Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 9.4

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \leq -1.396055320967211 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(t \cdot c - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 2.007154899331252 \cdot 10^{-94}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot \left(z \cdot b\right) + \sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)} \cdot \left(\sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)} \cdot \sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)}\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;j \leq 3.010292186307326 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 3.914296527345982 \cdot 10^{+59}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(a \cdot b\right) - z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(t \cdot c - y \cdot i\right)\right)\\ \end{array}\]

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 ((j <= -1.396055320967211e+107)) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))))) + ((double) (((double) (((double) cbrt(j)) * ((double) cbrt(j)))) * ((double) (((double) cbrt(j)) * ((double) (((double) (t * c)) - ((double) (y * i))))))))));
	} else {
		double VAR_1;
		if ((j <= 2.007154899331252e-94)) {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) - ((double) (((double) (c * ((double) (z * b)))) + ((double) (((double) cbrt(((double) (i * ((double) (a * ((double) -(b)))))))) * ((double) (((double) cbrt(((double) (i * ((double) (a * ((double) -(b)))))))) * ((double) cbrt(((double) (i * ((double) (a * ((double) -(b)))))))))))))))) + ((double) (((double) (t * ((double) (j * c)))) - ((double) (i * ((double) (j * y))))))));
		} else {
			double VAR_2;
			if ((j <= 3.010292186307326e-26)) {
				VAR_2 = ((double) (((double) (((double) (((double) (y * ((double) (x * z)))) - ((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_3;
				if ((j <= 3.914296527345982e+59)) {
					VAR_3 = ((double) (((double) (j * ((double) (((double) (t * c)) - ((double) (y * i)))))) + ((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (((double) (i * ((double) (a * b)))) - ((double) (z * ((double) (b * c))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) (x * ((double) (((double) (y * z)) - ((double) (t * a)))))) + ((double) (b * ((double) (((double) (a * i)) - ((double) (z * c)))))))) + ((double) (((double) (((double) cbrt(j)) * ((double) cbrt(j)))) * ((double) (((double) cbrt(j)) * ((double) (((double) (t * c)) - ((double) (y * i))))))))));
				}
				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	9.4

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

Derivation

1. Split input into 4 regimes

2. if j < -1.3960553209672111e107 or 3.9142965273459819e59 < j

   1. Initial program Error: 7.2 bits

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

   3. Applied add-cube-cbrt Error: 7.8 bits

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

   4. Applied associate-*l* Error: 7.8 bits

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

   5. Simplified Error: 7.8 bits

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

   if -1.3960553209672111e107 < j < 2.00715489933125203e-94

   1. Initial program Error: 14.8 bits

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

   3. Applied sub-neg Error: 14.8 bits

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

   4. Applied distribute-lft-in Error: 14.8 bits

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

   5. Simplified Error: 14.5 bits

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

   6. Simplified Error: 14.4 bits

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

   8. Applied add-cube-cbrt Error: 14.6 bits

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

   10. Applied sub-neg Error: 14.6 bits

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

   11. Applied distribute-lft-in Error: 14.6 bits

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

   12. Simplified Error: 12.2 bits

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

   14. Applied associate-*r* Error: 9.6 bits

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

   if 2.00715489933125203e-94 < j < 3.01029218630732577e-26

   1. Initial program Error: 13.4 bits

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

   3. Applied sub-neg Error: 13.4 bits

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

   4. Applied distribute-lft-in Error: 13.4 bits

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

   5. Simplified Error: 12.5 bits

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

   6. Simplified Error: 12.6 bits

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

   if 3.01029218630732577e-26 < j < 3.9142965273459819e59

   1. Initial program Error: 10.9 bits

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

   3. Applied sub-neg Error: 10.9 bits

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

   4. Applied distribute-lft-in Error: 10.9 bits

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

   5. Simplified Error: 10.3 bits

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

   6. Simplified Error: 9.5 bits

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

   8. Applied associate-*r* Error: 10.2 bits

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{\left(c \cdot b\right) \cdot z} + i \cdot \left(a \cdot \left(-b\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification Error: 9.4 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.396055320967211 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(t \cdot c - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 2.007154899331252 \cdot 10^{-94}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot \left(z \cdot b\right) + \sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)} \cdot \left(\sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)} \cdot \sqrt[3]{i \cdot \left(a \cdot \left(-b\right)\right)}\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;j \leq 3.010292186307326 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 3.914296527345982 \cdot 10^{+59}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(i \cdot \left(a \cdot b\right) - z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(t \cdot c - y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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)))))