Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 12.0

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.6013812807207604 \cdot 10^{+217}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + \left(t \cdot \left(c \cdot j\right) + y \cdot \left(i \cdot \left(-j\right)\right)\right)\\ \mathbf{elif}\;y \le -1.1526861879531309 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(\sqrt[3]{z} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)\right) + x \cdot \left(t \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;y \le 7.867274113144894 \cdot 10^{-38}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{elif}\;y \le 1.4865130668714244 \cdot 10^{+148}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + a \cdot \left(x \cdot \left(-t\right)\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(t \cdot \left(-a\right)\right) + z \cdot \left(y \cdot x\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(z \cdot c - a \cdot i\right) \cdot \sqrt[3]{b}\right)\right)\\ \end{array}\]

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 ((y <= -1.6013812807207604e+217)) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (a * i)))))))) + ((double) (((double) (t * ((double) (c * j)))) + ((double) (y * ((double) (i * ((double) -(j))))))))));
	} else {
		double VAR_1;
		if ((y <= -1.1526861879531309e-196)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) cbrt(z)) * ((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (((double) cbrt(x)) * ((double) (y * ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))))))))) + ((double) (x * ((double) (t * ((double) -(a)))))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (a * i)))))))) + ((double) (j * ((double) (((double) (t * c)) - ((double) (y * i))))))));
		} else {
			double VAR_2;
			if ((y <= 7.867274113144894e-38)) {
				VAR_2 = ((double) (((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))) + ((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (((double) (z * ((double) (b * c)))) + ((double) (a * ((double) (b * ((double) -(i))))))))))));
			} else {
				double VAR_3;
				if ((y <= 1.4865130668714244e+148)) {
					VAR_3 = ((double) (((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))) + ((double) (((double) (((double) (x * ((double) (y * z)))) + ((double) (a * ((double) (x * ((double) -(t)))))))) - ((double) (b * ((double) (((double) (z * c)) - ((double) (a * i))))))))));
				} else {
					VAR_3 = ((double) (((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))) + ((double) (((double) (((double) (x * ((double) (t * ((double) -(a)))))) + ((double) (z * ((double) (y * x)))))) - ((double) (((double) (((double) cbrt(b)) * ((double) cbrt(b)))) * ((double) (((double) (((double) (z * c)) - ((double) (a * i)))) * ((double) cbrt(b))))))))));
				}
				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	16.3
Herbie	12.0

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

Derivation

1. Split input into 5 regimes

2. if y < -1.6013812807207604e217

   1. Initial program 24.9

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

   3. Applied sub-neg 24.9

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

   4. Applied distribute-lft-in 24.9

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

   5. Simplified 23.7

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

   6. Simplified 15.4

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

   if -1.6013812807207604e217 < y < -1.152686187953131e-196

   1. Initial program 11.8

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

   3. Applied sub-neg 11.8

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

   4. Applied distribute-lft-in 11.8

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

   5. Simplified 11.8

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

   7. Applied associate-*r* 11.7

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

   9. Applied add-cube-cbrt 11.9

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

   10. Applied associate-*r* 11.9

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

   11. Simplified 11.1

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

   13. Applied add-cube-cbrt 11.2

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

   14. Applied associate-*l* 11.2

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

   if -1.152686187953131e-196 < y < 7.86727411314489398e-38

   1. Initial program 9.5

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

   3. Applied sub-neg 9.5

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

   4. Applied distribute-lft-in 9.5

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

   5. Simplified 9.1

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

   6. Simplified 9.3

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

   if 7.86727411314489398e-38 < y < 1.4865130668714244e148

   1. Initial program 11.4

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

   3. Applied sub-neg 11.4

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

   4. Applied distribute-lft-in 11.4

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

   5. Simplified 11.4

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

   7. Applied associate-*r* 11.6

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

   if 1.4865130668714244e148 < y

   1. Initial program 24.4

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

   3. Applied sub-neg 24.4

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

   4. Applied distribute-lft-in 24.4

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

   5. Simplified 24.4

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

   7. Applied associate-*r* 26.5

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

   9. Applied add-cube-cbrt 26.6

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

   10. Applied associate-*l* 26.6

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

   11. Simplified 26.6

       \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(t \cdot \left(-a\right)\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \color{blue}{\left(\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.6013812807207604 \cdot 10^{+217}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + \left(t \cdot \left(c \cdot j\right) + y \cdot \left(i \cdot \left(-j\right)\right)\right)\\ \mathbf{elif}\;y \le -1.1526861879531309 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(\sqrt[3]{z} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)\right) + x \cdot \left(t \cdot \left(-a\right)\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;y \le 7.867274113144894 \cdot 10^{-38}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + a \cdot \left(b \cdot \left(-i\right)\right)\right)\right)\\ \mathbf{elif}\;y \le 1.4865130668714244 \cdot 10^{+148}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + a \cdot \left(x \cdot \left(-t\right)\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(t \cdot \left(-a\right)\right) + z \cdot \left(y \cdot x\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(z \cdot c - a \cdot i\right) \cdot \sqrt[3]{b}\right)\right)\\ \end{array}\]

Reproduce

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