Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 9.7

Time: 23.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -2.961055619344420626519795079213745347097 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(\left(-a\right) \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 64272744.59900391101837158203125:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot a\right) \cdot x\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-t \cdot a\right) \cdot x\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83484 = x;
        double r83485 = y;
        double r83486 = z;
        double r83487 = r83485 * r83486;
        double r83488 = t;
        double r83489 = a;
        double r83490 = r83488 * r83489;
        double r83491 = r83487 - r83490;
        double r83492 = r83484 * r83491;
        double r83493 = b;
        double r83494 = c;
        double r83495 = r83494 * r83486;
        double r83496 = i;
        double r83497 = r83496 * r83489;
        double r83498 = r83495 - r83497;
        double r83499 = r83493 * r83498;
        double r83500 = r83492 - r83499;
        double r83501 = j;
        double r83502 = r83494 * r83488;
        double r83503 = r83496 * r83485;
        double r83504 = r83502 - r83503;
        double r83505 = r83501 * r83504;
        double r83506 = r83500 + r83505;
        return r83506;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83507 = j;
        double r83508 = -2.9610556193444206e-21;
        bool r83509 = r83507 <= r83508;
        double r83510 = y;
        double r83511 = z;
        double r83512 = x;
        double r83513 = r83511 * r83512;
        double r83514 = r83510 * r83513;
        double r83515 = t;
        double r83516 = a;
        double r83517 = -r83516;
        double r83518 = r83517 * r83512;
        double r83519 = r83515 * r83518;
        double r83520 = r83514 + r83519;
        double r83521 = b;
        double r83522 = c;
        double r83523 = r83522 * r83511;
        double r83524 = i;
        double r83525 = r83524 * r83516;
        double r83526 = r83523 - r83525;
        double r83527 = r83521 * r83526;
        double r83528 = r83520 - r83527;
        double r83529 = r83522 * r83515;
        double r83530 = r83524 * r83510;
        double r83531 = r83529 - r83530;
        double r83532 = r83507 * r83531;
        double r83533 = r83528 + r83532;
        double r83534 = 64272744.59900391;
        bool r83535 = r83507 <= r83534;
        double r83536 = r83510 * r83511;
        double r83537 = r83536 * r83512;
        double r83538 = r83515 * r83516;
        double r83539 = -r83538;
        double r83540 = r83539 * r83512;
        double r83541 = r83537 + r83540;
        double r83542 = r83541 - r83527;
        double r83543 = r83507 * r83522;
        double r83544 = r83515 * r83543;
        double r83545 = r83507 * r83510;
        double r83546 = r83524 * r83545;
        double r83547 = -r83546;
        double r83548 = r83544 + r83547;
        double r83549 = r83542 + r83548;
        double r83550 = r83514 + r83540;
        double r83551 = r83521 * r83522;
        double r83552 = r83511 * r83551;
        double r83553 = -r83525;
        double r83554 = r83553 * r83521;
        double r83555 = r83552 + r83554;
        double r83556 = r83550 - r83555;
        double r83557 = r83556 + r83532;
        double r83558 = r83535 ? r83549 : r83557;
        double r83559 = r83509 ? r83533 : r83558;
        return r83559;
}

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 < -2.9610556193444206e-21

   1. Initial program 7.3

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

   3. Applied sub-neg 7.3

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

   4. Applied distribute-lft-in 7.3

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

   5. Simplified 7.3

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

   6. Simplified 7.3

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

   8. Applied associate-*l* 7.9

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

   10. Applied pow1 7.9

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

   11. Applied pow1 7.9

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

   12. Applied pow-prod-down 7.9

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

   13. Simplified 8.8

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

   if -2.9610556193444206e-21 < j < 64272744.59900391

   1. Initial program 15.8

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

   3. Applied sub-neg 15.8

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

   4. Applied distribute-lft-in 15.8

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

   5. Simplified 15.8

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

   6. Simplified 15.8

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

   8. Applied sub-neg 15.8

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

   9. Applied distribute-lft-in 15.8

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

   10. Simplified 13.0

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

   11. Simplified 10.3

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

   if 64272744.59900391 < j

   1. Initial program 7.7

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

   3. Applied sub-neg 7.7

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

   4. Applied distribute-lft-in 7.7

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

   5. Simplified 7.7

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

   6. Simplified 7.7

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

   8. Applied associate-*l* 8.2

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

   10. Applied sub-neg 8.2

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

   11. Applied distribute-lft-in 8.2

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

   12. Simplified 8.7

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

   13. Simplified 8.7

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -2.961055619344420626519795079213745347097 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + t \cdot \left(\left(-a\right) \cdot x\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 64272744.59900391101837158203125:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot a\right) \cdot x\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-t \cdot a\right) \cdot x\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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)))))