Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 10.3

Time: 18.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -1.218516813516571829132864023880113981554 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(x \cdot t\right) \cdot \left(-a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 64272744.59900391101837158203125:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r408438 = x;
        double r408439 = y;
        double r408440 = z;
        double r408441 = r408439 * r408440;
        double r408442 = t;
        double r408443 = a;
        double r408444 = r408442 * r408443;
        double r408445 = r408441 - r408444;
        double r408446 = r408438 * r408445;
        double r408447 = b;
        double r408448 = c;
        double r408449 = r408448 * r408440;
        double r408450 = i;
        double r408451 = r408450 * r408443;
        double r408452 = r408449 - r408451;
        double r408453 = r408447 * r408452;
        double r408454 = r408446 - r408453;
        double r408455 = j;
        double r408456 = r408448 * r408442;
        double r408457 = r408450 * r408439;
        double r408458 = r408456 - r408457;
        double r408459 = r408455 * r408458;
        double r408460 = r408454 + r408459;
        return r408460;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r408461 = j;
        double r408462 = -1.2185168135165718e-215;
        bool r408463 = r408461 <= r408462;
        double r408464 = x;
        double r408465 = y;
        double r408466 = r408464 * r408465;
        double r408467 = z;
        double r408468 = r408466 * r408467;
        double r408469 = t;
        double r408470 = r408464 * r408469;
        double r408471 = a;
        double r408472 = -r408471;
        double r408473 = r408470 * r408472;
        double r408474 = r408468 + r408473;
        double r408475 = b;
        double r408476 = c;
        double r408477 = r408476 * r408467;
        double r408478 = i;
        double r408479 = r408478 * r408471;
        double r408480 = r408477 - r408479;
        double r408481 = r408475 * r408480;
        double r408482 = r408474 - r408481;
        double r408483 = r408476 * r408469;
        double r408484 = r408478 * r408465;
        double r408485 = r408483 - r408484;
        double r408486 = r408461 * r408485;
        double r408487 = r408482 + r408486;
        double r408488 = 64272744.59900391;
        bool r408489 = r408461 <= r408488;
        double r408490 = r408465 * r408467;
        double r408491 = r408469 * r408471;
        double r408492 = r408490 - r408491;
        double r408493 = r408464 * r408492;
        double r408494 = r408493 - r408481;
        double r408495 = r408461 * r408476;
        double r408496 = r408469 * r408495;
        double r408497 = r408461 * r408465;
        double r408498 = r408478 * r408497;
        double r408499 = -r408498;
        double r408500 = r408496 + r408499;
        double r408501 = r408494 + r408500;
        double r408502 = -r408491;
        double r408503 = r408464 * r408502;
        double r408504 = r408468 + r408503;
        double r408505 = r408475 * r408476;
        double r408506 = r408467 * r408505;
        double r408507 = -r408479;
        double r408508 = r408475 * r408507;
        double r408509 = r408506 + r408508;
        double r408510 = r408504 - r408509;
        double r408511 = r408510 + r408486;
        double r408512 = r408489 ? r408501 : r408511;
        double r408513 = r408463 ? r408487 : r408512;
        return r408513;
}

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	16.1
Herbie	10.3

\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if j < -1.2185168135165718e-215

   1. Initial program 10.9

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

   3. Applied sub-neg 10.9

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

   4. Applied distribute-lft-in 10.9

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

   6. Applied associate-*r* 11.0

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

   8. Applied distribute-rgt-neg-in 11.0

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

   9. Applied associate-*r* 11.4

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

   if -1.2185168135165718e-215 < j < 64272744.59900391

   1. Initial program 16.2

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

   3. Applied sub-neg 16.2

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

   4. Applied distribute-lft-in 16.2

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

   5. Simplified 13.2

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

   6. Simplified 10.3

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

   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. Using strategy rm

   6. Applied associate-*r* 7.9

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

   8. Applied sub-neg 7.9

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

   9. Applied distribute-lft-in 7.9

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

   10. Simplified 8.0

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.218516813516571829132864023880113981554 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(x \cdot t\right) \cdot \left(-a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \le 64272744.59900391101837158203125:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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 -8.1209789191959122e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.7125538182184851e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.63353334603158369e-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) (pow (* i y) 2))) (+ (* 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)))))