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

Average Error: 12.4 → 12.4

Time: 9.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.57520709208652848 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 4.4278067172035752 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

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 ((b <= -1.5752070920865285e-208)) {
		VAR = ((double) (((double) (((double) (((double) (((double) cbrt(((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))))) * ((double) cbrt(((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))))))) * ((double) (((double) cbrt(x)) * ((double) cbrt(((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))))))));
	} else {
		double VAR_1;
		if ((b <= 4.427806717203575e-169)) {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - 0.0)) + ((double) (j * ((double) (((double) (c * a)) - ((double) (y * i))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (((double) sqrt(b)) * ((double) (((double) sqrt(b)) * ((double) (((double) (c * z)) - ((double) (t * i)))))))))) + ((double) (j * ((double) (((double) (c * a)) - ((double) (y * i))))))));
		}
		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	12.4
Target	19.7
Herbie	12.4

\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -1.5752070920865285e-208

   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. Using strategy rm

   3. Applied add-cube-cbrt 10.7

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

   5. Applied cbrt-prod 10.6

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

   if -1.5752070920865285e-208 < b < 4.427806717203575e-169

   1. Initial program 18.4

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

   2. Taylor expanded around 0 18.2

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

   if 4.427806717203575e-169 < b

   1. Initial program 10.3

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

   3. Applied add-sqr-sqrt 10.4

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

   4. Applied associate-*l* 10.4

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.57520709208652848 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 4.4278067172035752 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(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) (pow (* t i) 2))) (+ (* 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) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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