Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.4 → 12.7

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.8016511393041913 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;x \le 1.1177337282084136 \cdot 10^{-156}:\\ \;\;\;\;\left(0 - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r466 = x;
        double r467 = y;
        double r468 = z;
        double r469 = r467 * r468;
        double r470 = t;
        double r471 = a;
        double r472 = r470 * r471;
        double r473 = r469 - r472;
        double r474 = r466 * r473;
        double r475 = b;
        double r476 = c;
        double r477 = r476 * r468;
        double r478 = i;
        double r479 = r478 * r471;
        double r480 = r477 - r479;
        double r481 = r475 * r480;
        double r482 = r474 - r481;
        double r483 = j;
        double r484 = r476 * r470;
        double r485 = r478 * r467;
        double r486 = r484 - r485;
        double r487 = r483 * r486;
        double r488 = r482 + r487;
        return r488;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r489 = x;
        double r490 = -6.801651139304191e-129;
        bool r491 = r489 <= r490;
        double r492 = c;
        double r493 = t;
        double r494 = r492 * r493;
        double r495 = i;
        double r496 = y;
        double r497 = r495 * r496;
        double r498 = r494 - r497;
        double r499 = j;
        double r500 = z;
        double r501 = r496 * r500;
        double r502 = a;
        double r503 = r493 * r502;
        double r504 = r501 - r503;
        double r505 = r489 * r504;
        double r506 = b;
        double r507 = r492 * r500;
        double r508 = r495 * r502;
        double r509 = r507 - r508;
        double r510 = r506 * r509;
        double r511 = r505 - r510;
        double r512 = fma(r498, r499, r511);
        double r513 = 1.1177337282084136e-156;
        bool r514 = r489 <= r513;
        double r515 = 0.0;
        double r516 = -r502;
        double r517 = r502 * r495;
        double r518 = fma(r516, r495, r517);
        double r519 = r506 * r518;
        double r520 = r510 + r519;
        double r521 = r515 - r520;
        double r522 = r496 * r495;
        double r523 = -r522;
        double r524 = fma(r492, r493, r523);
        double r525 = r499 * r524;
        double r526 = -r496;
        double r527 = fma(r526, r495, r522);
        double r528 = r499 * r527;
        double r529 = r525 + r528;
        double r530 = r521 + r529;
        double r531 = r502 * r493;
        double r532 = -r531;
        double r533 = fma(r496, r500, r532);
        double r534 = r489 * r533;
        double r535 = fma(r516, r493, r531);
        double r536 = r489 * r535;
        double r537 = r534 + r536;
        double r538 = r537 - r520;
        double r539 = r538 + r529;
        double r540 = r514 ? r530 : r539;
        double r541 = r491 ? r512 : r540;
        return r541;
}

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 x < -6.801651139304191e-129

   1. Initial program 9.6

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

   2. Simplified 9.6

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

   if -6.801651139304191e-129 < x < 1.1177337282084136e-156

   1. Initial program 17.8

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

   3. Applied prod-diff 17.8

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

   4. Applied distribute-lft-in 17.8

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

   5. Simplified 17.8

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

   7. Applied prod-diff 17.8

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

   8. Applied distribute-lft-in 17.8

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

   9. Taylor expanded around 0 18.6

      \[\leadsto \left(\color{blue}{0} - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]

   if 1.1177337282084136e-156 < x

   1. Initial program 9.7

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

   3. Applied prod-diff 9.7

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

   4. Applied distribute-lft-in 9.7

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

   5. Simplified 9.7

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

   7. Applied prod-diff 9.7

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

   8. Applied distribute-lft-in 9.7

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

   10. Applied prod-diff 9.7

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

   11. Applied distribute-lft-in 9.7

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.8016511393041913 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;x \le 1.1177337282084136 \cdot 10^{-156}:\\ \;\;\;\;\left(0 - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))))