Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 9.8

Time: 12.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.5669503841466718 \cdot 10^{111}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) - t \cdot \left(x \cdot a\right)\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;b \le 2.4111297055815879 \cdot 10^{119}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)} \cdot \left(\sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)}\right)\\ \end{array}\]

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 ((b <= -7.566950384146672e+111)) {
		VAR = ((double) (((double) (((double) (((double) (y * ((double) (z * x)))) - ((double) (t * ((double) (x * a)))))) + ((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))))) + ((double) (j * ((double) (((double) (t * c)) - ((double) (y * i))))))));
	} else {
		double VAR_1;
		if ((b <= 2.411129705581588e+119)) {
			VAR_1 = ((double) (((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))) + ((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (((double) (i * ((double) (b * a)))) - ((double) (z * ((double) (b * c))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))))) + ((double) (((double) cbrt(((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))))) * ((double) (((double) cbrt(((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))))) * ((double) cbrt(((double) (j * ((double) (((double) (t * c)) - ((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	12.2
Target	16.1
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -7.5669503841466718e111

   1. Initial program 6.0

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

   3. Applied sub-neg 6.0

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

   4. Applied distribute-lft-in 6.0

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

   5. Simplified 5.8

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

   6. Simplified 6.3

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

   if -7.5669503841466718e111 < b < 2.4111297055815879e119

   1. Initial program 13.8

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

   3. Applied sub-neg 13.8

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

   4. Applied distribute-lft-in 13.8

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

   5. Simplified 12.4

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

   6. Simplified 10.7

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

   8. Applied associate-*r* 10.6

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

   if 2.4111297055815879e119 < b

   1. Initial program 6.9

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

   3. Applied add-cube-cbrt 7.1

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.5669503841466718 \cdot 10^{111}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) - t \cdot \left(x \cdot a\right)\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;b \le 2.4111297055815879 \cdot 10^{119}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)} \cdot \left(\sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(t \cdot c - y \cdot i\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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)))))