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

Average Error: 12.5 → 11.1

Time: 10.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.6066223601240277 \cdot 10^{-104}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(x \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;y \le -1.3902995953278734 \cdot 10^{-275}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{elif}\;y \le 7.364773937325596 \cdot 10^{-181}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + \left(c \cdot \left(a \cdot j\right) + i \cdot \left(y \cdot \left(-j\right)\right)\right)\\ \mathbf{elif}\;y \le 2.1327073790487086 \cdot 10^{-47}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(x \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array}\]

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) (t * i)))))))) + ((double) (j * ((double) (((double) (c * a)) - ((double) (y * i))))))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double VAR;
	if ((y <= -4.6066223601240277e-104)) {
		VAR = ((double) (((double) (((double) (((double) (y * ((double) (z * x)))) + ((double) (t * ((double) (x * ((double) -(a)))))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (t * i)))))))) + ((double) (j * ((double) (((double) (a * c)) - ((double) (y * i))))))));
	} else {
		double VAR_1;
		if ((y <= -1.3902995953278734e-275)) {
			VAR_1 = ((double) (((double) (j * ((double) (((double) (a * c)) - ((double) (y * i)))))) + ((double) (((double) (((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))) * ((double) (x * ((double) (((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))) * ((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))))))))) - ((double) (((double) (z * ((double) (b * c)))) + ((double) (t * ((double) (b * ((double) -(i))))))))))));
		} else {
			double VAR_2;
			if ((y <= 7.364773937325596e-181)) {
				VAR_2 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (t * i)))))))) + ((double) (((double) (c * ((double) (a * j)))) + ((double) (i * ((double) (y * ((double) -(j))))))))));
			} else {
				double VAR_3;
				if ((y <= 2.1327073790487086e-47)) {
					VAR_3 = ((double) (((double) (j * ((double) (((double) (a * c)) - ((double) (y * i)))))) + ((double) (((double) (((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))) * ((double) (x * ((double) (((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))) * ((double) cbrt(((double) (((double) (y * z)) - ((double) (t * a)))))))))))) - ((double) (((double) (z * ((double) (b * c)))) + ((double) (t * ((double) (b * ((double) -(i))))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) (((double) (y * ((double) (z * x)))) + ((double) (t * ((double) (x * ((double) -(a)))))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (t * i)))))))) + ((double) (j * ((double) (((double) (a * c)) - ((double) (y * i))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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

Target

Original	12.5
Target	20.0
Herbie	11.1

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

Derivation

1. Split input into 3 regimes

2. if y < -4.6066223601240277e-104 or 2.13270737904870859e-47 < y

   1. Initial program 14.8

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
   2. Using strategy rm

   3. Applied sub-neg 14.8

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

   4. Applied distribute-lft-in 14.8

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

   5. Simplified 15.6

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

   6. Simplified 14.8

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

   8. Applied associate-*r* 11.9

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

   if -4.6066223601240277e-104 < y < -1.39029959532787341e-275 or 7.364773937325596e-181 < y < 2.13270737904870859e-47

   1. Initial program 9.7

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 9.9

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

   4. Applied associate-*r* 9.9

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

   5. Simplified 9.9

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

   7. Applied sub-neg 9.9

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

   8. Applied distribute-lft-in 9.9

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

   9. Simplified 9.5

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

   10. Simplified 10.0

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

   if -1.39029959532787341e-275 < y < 7.364773937325596e-181

   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. Using strategy rm

   3. Applied sub-neg 10.1

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

   4. Applied distribute-lft-in 10.1

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

   5. Simplified 10.5

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

   6. Simplified 10.2

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.6066223601240277 \cdot 10^{-104}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(x \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;y \le -1.3902995953278734 \cdot 10^{-275}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{elif}\;y \le 7.364773937325596 \cdot 10^{-181}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + \left(c \cdot \left(a \cdot j\right) + i \cdot \left(y \cdot \left(-j\right)\right)\right)\\ \mathbf{elif}\;y \le 2.1327073790487086 \cdot 10^{-47}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(x \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - t \cdot i\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))))