Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 9.1

Time: 12.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -9.340382156181511 \cdot 10^{58}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \mathsf{fma}\left(c, z, -i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 1.13821747314357863 \cdot 10^{34}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, -a \cdot t\right)\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \mathsf{fma}\left(c, z, -i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double temp;
	if ((j <= -9.340382156181511e+58)) {
		temp = ((((x * fma((y * (cbrt(z) * cbrt(z))), cbrt(z), -(a * t))) + (x * fma(-a, t, (a * t)))) - ((b * fma(c, z, -(i * a))) + (b * fma(-a, i, (a * i))))) + (j * ((c * t) - (i * y))));
	} else {
		double temp_1;
		if ((j <= 1.1382174731435786e+34)) {
			temp_1 = (((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((t * (j * c)) - (i * (j * y))));
		} else {
			temp_1 = (((((cbrt(x) * cbrt(x)) * (cbrt(x) * fma((y * (cbrt(z) * cbrt(z))), cbrt(z), -(a * t)))) + (x * fma(-a, t, (a * t)))) - ((b * fma(c, z, -(i * a))) + (b * fma(-a, i, (a * i))))) + (j * ((c * t) - (i * y))));
		}
		temp = temp_1;
	}
	return temp;
}

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

Derivation

1. Split input into 3 regimes

2. if j < -9.340382156181511e+58

   1. Initial program 6.4

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

   3. Applied prod-diff 6.4

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

   4. Applied distribute-lft-in 6.4

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

   5. Simplified 6.4

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

   7. Applied add-cube-cbrt 6.5

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

   8. Applied associate-*r* 6.5

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

   9. Applied prod-diff 6.5

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

   10. Applied distribute-lft-in 6.5

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

   if -9.340382156181511e+58 < j < 1.1382174731435786e+34

   1. Initial program 14.4

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

   2. Taylor expanded around inf 10.0

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

   if 1.1382174731435786e+34 < j

   1. Initial program 7.4

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

   3. Applied prod-diff 7.4

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

   4. Applied distribute-lft-in 7.4

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

   5. Simplified 7.4

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

   7. Applied add-cube-cbrt 7.4

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

   8. Applied associate-*r* 7.5

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

   9. Applied prod-diff 7.5

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

   10. Applied distribute-lft-in 7.5

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

   12. Applied add-cube-cbrt 7.6

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

   13. Applied associate-*l* 7.6

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -9.340382156181511 \cdot 10^{58}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \mathsf{fma}\left(c, z, -i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 1.13821747314357863 \cdot 10^{34}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, -a \cdot t\right)\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \mathsf{fma}\left(c, z, -i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))