Average Error: 12.5 → 12.5
Time: 10.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.8066359493368801 \cdot 10^{-143}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(\left(\sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)} \cdot \sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)} + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{elif}\;b \le 2.01753666676692102 \cdot 10^{-218}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]
\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 -8.8066359493368801 \cdot 10^{-143}:\\
\;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(\left(\sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)} \cdot \sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)}\right) \cdot \sqrt[3]{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 2.01753666676692102 \cdot 10^{-218}:\\
\;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\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{else}:\\
\;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - i \cdot a\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 r333543 = x;
        double r333544 = y;
        double r333545 = z;
        double r333546 = r333544 * r333545;
        double r333547 = t;
        double r333548 = a;
        double r333549 = r333547 * r333548;
        double r333550 = r333546 - r333549;
        double r333551 = r333543 * r333550;
        double r333552 = b;
        double r333553 = c;
        double r333554 = r333553 * r333545;
        double r333555 = i;
        double r333556 = r333555 * r333548;
        double r333557 = r333554 - r333556;
        double r333558 = r333552 * r333557;
        double r333559 = r333551 - r333558;
        double r333560 = j;
        double r333561 = r333553 * r333547;
        double r333562 = r333555 * r333544;
        double r333563 = r333561 - r333562;
        double r333564 = r333560 * r333563;
        double r333565 = r333559 + r333564;
        return r333565;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r333566 = b;
        double r333567 = -8.80663594933688e-143;
        bool r333568 = r333566 <= r333567;
        double r333569 = x;
        double r333570 = y;
        double r333571 = z;
        double r333572 = a;
        double r333573 = t;
        double r333574 = r333572 * r333573;
        double r333575 = -r333574;
        double r333576 = fma(r333570, r333571, r333575);
        double r333577 = r333569 * r333576;
        double r333578 = -r333572;
        double r333579 = fma(r333578, r333573, r333574);
        double r333580 = r333569 * r333579;
        double r333581 = r333577 + r333580;
        double r333582 = c;
        double r333583 = r333582 * r333571;
        double r333584 = i;
        double r333585 = r333584 * r333572;
        double r333586 = r333583 - r333585;
        double r333587 = r333566 * r333586;
        double r333588 = r333572 * r333584;
        double r333589 = fma(r333578, r333584, r333588);
        double r333590 = r333566 * r333589;
        double r333591 = r333587 + r333590;
        double r333592 = r333581 - r333591;
        double r333593 = j;
        double r333594 = r333570 * r333584;
        double r333595 = -r333594;
        double r333596 = fma(r333582, r333573, r333595);
        double r333597 = r333593 * r333596;
        double r333598 = cbrt(r333597);
        double r333599 = r333598 * r333598;
        double r333600 = r333599 * r333598;
        double r333601 = -r333570;
        double r333602 = fma(r333601, r333584, r333594);
        double r333603 = r333593 * r333602;
        double r333604 = r333600 + r333603;
        double r333605 = r333592 + r333604;
        double r333606 = 2.017536666766921e-218;
        bool r333607 = r333566 <= r333606;
        double r333608 = 0.0;
        double r333609 = r333581 - r333608;
        double r333610 = r333597 + r333603;
        double r333611 = r333609 + r333610;
        double r333612 = r333570 * r333571;
        double r333613 = r333573 * r333572;
        double r333614 = r333612 - r333613;
        double r333615 = cbrt(r333614);
        double r333616 = r333615 * r333615;
        double r333617 = r333569 * r333616;
        double r333618 = r333617 * r333615;
        double r333619 = r333618 - r333587;
        double r333620 = r333582 * r333573;
        double r333621 = r333584 * r333570;
        double r333622 = r333620 - r333621;
        double r333623 = r333593 * r333622;
        double r333624 = r333619 + r333623;
        double r333625 = r333607 ? r333611 : r333624;
        double r333626 = r333568 ? r333605 : r333625;
        return r333626;
}

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

Target

Original12.5
Target16.6
Herbie12.5
\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.80663594933688e-143

    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 prod-diff9.5

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

    4. Applied distribute-lft-in9.5

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

    6. Applied prod-diff9.5

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

    7. Applied distribute-lft-in9.5

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

    9. Applied prod-diff9.5

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

    10. Applied distribute-lft-in9.4

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

    11. Simplified9.4

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

    13. Applied add-cube-cbrt9.7

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

    if -8.80663594933688e-143 < b < 2.017536666766921e-218

    1. Initial program 18.1

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

    3. Applied prod-diff18.1

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

    4. Applied distribute-lft-in18.1

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

    6. Applied prod-diff18.1

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

    7. Applied distribute-lft-in18.1

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

    8. Taylor expanded around 0 17.5

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

    if 2.017536666766921e-218 < b

    1. Initial program 11.3

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

    3. Applied add-cube-cbrt11.6

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

    4. Applied associate-*r*11.6

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.8066359493368801 \cdot 10^{-143}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(\left(\sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)} \cdot \sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right)} + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{elif}\;b \le 2.01753666676692102 \cdot 10^{-218}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))