Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 10.0

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.31316871954506083868573310292063806129 \cdot 10^{-109}:\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;x \le 3.217371995632368362710845571958895917258 \cdot 10^{-105}:\\ \;\;\;\;\left(i \cdot \left(a \cdot b\right) - c \cdot \left(b \cdot z\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(x \cdot z\right) \cdot y - a \cdot \left(t \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \le 1.68252005080698548650945614352565896303 \cdot 10^{50}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(j \cdot c\right) \cdot t\right)\right) + \left(i \cdot \left(a \cdot b\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(a \cdot b\right) - \left(\left(\sqrt[3]{z} \cdot b\right) \cdot c\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r102368 = x;
        double r102369 = y;
        double r102370 = z;
        double r102371 = r102369 * r102370;
        double r102372 = t;
        double r102373 = a;
        double r102374 = r102372 * r102373;
        double r102375 = r102371 - r102374;
        double r102376 = r102368 * r102375;
        double r102377 = b;
        double r102378 = c;
        double r102379 = r102378 * r102370;
        double r102380 = i;
        double r102381 = r102380 * r102373;
        double r102382 = r102379 - r102381;
        double r102383 = r102377 * r102382;
        double r102384 = r102376 - r102383;
        double r102385 = j;
        double r102386 = r102378 * r102372;
        double r102387 = r102380 * r102369;
        double r102388 = r102386 - r102387;
        double r102389 = r102385 * r102388;
        double r102390 = r102384 + r102389;
        return r102390;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r102391 = x;
        double r102392 = -4.313168719545061e-109;
        bool r102393 = r102391 <= r102392;
        double r102394 = j;
        double r102395 = c;
        double r102396 = t;
        double r102397 = r102395 * r102396;
        double r102398 = i;
        double r102399 = y;
        double r102400 = r102398 * r102399;
        double r102401 = r102397 - r102400;
        double r102402 = r102394 * r102401;
        double r102403 = z;
        double r102404 = r102399 * r102403;
        double r102405 = a;
        double r102406 = r102396 * r102405;
        double r102407 = r102404 - r102406;
        double r102408 = r102391 * r102407;
        double r102409 = r102402 + r102408;
        double r102410 = b;
        double r102411 = r102410 * r102398;
        double r102412 = r102405 * r102411;
        double r102413 = r102410 * r102395;
        double r102414 = r102403 * r102413;
        double r102415 = r102412 - r102414;
        double r102416 = r102409 + r102415;
        double r102417 = 3.2173719956323684e-105;
        bool r102418 = r102391 <= r102417;
        double r102419 = r102405 * r102410;
        double r102420 = r102398 * r102419;
        double r102421 = r102410 * r102403;
        double r102422 = r102395 * r102421;
        double r102423 = r102420 - r102422;
        double r102424 = r102391 * r102403;
        double r102425 = r102424 * r102399;
        double r102426 = r102396 * r102391;
        double r102427 = r102405 * r102426;
        double r102428 = r102425 - r102427;
        double r102429 = r102402 + r102428;
        double r102430 = r102423 + r102429;
        double r102431 = 1.6825200508069855e+50;
        bool r102432 = r102391 <= r102431;
        double r102433 = r102399 * r102394;
        double r102434 = -r102398;
        double r102435 = r102433 * r102434;
        double r102436 = r102394 * r102395;
        double r102437 = r102436 * r102396;
        double r102438 = r102435 + r102437;
        double r102439 = r102408 + r102438;
        double r102440 = r102420 - r102414;
        double r102441 = r102439 + r102440;
        double r102442 = cbrt(r102403);
        double r102443 = r102442 * r102410;
        double r102444 = r102443 * r102395;
        double r102445 = r102442 * r102442;
        double r102446 = r102444 * r102445;
        double r102447 = r102420 - r102446;
        double r102448 = r102447 + r102409;
        double r102449 = r102432 ? r102441 : r102448;
        double r102450 = r102418 ? r102430 : r102449;
        double r102451 = r102393 ? r102416 : r102450;
        return r102451;
}

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 4 regimes

2. if x < -4.313168719545061e-109

   1. Initial program 9.9

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

   2. Simplified 9.9

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

   3. Taylor expanded around inf 10.3

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

   4. Simplified 10.0

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

   6. Applied associate-*r* 9.9

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

   7. Simplified 9.9

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

   9. Applied associate-*l* 10.3

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

   10. Simplified 10.3

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

   12. Applied associate-*l* 10.3

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

   if -4.313168719545061e-109 < x < 3.2173719956323684e-105

   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. Simplified 16.2

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

   3. Taylor expanded around inf 16.3

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

   4. Simplified 16.6

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

   6. Applied associate-*r* 16.7

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

   7. Simplified 16.7

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

   8. Taylor expanded around inf 13.9

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

   9. Simplified 10.9

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

   if 3.2173719956323684e-105 < x < 1.6825200508069855e+50

   1. Initial program 11.5

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

   2. Simplified 11.5

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

   3. Taylor expanded around inf 9.9

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

   4. Simplified 11.6

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

   6. Applied associate-*r* 11.2

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

   7. Simplified 11.2

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

   9. Applied associate-*l* 10.1

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

   10. Simplified 10.1

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

   12. Applied sub-neg 10.1

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

   13. Applied distribute-lft-in 10.1

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

   14. Simplified 10.4

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

   15. Simplified 9.7

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

   if 1.6825200508069855e+50 < x

   1. Initial program 7.0

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

   2. Simplified 7.0

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

   3. Taylor expanded around inf 8.1

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

   4. Simplified 6.6

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

   6. Applied associate-*r* 7.1

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

   7. Simplified 7.1

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

   9. Applied associate-*l* 8.2

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

   10. Simplified 8.2

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

   12. Applied add-cube-cbrt 8.3

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

   13. Applied associate-*l* 8.3

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

   14. Simplified 7.1

       \[\leadsto \left(\left(a \cdot b\right) \cdot i - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \color{blue}{\left(c \cdot \left(b \cdot \sqrt[3]{z}\right)\right)}\right) + \left(\left(z \cdot y - t \cdot a\right) \cdot x + j \cdot \left(t \cdot c - i \cdot y\right)\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.31316871954506083868573310292063806129 \cdot 10^{-109}:\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right) + \left(a \cdot \left(b \cdot i\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;x \le 3.217371995632368362710845571958895917258 \cdot 10^{-105}:\\ \;\;\;\;\left(i \cdot \left(a \cdot b\right) - c \cdot \left(b \cdot z\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(x \cdot z\right) \cdot y - a \cdot \left(t \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \le 1.68252005080698548650945614352565896303 \cdot 10^{50}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(j \cdot c\right) \cdot t\right)\right) + \left(i \cdot \left(a \cdot b\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(a \cdot b\right) - \left(\left(\sqrt[3]{z} \cdot b\right) \cdot c\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))