Average Error: 11.3 → 11.8
Time: 50.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -2.5260943993090073 \cdot 10^{-173}:\\ \;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{b}\right))_*\\ \mathbf{elif}\;j \le 2.5280853440332446 \cdot 10^{-137}:\\ \;\;\;\;(j \cdot 0 + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \mathbf{else}:\\ \;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{b}\right))_*\\ \end{array}\]
\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}\;j \le -2.5260943993090073 \cdot 10^{-173}:\\
\;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{b}\right))_*\\

\mathbf{elif}\;j \le 2.5280853440332446 \cdot 10^{-137}:\\
\;\;\;\;(j \cdot 0 + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\

\mathbf{else}:\\
\;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{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 r15730542 = x;
        double r15730543 = y;
        double r15730544 = z;
        double r15730545 = r15730543 * r15730544;
        double r15730546 = t;
        double r15730547 = a;
        double r15730548 = r15730546 * r15730547;
        double r15730549 = r15730545 - r15730548;
        double r15730550 = r15730542 * r15730549;
        double r15730551 = b;
        double r15730552 = c;
        double r15730553 = r15730552 * r15730544;
        double r15730554 = i;
        double r15730555 = r15730554 * r15730547;
        double r15730556 = r15730553 - r15730555;
        double r15730557 = r15730551 * r15730556;
        double r15730558 = r15730550 - r15730557;
        double r15730559 = j;
        double r15730560 = r15730552 * r15730546;
        double r15730561 = r15730554 * r15730543;
        double r15730562 = r15730560 - r15730561;
        double r15730563 = r15730559 * r15730562;
        double r15730564 = r15730558 + r15730563;
        return r15730564;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r15730565 = j;
        double r15730566 = -2.5260943993090073e-173;
        bool r15730567 = r15730565 <= r15730566;
        double r15730568 = t;
        double r15730569 = c;
        double r15730570 = y;
        double r15730571 = -r15730570;
        double r15730572 = i;
        double r15730573 = r15730571 * r15730572;
        double r15730574 = fma(r15730568, r15730569, r15730573);
        double r15730575 = x;
        double r15730576 = z;
        double r15730577 = r15730570 * r15730576;
        double r15730578 = a;
        double r15730579 = r15730578 * r15730568;
        double r15730580 = r15730577 - r15730579;
        double r15730581 = r15730575 * r15730580;
        double r15730582 = b;
        double r15730583 = cbrt(r15730582);
        double r15730584 = r15730569 * r15730576;
        double r15730585 = r15730578 * r15730572;
        double r15730586 = r15730584 - r15730585;
        double r15730587 = r15730583 * r15730586;
        double r15730588 = r15730583 * r15730587;
        double r15730589 = r15730588 * r15730583;
        double r15730590 = r15730581 - r15730589;
        double r15730591 = fma(r15730565, r15730574, r15730590);
        double r15730592 = 2.5280853440332446e-137;
        bool r15730593 = r15730565 <= r15730592;
        double r15730594 = 0.0;
        double r15730595 = r15730586 * r15730582;
        double r15730596 = r15730581 - r15730595;
        double r15730597 = fma(r15730565, r15730594, r15730596);
        double r15730598 = r15730593 ? r15730597 : r15730591;
        double r15730599 = r15730567 ? r15730591 : r15730598;
        return r15730599;
}

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 < -2.5260943993090073e-173 or 2.5280853440332446e-137 < j

    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

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

    4. Applied fma-neg9.1

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

    6. Applied add-cube-cbrt9.4

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

    7. Applied associate-*r*9.4

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

    9. Applied associate-*r*9.4

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

    if -2.5260943993090073e-173 < j < 2.5280853440332446e-137

    1. Initial program 16.2

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

    2. Simplified16.2

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

    3. Taylor expanded around 0 17.3

      \[\leadsto (j \cdot \color{blue}{0} + \left(\left(z \cdot y - t \cdot a\right) \cdot x - \left(z \cdot c - i \cdot a\right) \cdot b\right))_*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -2.5260943993090073 \cdot 10^{-173}:\\ \;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{b}\right))_*\\ \mathbf{elif}\;j \le 2.5280853440332446 \cdot 10^{-137}:\\ \;\;\;\;(j \cdot 0 + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \mathbf{else}:\\ \;\;\;\;(j \cdot \left((t \cdot c + \left(\left(-y\right) \cdot i\right))_*\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \cdot \sqrt[3]{b}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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)))))