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

Average Error: 12.2 → 9.2

Time: 15.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 3.50292219667431216 \cdot 10^{-58}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) - t \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

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 <= 3.502922196674312e-58)) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (((double) (z * ((double) (b * c)))) - ((double) (t * ((double) (i * b)))))))) + ((double) (j * ((double) (((double) (c * a)) - ((double) (y * i))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (((double) cbrt(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))))))));
	}
	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	12.2
Target	20.3
Herbie	9.2

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

Derivation

1. Split input into 2 regimes

2. if b < 3.50292219667431216e-58

   1. Initial program 16.3

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

   3. Applied add-sqr-sqrt 16.3

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

   4. Applied associate-*l* 16.3

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

   6. Applied add-sqr-sqrt 16.3

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

   7. Applied sqrt-prod 16.3

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

   8. Applied associate-*l* 16.3

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

   10. Applied add-cube-cbrt 16.3

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

   11. Applied sqrt-prod 16.3

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

   12. Applied associate-*l* 16.3

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

   13. Simplified 16.3

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

   14. Taylor expanded around inf 10.1

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

   if 3.50292219667431216e-58 < b

   1. Initial program 8.0

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

   3. Applied add-cube-cbrt 8.3

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

   4. Applied associate-*l* 8.3

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.50292219667431216 \cdot 10^{-58}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) - t \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))))