Average Error: 12.5 → 12.5
Time: 10.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.4404428017653185 \cdot 10^{-231}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{elif}\;b \le 8.0665543386957723 \cdot 10^{-207}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]
\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}\;b \le -3.4404428017653185 \cdot 10^{-231}:\\
\;\;\;\;\left(\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) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\

\mathbf{elif}\;b \le 8.0665543386957723 \cdot 10^{-207}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r98531 = x;
        double r98532 = y;
        double r98533 = z;
        double r98534 = r98532 * r98533;
        double r98535 = t;
        double r98536 = a;
        double r98537 = r98535 * r98536;
        double r98538 = r98534 - r98537;
        double r98539 = r98531 * r98538;
        double r98540 = b;
        double r98541 = c;
        double r98542 = r98541 * r98533;
        double r98543 = i;
        double r98544 = r98543 * r98536;
        double r98545 = r98542 - r98544;
        double r98546 = r98540 * r98545;
        double r98547 = r98539 - r98546;
        double r98548 = j;
        double r98549 = r98541 * r98535;
        double r98550 = r98543 * r98532;
        double r98551 = r98549 - r98550;
        double r98552 = r98548 * r98551;
        double r98553 = r98547 + r98552;
        return r98553;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r98554 = b;
        double r98555 = -3.4404428017653185e-231;
        bool r98556 = r98554 <= r98555;
        double r98557 = x;
        double r98558 = cbrt(r98557);
        double r98559 = r98558 * r98558;
        double r98560 = y;
        double r98561 = z;
        double r98562 = r98560 * r98561;
        double r98563 = t;
        double r98564 = a;
        double r98565 = r98563 * r98564;
        double r98566 = r98562 - r98565;
        double r98567 = r98558 * r98566;
        double r98568 = r98559 * r98567;
        double r98569 = c;
        double r98570 = r98569 * r98561;
        double r98571 = i;
        double r98572 = r98571 * r98564;
        double r98573 = r98570 - r98572;
        double r98574 = r98554 * r98573;
        double r98575 = r98568 - r98574;
        double r98576 = j;
        double r98577 = r98560 * r98571;
        double r98578 = -r98577;
        double r98579 = fma(r98569, r98563, r98578);
        double r98580 = r98576 * r98579;
        double r98581 = -r98560;
        double r98582 = fma(r98581, r98571, r98577);
        double r98583 = r98576 * r98582;
        double r98584 = r98580 + r98583;
        double r98585 = r98575 + r98584;
        double r98586 = 8.066554338695772e-207;
        bool r98587 = r98554 <= r98586;
        double r98588 = r98557 * r98566;
        double r98589 = 0.0;
        double r98590 = r98588 - r98589;
        double r98591 = r98569 * r98563;
        double r98592 = r98571 * r98560;
        double r98593 = r98591 - r98592;
        double r98594 = r98576 * r98593;
        double r98595 = r98590 + r98594;
        double r98596 = sqrt(r98554);
        double r98597 = r98596 * r98573;
        double r98598 = r98596 * r98597;
        double r98599 = r98588 - r98598;
        double r98600 = r98599 + r98594;
        double r98601 = r98587 ? r98595 : r98600;
        double r98602 = r98556 ? r98585 : r98601;
        return r98602;
}

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 b < -3.4404428017653185e-231

    1. Initial program 11.5

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

    3. Applied prod-diff11.5

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

    4. Applied distribute-lft-in11.5

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

    6. Applied add-cube-cbrt11.7

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

    7. Applied associate-*l*11.7

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

    if -3.4404428017653185e-231 < b < 8.066554338695772e-207

    1. Initial program 17.0

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

    2. Taylor expanded around 0 16.4

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

    if 8.066554338695772e-207 < b

    1. Initial program 11.4

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

    3. Applied add-sqr-sqrt11.5

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

    4. Applied associate-*l*11.5

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.4404428017653185 \cdot 10^{-231}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{elif}\;b \le 8.0665543386957723 \cdot 10^{-207}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))))