Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 12.0

Time: 11.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.62289413396292431 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-i \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \le 3.2098948594341111 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + -1 \cdot \left(\left(a \cdot \left(x \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \cdot \sqrt[3]{t}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double VAR;
	if ((b <= -8.622894133962924e-262)) {
		VAR = ((double) fma(((double) (((double) (c * t)) - ((double) (i * y)))), j, ((double) (((double) (((double) (x * ((double) (y * z)))) + ((double) (x * ((double) -(((double) (t * a)))))))) - ((double) (((double) (b * ((double) (c * z)))) + ((double) (b * ((double) -(((double) (i * a))))))))))));
	} else {
		double VAR_1;
		if ((b <= 3.209894859434111e-287)) {
			VAR_1 = ((double) fma(((double) (((double) (c * t)) - ((double) (i * y)))), j, ((double) (((double) (((double) (x * ((double) (y * z)))) + ((double) (-1.0 * ((double) (a * ((double) (x * t)))))))) - 0.0))));
		} else {
			VAR_1 = ((double) fma(((double) (((double) (c * t)) - ((double) (i * y)))), j, ((double) (((double) (((double) (x * ((double) (y * z)))) + ((double) (-1.0 * ((double) (((double) (a * ((double) (x * ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))))) * ((double) cbrt(t)))))))) - ((double) (b * ((double) (((double) (c * z)) - ((double) (i * a))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Derivation

1. Split input into 3 regimes

2. if b < -8.622894133962924e-262

   1. Initial program 11.9

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

   2. Simplified 11.9

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

   4. Applied sub-neg 11.9

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

   5. Applied distribute-lft-in 11.9

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

   7. Applied sub-neg 11.9

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

   8. Applied distribute-lft-in 11.9

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

   if -8.622894133962924e-262 < b < 3.209894859434111e-287

   1. Initial program 18.4

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

   2. Simplified 18.4

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

   4. Applied sub-neg 18.4

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

   5. Applied distribute-lft-in 18.4

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

   6. Taylor expanded around inf 17.9

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

   7. Taylor expanded around 0 14.8

      \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - \color{blue}{0}\right)\]

   if 3.209894859434111e-287 < b

   1. Initial program 11.4

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

   2. Simplified 11.4

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

   4. Applied sub-neg 11.4

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

   5. Applied distribute-lft-in 11.4

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

   6. Taylor expanded around inf 12.3

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

   8. Applied add-cube-cbrt 12.4

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

   9. Applied associate-*r* 12.4

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

   10. Applied associate-*r* 11.7

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.62289413396292431 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-i \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \le 3.2098948594341111 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + -1 \cdot \left(\left(a \cdot \left(x \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \cdot \sqrt[3]{t}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))