.1

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

   2. Simplified 11.6

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

   3. Taylor expanded in y around 0 23.4

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

   4. Simplified 22.8

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

   5. Applied log1p-expm1-u_binary64 30.6

      \[\leadsto c \cdot \left(a \cdot j - z \cdot b\right) + t \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(i \cdot b - a \cdot x\right)\right)} \]

   if 1.4276677929317117e-52 < y

   1. Initial program 20.3

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

   2. Simplified 17.9

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

   3. Taylor expanded in a around 0 18.8

      \[\leadsto \mathsf{fma}\left(b, t \cdot i - z \cdot c, \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{-1 \cdot \left(y \cdot \left(i \cdot j\right)\right)}\right)\right) \]
3. Recombined 9 regimes into one program.

4. Final simplification 19.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4146347982611155 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(t, i \cdot b - a \cdot x, c \cdot \left(a \cdot j - b \cdot z\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq -5.583327439550542 \cdot 10^{+166}:\\ \;\;\;\;\left(i \cdot \left(t \cdot b\right) + y \cdot \left(x \cdot z\right)\right) - \left(y \cdot \left(i \cdot j\right) + a \cdot \left(t \cdot x\right)\right)\\ \mathbf{elif}\;y \leq -6.587183720127357 \cdot 10^{+113}:\\ \;\;\;\;\left(y \cdot \left(x \cdot z\right) + c \cdot \left(a \cdot j\right)\right) - \left(y \cdot \left(i \cdot j\right) + c \cdot \left(b \cdot z\right)\right)\\ \mathbf{elif}\;y \leq -746741518026.9556:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right) + i \cdot \left(t \cdot b - y \cdot j\right)\\ \mathbf{elif}\;y \leq -5.418483375313162 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(a \cdot c - y \cdot i\right)\right) - c \cdot \left(b \cdot z\right)\\ \mathbf{elif}\;y \leq 3.5212638255044147 \cdot 10^{-245}:\\ \;\;\;\;b \cdot \left(t \cdot i\right) + \left(a \cdot \left(c \cdot j - t \cdot x\right) - b \cdot \left(c \cdot z\right)\right)\\ \mathbf{elif}\;y \leq 1.0736222073065391 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, \mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -t \cdot a\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.4276677929317117 \cdot 10^{-52}:\\ \;\;\;\;c \cdot \left(a \cdot j - b \cdot z\right) + t \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(i \cdot b - a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, \mathsf{fma}\left(x, y \cdot z - t \cdot a, -y \cdot \left(i \cdot j\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))))