Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 11.8

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.2733319676195215 \cdot 10^{-175}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\\ \mathbf{elif}\;y \le 3.6987829761029658 \cdot 10^{118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t\right)\right) + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \left(j \cdot i\right) \cdot \left(-y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r111475 = x;
        double r111476 = y;
        double r111477 = z;
        double r111478 = r111476 * r111477;
        double r111479 = t;
        double r111480 = a;
        double r111481 = r111479 * r111480;
        double r111482 = r111478 - r111481;
        double r111483 = r111475 * r111482;
        double r111484 = b;
        double r111485 = c;
        double r111486 = r111485 * r111477;
        double r111487 = i;
        double r111488 = r111487 * r111480;
        double r111489 = r111486 - r111488;
        double r111490 = r111484 * r111489;
        double r111491 = r111483 - r111490;
        double r111492 = j;
        double r111493 = r111485 * r111479;
        double r111494 = r111487 * r111476;
        double r111495 = r111493 - r111494;
        double r111496 = r111492 * r111495;
        double r111497 = r111491 + r111496;
        return r111497;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r111498 = y;
        double r111499 = -2.2733319676195215e-175;
        bool r111500 = r111498 <= r111499;
        double r111501 = x;
        double r111502 = z;
        double r111503 = r111498 * r111502;
        double r111504 = t;
        double r111505 = a;
        double r111506 = r111504 * r111505;
        double r111507 = r111503 - r111506;
        double r111508 = r111501 * r111507;
        double r111509 = b;
        double r111510 = c;
        double r111511 = r111510 * r111502;
        double r111512 = i;
        double r111513 = r111512 * r111505;
        double r111514 = r111511 - r111513;
        double r111515 = r111509 * r111514;
        double r111516 = r111508 - r111515;
        double r111517 = j;
        double r111518 = r111517 * r111510;
        double r111519 = r111518 * r111504;
        double r111520 = -1.0;
        double r111521 = r111498 * r111517;
        double r111522 = r111512 * r111521;
        double r111523 = r111520 * r111522;
        double r111524 = 1.0;
        double r111525 = pow(r111523, r111524);
        double r111526 = r111519 + r111525;
        double r111527 = r111516 + r111526;
        double r111528 = 3.698782976102966e+118;
        bool r111529 = r111498 <= r111528;
        double r111530 = cbrt(r111517);
        double r111531 = r111530 * r111530;
        double r111532 = r111510 * r111504;
        double r111533 = r111530 * r111532;
        double r111534 = r111531 * r111533;
        double r111535 = r111534 + r111525;
        double r111536 = r111516 + r111535;
        double r111537 = r111517 * r111532;
        double r111538 = r111517 * r111512;
        double r111539 = -r111498;
        double r111540 = r111538 * r111539;
        double r111541 = r111537 + r111540;
        double r111542 = r111516 + r111541;
        double r111543 = r111529 ? r111536 : r111542;
        double r111544 = r111500 ? r111527 : r111543;
        return r111544;
}

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 y < -2.2733319676195215e-175

   1. Initial program 12.8

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

   3. Applied sub-neg 12.8

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

   4. Applied distribute-lft-in 12.8

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

   6. Applied pow1 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \color{blue}{{\left(-i \cdot y\right)}^{1}}\right)\]

   7. Applied pow1 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{{j}^{1}} \cdot {\left(-i \cdot y\right)}^{1}\right)\]

   8. Applied pow-prod-down 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{{\left(j \cdot \left(-i \cdot y\right)\right)}^{1}}\right)\]

   9. Simplified 13.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + {\color{blue}{\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}}^{1}\right)\]
   10. Using strategy rm

   11. Applied associate-*r* 13.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(j \cdot c\right) \cdot t} + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\]

   if -2.2733319676195215e-175 < y < 3.698782976102966e+118

   1. Initial program 10.0

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

   3. Applied sub-neg 10.0

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

   4. Applied distribute-lft-in 10.0

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

   6. Applied pow1 10.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \color{blue}{{\left(-i \cdot y\right)}^{1}}\right)\]

   7. Applied pow1 10.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{{j}^{1}} \cdot {\left(-i \cdot y\right)}^{1}\right)\]

   8. Applied pow-prod-down 10.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{{\left(j \cdot \left(-i \cdot y\right)\right)}^{1}}\right)\]

   9. Simplified 10.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + {\color{blue}{\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}}^{1}\right)\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 10.3

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \sqrt[3]{j}\right)} \cdot \left(c \cdot t\right) + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\]

   12. Applied associate-*l* 10.3

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t\right)\right)} + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\]

   if 3.698782976102966e+118 < y

   1. Initial program 22.1

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

   3. Applied sub-neg 22.1

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

   4. Applied distribute-lft-in 22.1

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

   6. Applied distribute-rgt-neg-in 22.1

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

   7. Applied associate-*r* 15.2

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.2733319676195215 \cdot 10^{-175}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\\ \mathbf{elif}\;y \le 3.6987829761029658 \cdot 10^{118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t\right)\right) + {\left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \left(j \cdot i\right) \cdot \left(-y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))