Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 12.5

Time: 21.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83473 = x;
        double r83474 = y;
        double r83475 = z;
        double r83476 = r83474 * r83475;
        double r83477 = t;
        double r83478 = a;
        double r83479 = r83477 * r83478;
        double r83480 = r83476 - r83479;
        double r83481 = r83473 * r83480;
        double r83482 = b;
        double r83483 = c;
        double r83484 = r83483 * r83475;
        double r83485 = i;
        double r83486 = r83485 * r83478;
        double r83487 = r83484 - r83486;
        double r83488 = r83482 * r83487;
        double r83489 = r83481 - r83488;
        double r83490 = j;
        double r83491 = r83483 * r83477;
        double r83492 = r83485 * r83474;
        double r83493 = r83491 - r83492;
        double r83494 = r83490 * r83493;
        double r83495 = r83489 + r83494;
        return r83495;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83496 = j;
        double r83497 = c;
        double r83498 = t;
        double r83499 = r83497 * r83498;
        double r83500 = i;
        double r83501 = y;
        double r83502 = r83500 * r83501;
        double r83503 = r83499 - r83502;
        double r83504 = r83496 * r83503;
        double r83505 = x;
        double r83506 = z;
        double r83507 = r83501 * r83506;
        double r83508 = a;
        double r83509 = r83498 * r83508;
        double r83510 = r83507 - r83509;
        double r83511 = r83505 * r83510;
        double r83512 = b;
        double r83513 = r83497 * r83506;
        double r83514 = r83500 * r83508;
        double r83515 = r83513 - r83514;
        double r83516 = r83512 * r83515;
        double r83517 = r83511 - r83516;
        double r83518 = r83504 + r83517;
        return r83518;
}

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 2 regimes

2. if j < -2.889731688948513e+20 or 8.743680770649006e+97 < j

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.4

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

   4. Applied associate-*l* 7.4

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

   if -2.889731688948513e+20 < j < 8.743680770649006e+97

   1. Initial program 14.7

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

   3. Applied add-cube-cbrt 14.8

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

   4. Applied associate-*l* 14.8

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

   6. Applied sub-neg 14.8

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

   7. Applied distribute-lft-in 14.8

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

   8. Applied distribute-lft-in 14.8

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

   9. Simplified 12.7

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

   10. Simplified 12.6

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

   12. Applied associate-*r* 10.6

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

   14. Applied associate-*r* 10.7

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + \left(\left(-j\right) \cdot i\right) \cdot y\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.5

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

Reproduce

herbie shell --seed 2019303 
(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)))))