Average Error: 11.3 → 11.0
Time: 32.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.774121744662978 \cdot 10^{+79}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;z \le -4.1669150400006257 \cdot 10^{-116}:\\ \;\;\;\;\left(\left(j \cdot c\right) \cdot t - \sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \left(\sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \sqrt[3]{i \cdot \left(y \cdot j\right)}\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le -8.984309356821545 \cdot 10^{-152}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le -1.8469584829062982 \cdot 10^{-273}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) - i \cdot \left(y \cdot j\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le 9.976262185724083 \cdot 10^{-303}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le 2.3497846263581813 \cdot 10^{-253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) - i \cdot \left(y \cdot j\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \end{array}\]
\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}\;z \le -2.774121744662978 \cdot 10^{+79}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\

\mathbf{elif}\;z \le -4.1669150400006257 \cdot 10^{-116}:\\
\;\;\;\;\left(\left(j \cdot c\right) \cdot t - \sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \left(\sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \sqrt[3]{i \cdot \left(y \cdot j\right)}\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\

\mathbf{elif}\;z \le -8.984309356821545 \cdot 10^{-152}:\\
\;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\

\mathbf{elif}\;z \le -1.8469584829062982 \cdot 10^{-273}:\\
\;\;\;\;\left(c \cdot \left(t \cdot j\right) - i \cdot \left(y \cdot j\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\

\mathbf{elif}\;z \le 9.976262185724083 \cdot 10^{-303}:\\
\;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\

\mathbf{elif}\;z \le 2.3497846263581813 \cdot 10^{-253}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1692535 = x;
        double r1692536 = y;
        double r1692537 = z;
        double r1692538 = r1692536 * r1692537;
        double r1692539 = t;
        double r1692540 = a;
        double r1692541 = r1692539 * r1692540;
        double r1692542 = r1692538 - r1692541;
        double r1692543 = r1692535 * r1692542;
        double r1692544 = b;
        double r1692545 = c;
        double r1692546 = r1692545 * r1692537;
        double r1692547 = i;
        double r1692548 = r1692547 * r1692540;
        double r1692549 = r1692546 - r1692548;
        double r1692550 = r1692544 * r1692549;
        double r1692551 = r1692543 - r1692550;
        double r1692552 = j;
        double r1692553 = r1692545 * r1692539;
        double r1692554 = r1692547 * r1692536;
        double r1692555 = r1692553 - r1692554;
        double r1692556 = r1692552 * r1692555;
        double r1692557 = r1692551 + r1692556;
        return r1692557;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1692558 = z;
        double r1692559 = -2.774121744662978e+79;
        bool r1692560 = r1692558 <= r1692559;
        double r1692561 = x;
        double r1692562 = y;
        double r1692563 = r1692562 * r1692558;
        double r1692564 = a;
        double r1692565 = t;
        double r1692566 = r1692564 * r1692565;
        double r1692567 = r1692563 - r1692566;
        double r1692568 = r1692561 * r1692567;
        double r1692569 = b;
        double r1692570 = c;
        double r1692571 = r1692569 * r1692570;
        double r1692572 = r1692571 * r1692558;
        double r1692573 = i;
        double r1692574 = r1692569 * r1692573;
        double r1692575 = r1692564 * r1692574;
        double r1692576 = r1692572 - r1692575;
        double r1692577 = r1692568 - r1692576;
        double r1692578 = r1692570 * r1692565;
        double r1692579 = r1692562 * r1692573;
        double r1692580 = r1692578 - r1692579;
        double r1692581 = j;
        double r1692582 = cbrt(r1692581);
        double r1692583 = r1692580 * r1692582;
        double r1692584 = r1692582 * r1692582;
        double r1692585 = r1692583 * r1692584;
        double r1692586 = r1692577 + r1692585;
        double r1692587 = -4.1669150400006257e-116;
        bool r1692588 = r1692558 <= r1692587;
        double r1692589 = r1692581 * r1692570;
        double r1692590 = r1692589 * r1692565;
        double r1692591 = r1692562 * r1692581;
        double r1692592 = r1692573 * r1692591;
        double r1692593 = cbrt(r1692592);
        double r1692594 = r1692593 * r1692593;
        double r1692595 = r1692593 * r1692594;
        double r1692596 = r1692590 - r1692595;
        double r1692597 = r1692570 * r1692558;
        double r1692598 = r1692573 * r1692564;
        double r1692599 = r1692597 - r1692598;
        double r1692600 = r1692599 * r1692569;
        double r1692601 = r1692568 - r1692600;
        double r1692602 = r1692596 + r1692601;
        double r1692603 = -8.984309356821545e-152;
        bool r1692604 = r1692558 <= r1692603;
        double r1692605 = r1692580 * r1692581;
        double r1692606 = r1692561 * r1692563;
        double r1692607 = r1692561 * r1692565;
        double r1692608 = r1692607 * r1692564;
        double r1692609 = r1692606 - r1692608;
        double r1692610 = r1692609 - r1692600;
        double r1692611 = r1692605 + r1692610;
        double r1692612 = -1.8469584829062982e-273;
        bool r1692613 = r1692558 <= r1692612;
        double r1692614 = r1692565 * r1692581;
        double r1692615 = r1692570 * r1692614;
        double r1692616 = r1692615 - r1692592;
        double r1692617 = r1692616 + r1692601;
        double r1692618 = 9.976262185724083e-303;
        bool r1692619 = r1692558 <= r1692618;
        double r1692620 = 2.3497846263581813e-253;
        bool r1692621 = r1692558 <= r1692620;
        double r1692622 = r1692621 ? r1692586 : r1692617;
        double r1692623 = r1692619 ? r1692611 : r1692622;
        double r1692624 = r1692613 ? r1692617 : r1692623;
        double r1692625 = r1692604 ? r1692611 : r1692624;
        double r1692626 = r1692588 ? r1692602 : r1692625;
        double r1692627 = r1692560 ? r1692586 : r1692626;
        return r1692627;
}

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 z < -2.774121744662978e+79 or 9.976262185724083e-303 < z < 2.3497846263581813e-253

    1. Initial program 15.5

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

    3. Applied add-cube-cbrt15.8

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

    4. Applied associate-*l*15.8

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

    5. Taylor expanded around -inf 12.5

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

    if -2.774121744662978e+79 < z < -4.1669150400006257e-116

    1. Initial program 9.5

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

    3. Applied add-cube-cbrt9.8

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

    4. Applied associate-*l*9.8

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

    5. Taylor expanded around -inf 9.9

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

    7. Applied add-cube-cbrt10.1

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

    if -4.1669150400006257e-116 < z < -8.984309356821545e-152 or -1.8469584829062982e-273 < z < 9.976262185724083e-303

    1. Initial program 8.3

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

    2. Taylor expanded around -inf 8.7

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

    if -8.984309356821545e-152 < z < -1.8469584829062982e-273 or 2.3497846263581813e-253 < z

    1. Initial program 11.0

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

    3. Applied add-cube-cbrt11.3

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

    4. Applied associate-*l*11.3

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

    5. Taylor expanded around -inf 11.7

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

    7. Applied associate-*r*11.1

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.774121744662978 \cdot 10^{+79}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;z \le -4.1669150400006257 \cdot 10^{-116}:\\ \;\;\;\;\left(\left(j \cdot c\right) \cdot t - \sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \left(\sqrt[3]{i \cdot \left(y \cdot j\right)} \cdot \sqrt[3]{i \cdot \left(y \cdot j\right)}\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le -8.984309356821545 \cdot 10^{-152}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le -1.8469584829062982 \cdot 10^{-273}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) - i \cdot \left(y \cdot j\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le 9.976262185724083 \cdot 10^{-303}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;z \le 2.3497846263581813 \cdot 10^{-253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\left(b \cdot c\right) \cdot z - a \cdot \left(b \cdot i\right)\right)\right) + \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) - i \cdot \left(y \cdot j\right)\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(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)))))