Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 11.3

Time: 13.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.71054249675121781 \cdot 10^{34}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{elif}\;x \le 1.419491066943148 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r100511 = x;
        double r100512 = y;
        double r100513 = z;
        double r100514 = r100512 * r100513;
        double r100515 = t;
        double r100516 = a;
        double r100517 = r100515 * r100516;
        double r100518 = r100514 - r100517;
        double r100519 = r100511 * r100518;
        double r100520 = b;
        double r100521 = c;
        double r100522 = r100521 * r100513;
        double r100523 = i;
        double r100524 = r100523 * r100516;
        double r100525 = r100522 - r100524;
        double r100526 = r100520 * r100525;
        double r100527 = r100519 - r100526;
        double r100528 = j;
        double r100529 = r100521 * r100515;
        double r100530 = r100523 * r100512;
        double r100531 = r100529 - r100530;
        double r100532 = r100528 * r100531;
        double r100533 = r100527 + r100532;
        return r100533;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r100534 = x;
        double r100535 = -5.710542496751218e+34;
        bool r100536 = r100534 <= r100535;
        double r100537 = y;
        double r100538 = z;
        double r100539 = r100537 * r100538;
        double r100540 = t;
        double r100541 = a;
        double r100542 = r100540 * r100541;
        double r100543 = r100539 - r100542;
        double r100544 = r100534 * r100543;
        double r100545 = b;
        double r100546 = c;
        double r100547 = r100546 * r100538;
        double r100548 = i;
        double r100549 = r100548 * r100541;
        double r100550 = r100547 - r100549;
        double r100551 = r100545 * r100550;
        double r100552 = r100544 - r100551;
        double r100553 = j;
        double r100554 = r100540 * r100553;
        double r100555 = r100554 * r100546;
        double r100556 = r100548 * r100553;
        double r100557 = r100556 * r100537;
        double r100558 = -r100557;
        double r100559 = r100555 + r100558;
        double r100560 = r100552 + r100559;
        double r100561 = 1.419491066943148e-153;
        bool r100562 = r100534 <= r100561;
        double r100563 = r100539 * r100534;
        double r100564 = r100534 * r100540;
        double r100565 = r100541 * r100564;
        double r100566 = -r100565;
        double r100567 = r100563 + r100566;
        double r100568 = r100567 - r100551;
        double r100569 = r100553 * r100537;
        double r100570 = r100548 * r100569;
        double r100571 = -r100570;
        double r100572 = r100555 + r100571;
        double r100573 = r100568 + r100572;
        double r100574 = r100545 * r100546;
        double r100575 = r100538 * r100574;
        double r100576 = -r100549;
        double r100577 = r100576 * r100545;
        double r100578 = r100575 + r100577;
        double r100579 = r100544 - r100578;
        double r100580 = r100579 + r100572;
        double r100581 = r100562 ? r100573 : r100580;
        double r100582 = r100536 ? r100560 : r100581;
        return r100582;
}

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 < -5.710542496751218e+34

   1. Initial program 6.6

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

   3. Applied sub-neg 6.6

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

   4. Applied distribute-lft-in 6.6

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

   5. Simplified 8.1

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

   6. Simplified 8.0

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

   8. Applied associate-*r* 8.0

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

   10. Applied associate-*r* 8.4

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

   if -5.710542496751218e+34 < x < 1.419491066943148e-153

   1. Initial program 15.7

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

   3. Applied sub-neg 15.7

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

   4. Applied distribute-lft-in 15.7

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

   5. Simplified 15.7

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

   6. Simplified 15.4

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

   8. Applied associate-*r* 15.5

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

   10. Applied sub-neg 15.5

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

   11. Applied distribute-lft-in 15.5

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

   12. Simplified 15.5

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

   13. Simplified 13.0

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

   if 1.419491066943148e-153 < x

   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 10.0

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

   5. Simplified 10.8

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

   6. Simplified 10.5

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

   8. Applied associate-*r* 10.1

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

   10. Applied sub-neg 10.1

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

   11. Applied distribute-lft-in 10.1

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

   12. Simplified 10.1

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

   13. Simplified 10.1

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.71054249675121781 \cdot 10^{34}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{elif}\;x \le 1.419491066943148 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))