Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.4 → 12.2

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.346408381266666 \cdot 10^{-175}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;x \leq 1.1233370841637992 \cdot 10^{-208}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right) + \left(t \cdot \left(c \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;x \leq 8.216030825806673 \cdot 10^{+27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(c \cdot \left(j \cdot \sqrt[3]{t}\right)\right) - y \cdot \left(i \cdot j\right)\right)\\ \end{array}\]

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 ((x <= -4.346408381266666e-175)) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (((double) (a * ((double) (b * i)))) - ((double) (z * ((double) (b * c)))))))) + ((double) (j * ((double) (((double) (t * c)) - ((double) (y * i))))))));
	} else {
		double VAR_1;
		if ((x <= 1.1233370841637992e-208)) {
			VAR_1 = ((double) (((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))) + ((double) (((double) (t * ((double) (c * j)))) - ((double) (y * ((double) (i * j))))))));
		} else {
			double VAR_2;
			if ((x <= 8.216030825806673e+27)) {
				VAR_2 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (((double) (a * ((double) (b * i)))) - ((double) (z * ((double) (b * c)))))))) + ((double) (j * ((double) (((double) (t * c)) - ((double) (y * i))))))));
			} else {
				VAR_2 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))))) + ((double) (((double) (((double) (((double) cbrt(t)) * ((double) cbrt(t)))) * ((double) (c * ((double) (j * ((double) cbrt(t)))))))) - ((double) (y * ((double) (i * j))))))));
			}
			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	12.4
Target	16.1
Herbie	12.2

\[\begin{array}{l} \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -4.34640838126666601e-175 or 1.1233370841637992e-208 < x < 8.21603082580667259e27

   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. Using strategy rm

   3. Applied sub-neg 11.9

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

   4. Applied distribute-lft-in 11.9

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

   5. Simplified 12.0

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

   6. Simplified 11.5

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

   if -4.34640838126666601e-175 < x < 1.1233370841637992e-208

   1. Initial program 17.3

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

   3. Applied sub-neg 17.3

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

   4. Applied distribute-lft-in 17.3

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

   5. Simplified 17.8

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

   6. Simplified 17.6

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

   7. Taylor expanded around 0 16.9

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

   if 8.21603082580667259e27 < x

   1. Initial program 7.0

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

   3. Applied sub-neg 7.0

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

   4. Applied distribute-lft-in 7.0

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

   5. Simplified 8.1

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

   6. Simplified 8.9

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

   8. Applied add-cube-cbrt 9.0

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

   9. Applied associate-*l* 9.0

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

   10. Simplified 8.3

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

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.346408381266666 \cdot 10^{-175}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;x \leq 1.1233370841637992 \cdot 10^{-208}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right) + \left(t \cdot \left(c \cdot j\right) - y \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;x \leq 8.216030825806673 \cdot 10^{+27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(c \cdot \left(j \cdot \sqrt[3]{t}\right)\right) - y \cdot \left(i \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

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