Average Error: 11.8 → 10.0
Time: 25.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.913925828335769 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;z \le 1.9450455774818727 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, c, -y \cdot i\right), j, \mathsf{fma}\left(\mathsf{fma}\left(i, a, z \cdot \left(-c\right)\right), b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\right)\right)\\ \end{array}\]
\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}\;z \le -1.913925828335769 \cdot 10^{+65}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\right)\right)\\

\mathbf{elif}\;z \le 1.9450455774818727 \cdot 10^{+77}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, c, -y \cdot i\right), j, \mathsf{fma}\left(\mathsf{fma}\left(i, a, z \cdot \left(-c\right)\right), b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\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 r1970467 = x;
        double r1970468 = y;
        double r1970469 = z;
        double r1970470 = r1970468 * r1970469;
        double r1970471 = t;
        double r1970472 = a;
        double r1970473 = r1970471 * r1970472;
        double r1970474 = r1970470 - r1970473;
        double r1970475 = r1970467 * r1970474;
        double r1970476 = b;
        double r1970477 = c;
        double r1970478 = r1970477 * r1970469;
        double r1970479 = i;
        double r1970480 = r1970479 * r1970472;
        double r1970481 = r1970478 - r1970480;
        double r1970482 = r1970476 * r1970481;
        double r1970483 = r1970475 - r1970482;
        double r1970484 = j;
        double r1970485 = r1970477 * r1970471;
        double r1970486 = r1970479 * r1970468;
        double r1970487 = r1970485 - r1970486;
        double r1970488 = r1970484 * r1970487;
        double r1970489 = r1970483 + r1970488;
        return r1970489;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1970490 = z;
        double r1970491 = -1.913925828335769e+65;
        bool r1970492 = r1970490 <= r1970491;
        double r1970493 = t;
        double r1970494 = c;
        double r1970495 = r1970493 * r1970494;
        double r1970496 = y;
        double r1970497 = i;
        double r1970498 = r1970496 * r1970497;
        double r1970499 = r1970495 - r1970498;
        double r1970500 = j;
        double r1970501 = x;
        double r1970502 = r1970501 * r1970496;
        double r1970503 = r1970502 * r1970490;
        double r1970504 = r1970493 * r1970501;
        double r1970505 = a;
        double r1970506 = b;
        double r1970507 = r1970506 * r1970494;
        double r1970508 = r1970490 * r1970507;
        double r1970509 = fma(r1970504, r1970505, r1970508);
        double r1970510 = r1970503 - r1970509;
        double r1970511 = fma(r1970499, r1970500, r1970510);
        double r1970512 = 1.9450455774818727e+77;
        bool r1970513 = r1970490 <= r1970512;
        double r1970514 = -r1970498;
        double r1970515 = fma(r1970493, r1970494, r1970514);
        double r1970516 = -r1970494;
        double r1970517 = r1970490 * r1970516;
        double r1970518 = fma(r1970497, r1970505, r1970517);
        double r1970519 = r1970496 * r1970490;
        double r1970520 = r1970505 * r1970493;
        double r1970521 = r1970519 - r1970520;
        double r1970522 = cbrt(r1970521);
        double r1970523 = r1970501 * r1970522;
        double r1970524 = r1970522 * r1970522;
        double r1970525 = r1970523 * r1970524;
        double r1970526 = fma(r1970518, r1970506, r1970525);
        double r1970527 = fma(r1970515, r1970500, r1970526);
        double r1970528 = r1970513 ? r1970527 : r1970511;
        double r1970529 = r1970492 ? r1970511 : r1970528;
        return r1970529;
}

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 z < -1.913925828335769e+65 or 1.9450455774818727e+77 < z

    1. Initial program 19.0

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

    2. Simplified19.1

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

    3. Taylor expanded around inf 17.5

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

    4. Simplified11.7

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

    if -1.913925828335769e+65 < z < 1.9450455774818727e+77

    1. Initial program 9.1

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

    2. Simplified9.1

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

    4. Applied fma-neg9.1

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

    6. Applied add-cube-cbrt9.3

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

    7. Applied associate-*l*9.3

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.913925828335769 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\right)\right)\\ \mathbf{elif}\;z \le 1.9450455774818727 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, c, -y \cdot i\right), j, \mathsf{fma}\left(\mathsf{fma}\left(i, a, z \cdot \left(-c\right)\right), b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y\right) \cdot z - \mathsf{fma}\left(t \cdot x, a, z \cdot \left(b \cdot c\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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)))))