Average Error: 12.4 → 11.1
Time: 16.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.20987865003621501 \cdot 10^{56}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;x \le -3.8550811051740131 \cdot 10^{-175}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{elif}\;x \le 4.0236427972675005 \cdot 10^{-308}:\\ \;\;\;\;\left(-b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 5.48540352948354912 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]
\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 -3.20987865003621501 \cdot 10^{56}:\\
\;\;\;\;\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 \left(c \cdot t - i \cdot y\right)\right)\\

\mathbf{elif}\;x \le -3.8550811051740131 \cdot 10^{-175}:\\
\;\;\;\;\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(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\

\mathbf{elif}\;x \le 4.0236427972675005 \cdot 10^{-308}:\\
\;\;\;\;\left(-b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

\mathbf{elif}\;x \le 5.48540352948354912 \cdot 10^{-65}:\\
\;\;\;\;\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(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r113449 = x;
        double r113450 = y;
        double r113451 = z;
        double r113452 = r113450 * r113451;
        double r113453 = t;
        double r113454 = a;
        double r113455 = r113453 * r113454;
        double r113456 = r113452 - r113455;
        double r113457 = r113449 * r113456;
        double r113458 = b;
        double r113459 = c;
        double r113460 = r113459 * r113451;
        double r113461 = i;
        double r113462 = r113461 * r113454;
        double r113463 = r113460 - r113462;
        double r113464 = r113458 * r113463;
        double r113465 = r113457 - r113464;
        double r113466 = j;
        double r113467 = r113459 * r113453;
        double r113468 = r113461 * r113450;
        double r113469 = r113467 - r113468;
        double r113470 = r113466 * r113469;
        double r113471 = r113465 + r113470;
        return r113471;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r113472 = x;
        double r113473 = -3.209878650036215e+56;
        bool r113474 = r113472 <= r113473;
        double r113475 = y;
        double r113476 = z;
        double r113477 = r113475 * r113476;
        double r113478 = t;
        double r113479 = a;
        double r113480 = r113478 * r113479;
        double r113481 = r113477 - r113480;
        double r113482 = r113472 * r113481;
        double r113483 = b;
        double r113484 = c;
        double r113485 = r113484 * r113476;
        double r113486 = i;
        double r113487 = r113486 * r113479;
        double r113488 = r113485 - r113487;
        double r113489 = r113483 * r113488;
        double r113490 = r113482 - r113489;
        double r113491 = j;
        double r113492 = cbrt(r113491);
        double r113493 = r113492 * r113492;
        double r113494 = r113484 * r113478;
        double r113495 = r113486 * r113475;
        double r113496 = r113494 - r113495;
        double r113497 = r113492 * r113496;
        double r113498 = r113493 * r113497;
        double r113499 = r113490 + r113498;
        double r113500 = -3.855081105174013e-175;
        bool r113501 = r113472 <= r113500;
        double r113502 = r113477 * r113472;
        double r113503 = r113472 * r113478;
        double r113504 = r113479 * r113503;
        double r113505 = -r113504;
        double r113506 = r113502 + r113505;
        double r113507 = r113506 - r113489;
        double r113508 = r113491 * r113484;
        double r113509 = r113478 * r113508;
        double r113510 = r113486 * r113491;
        double r113511 = r113510 * r113475;
        double r113512 = -r113511;
        double r113513 = r113509 + r113512;
        double r113514 = r113507 + r113513;
        double r113515 = 4.0236427972675005e-308;
        bool r113516 = r113472 <= r113515;
        double r113517 = -r113489;
        double r113518 = r113491 * r113496;
        double r113519 = r113517 + r113518;
        double r113520 = 5.485403529483549e-65;
        bool r113521 = r113472 <= r113520;
        double r113522 = sqrt(r113472);
        double r113523 = r113522 * r113481;
        double r113524 = r113522 * r113523;
        double r113525 = r113524 - r113489;
        double r113526 = r113525 + r113513;
        double r113527 = r113521 ? r113514 : r113526;
        double r113528 = r113516 ? r113519 : r113527;
        double r113529 = r113501 ? r113514 : r113528;
        double r113530 = r113474 ? r113499 : r113529;
        return r113530;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if x < -3.209878650036215e+56

    1. Initial program 7.3

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

    3. Applied add-cube-cbrt7.4

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

    4. Applied associate-*l*7.4

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

    if -3.209878650036215e+56 < x < -3.855081105174013e-175 or 4.0236427972675005e-308 < x < 5.485403529483549e-65

    1. Initial program 14.3

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

    3. Applied sub-neg14.3

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

    4. Applied distribute-lft-in14.3

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

    5. Simplified15.0

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

    6. Simplified14.5

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

    8. Applied associate-*r*14.1

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

    10. Applied sub-neg14.1

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

    11. Applied distribute-lft-in14.1

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

    12. Simplified14.1

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

    13. Simplified11.5

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

    if -3.855081105174013e-175 < x < 4.0236427972675005e-308

    1. Initial program 17.7

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

    2. Taylor expanded around 0 16.4

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    if 5.485403529483549e-65 < x

    1. Initial program 8.9

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

    3. Applied sub-neg8.9

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

    4. Applied distribute-lft-in8.9

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

    5. Simplified9.1

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

    6. Simplified8.9

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

    8. Applied associate-*r*9.3

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

    10. Applied add-sqr-sqrt9.5

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

    11. Applied associate-*l*9.5

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.20987865003621501 \cdot 10^{56}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;x \le -3.8550811051740131 \cdot 10^{-175}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{elif}\;x \le 4.0236427972675005 \cdot 10^{-308}:\\ \;\;\;\;\left(-b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 5.48540352948354912 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(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)))))