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

Average Error: 11.8 → 9.2

Time: 12.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.521938149705369 \cdot 10^{+39} \lor \neg \left(x \leq 5.1255690185328515 \cdot 10^{-53}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(t \cdot b\right) - c \cdot \left(z \cdot b\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\left(y \cdot \left(x \cdot z\right) - \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(t \cdot \left(x \cdot \sqrt[3]{a}\right)\right)\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \end{array}\]

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) (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 (((x <= -1.521938149705369e+39) || !(x <= 5.1255690185328515e-53))) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (((double) (i * ((double) (t * b)))) - ((double) (c * ((double) (z * b)))))))) + ((double) (j * ((double) (((double) (a * c)) - ((double) (y * i))))))));
	} else {
		VAR = ((double) (((double) (j * ((double) (((double) (a * c)) - ((double) (y * i)))))) + ((double) (((double) (((double) (y * ((double) (x * z)))) - ((double) (((double) (((double) cbrt(a)) * ((double) cbrt(a)))) * ((double) (t * ((double) (x * ((double) cbrt(a)))))))))) + ((double) (b * ((double) (((double) (t * i)) - ((double) (z * c))))))))));
	}
	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	11.8
Target	19.6
Herbie	9.2

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

Derivation

1. Split input into 2 regimes

2. if x < -1.5219381497053691e39 or 5.12556901853285147e-53 < x

   1. Initial program 7.8

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

   3. Applied sub-neg 7.8

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

   4. Applied distribute-lft-in 7.8

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

   5. Simplified 7.9

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

   6. Simplified 8.3

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

   if -1.5219381497053691e39 < x < 5.12556901853285147e-53

   1. Initial program 14.6

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

   3. Applied sub-neg 14.6

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

   4. Applied distribute-lft-in 14.6

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

   5. Simplified 12.5

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

   6. Simplified 9.9

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

   8. Applied add-cube-cbrt 10.0

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

   9. Applied associate-*l* 10.0

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

   10. Simplified 9.9

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.521938149705369 \cdot 10^{+39} \lor \neg \left(x \leq 5.1255690185328515 \cdot 10^{-53}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(t \cdot b\right) - c \cdot \left(z \cdot b\right)\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) + \left(\left(y \cdot \left(x \cdot z\right) - \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(t \cdot \left(x \cdot \sqrt[3]{a}\right)\right)\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))))