Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 11.7

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -5.399805233907068252297795713753297330209 \cdot 10^{-132}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + 1 \cdot \left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)\right)\\ \mathbf{elif}\;i \le -6.066601058953688871087898362073558578406 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;i \le 5.35820216338663462891582209778136737461 \cdot 10^{-130}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;i \le 9.764633783766700810211076280405055225809 \cdot 10^{238}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \left(j \cdot i\right) \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r694448 = x;
        double r694449 = y;
        double r694450 = z;
        double r694451 = r694449 * r694450;
        double r694452 = t;
        double r694453 = a;
        double r694454 = r694452 * r694453;
        double r694455 = r694451 - r694454;
        double r694456 = r694448 * r694455;
        double r694457 = b;
        double r694458 = c;
        double r694459 = r694458 * r694450;
        double r694460 = i;
        double r694461 = r694460 * r694453;
        double r694462 = r694459 - r694461;
        double r694463 = r694457 * r694462;
        double r694464 = r694456 - r694463;
        double r694465 = j;
        double r694466 = r694458 * r694452;
        double r694467 = r694460 * r694449;
        double r694468 = r694466 - r694467;
        double r694469 = r694465 * r694468;
        double r694470 = r694464 + r694469;
        return r694470;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r694471 = i;
        double r694472 = -5.399805233907068e-132;
        bool r694473 = r694471 <= r694472;
        double r694474 = x;
        double r694475 = y;
        double r694476 = z;
        double r694477 = r694475 * r694476;
        double r694478 = t;
        double r694479 = a;
        double r694480 = r694478 * r694479;
        double r694481 = r694477 - r694480;
        double r694482 = r694474 * r694481;
        double r694483 = b;
        double r694484 = c;
        double r694485 = r694484 * r694476;
        double r694486 = r694471 * r694479;
        double r694487 = r694485 - r694486;
        double r694488 = r694483 * r694487;
        double r694489 = r694482 - r694488;
        double r694490 = j;
        double r694491 = r694484 * r694478;
        double r694492 = r694490 * r694491;
        double r694493 = 1.0;
        double r694494 = -1.0;
        double r694495 = r694475 * r694490;
        double r694496 = r694471 * r694495;
        double r694497 = r694494 * r694496;
        double r694498 = r694493 * r694497;
        double r694499 = r694492 + r694498;
        double r694500 = r694489 + r694499;
        double r694501 = -6.066601058953689e-271;
        bool r694502 = r694471 <= r694501;
        double r694503 = r694474 * r694477;
        double r694504 = r694474 * r694478;
        double r694505 = r694479 * r694504;
        double r694506 = r694494 * r694505;
        double r694507 = r694493 * r694506;
        double r694508 = r694503 + r694507;
        double r694509 = r694508 - r694488;
        double r694510 = r694471 * r694475;
        double r694511 = -r694510;
        double r694512 = r694490 * r694511;
        double r694513 = r694492 + r694512;
        double r694514 = r694509 + r694513;
        double r694515 = 5.358202163386635e-130;
        bool r694516 = r694471 <= r694515;
        double r694517 = r694474 * r694475;
        double r694518 = r694517 * r694476;
        double r694519 = -r694480;
        double r694520 = r694474 * r694519;
        double r694521 = r694518 + r694520;
        double r694522 = r694521 - r694488;
        double r694523 = r694522 + r694513;
        double r694524 = 9.764633783766701e+238;
        bool r694525 = r694471 <= r694524;
        double r694526 = r694490 * r694471;
        double r694527 = -r694475;
        double r694528 = r694526 * r694527;
        double r694529 = r694492 + r694528;
        double r694530 = r694489 + r694529;
        double r694531 = r694503 + r694520;
        double r694532 = r694531 - r694488;
        double r694533 = r694490 * r694484;
        double r694534 = r694533 * r694478;
        double r694535 = r694534 + r694512;
        double r694536 = r694532 + r694535;
        double r694537 = r694525 ? r694530 : r694536;
        double r694538 = r694516 ? r694523 : r694537;
        double r694539 = r694502 ? r694514 : r694538;
        double r694540 = r694473 ? r694500 : r694539;
        return r694540;
}

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	12.3
Target	16.2
Herbie	11.7

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

Derivation

1. Split input into 5 regimes

2. if i < -5.399805233907068e-132

   1. Initial program 13.5

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

   3. Applied sub-neg 13.5

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

   4. Applied distribute-lft-in 13.5

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

   6. Applied *-un-lft-identity 13.5

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

   7. Applied associate-*l* 13.5

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

   8. Simplified 11.8

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

   if -5.399805233907068e-132 < i < -6.066601058953689e-271

   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 9.9

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

   6. Applied sub-neg 9.9

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

   7. Applied distribute-lft-in 9.9

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

   9. Applied *-un-lft-identity 9.9

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

   10. Applied associate-*l* 9.9

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

   11. Simplified 9.5

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

   if -6.066601058953689e-271 < i < 5.358202163386635e-130

   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 sub-neg 9.7

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

   4. Applied distribute-lft-in 9.7

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

   6. Applied sub-neg 9.7

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

   7. Applied distribute-lft-in 9.7

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

   9. Applied associate-*r* 10.4

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

   if 5.358202163386635e-130 < i < 9.764633783766701e+238

   1. Initial program 12.1

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

   3. Applied sub-neg 12.1

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

   4. Applied distribute-lft-in 12.1

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

   6. Applied distribute-rgt-neg-in 12.1

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

   7. Applied associate-*r* 11.9

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

   if 9.764633783766701e+238 < i

   1. Initial program 30.2

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

   3. Applied sub-neg 30.2

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

   4. Applied distribute-lft-in 30.2

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

   6. Applied sub-neg 30.2

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

   7. Applied distribute-lft-in 30.2

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

   9. Applied associate-*r* 28.9

      \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(j \cdot c\right) \cdot t} + j \cdot \left(-i \cdot y\right)\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -5.399805233907068252297795713753297330209 \cdot 10^{-132}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + 1 \cdot \left(-1 \cdot \left(i \cdot \left(y \cdot j\right)\right)\right)\right)\\ \mathbf{elif}\;i \le -6.066601058953688871087898362073558578406 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;i \le 5.35820216338663462891582209778136737461 \cdot 10^{-130}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;i \le 9.764633783766700810211076280405055225809 \cdot 10^{238}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \left(j \cdot i\right) \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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 -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-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 -7.633533346031584e-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)))))