Average Error: 12.0 → 10.8
Time: 9.9s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5693231140139828 \cdot 10^{40}:\\ \;\;\;\;\left(x \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + -1 \cdot \left(\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.5693231140139828 \cdot 10^{40}:\\
\;\;\;\;\left(x \cdot \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 - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + -1 \cdot \left(\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1069602 = x;
        double r1069603 = y;
        double r1069604 = z;
        double r1069605 = r1069603 * r1069604;
        double r1069606 = t;
        double r1069607 = a;
        double r1069608 = r1069606 * r1069607;
        double r1069609 = r1069605 - r1069608;
        double r1069610 = r1069602 * r1069609;
        double r1069611 = b;
        double r1069612 = c;
        double r1069613 = r1069612 * r1069604;
        double r1069614 = i;
        double r1069615 = r1069606 * r1069614;
        double r1069616 = r1069613 - r1069615;
        double r1069617 = r1069611 * r1069616;
        double r1069618 = r1069610 - r1069617;
        double r1069619 = j;
        double r1069620 = r1069612 * r1069607;
        double r1069621 = r1069603 * r1069614;
        double r1069622 = r1069620 - r1069621;
        double r1069623 = r1069619 * r1069622;
        double r1069624 = r1069618 + r1069623;
        return r1069624;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1069625 = b;
        double r1069626 = -1.5693231140139828e+40;
        bool r1069627 = r1069625 <= r1069626;
        double r1069628 = x;
        double r1069629 = y;
        double r1069630 = z;
        double r1069631 = r1069629 * r1069630;
        double r1069632 = t;
        double r1069633 = a;
        double r1069634 = r1069632 * r1069633;
        double r1069635 = r1069631 - r1069634;
        double r1069636 = cbrt(r1069635);
        double r1069637 = r1069636 * r1069636;
        double r1069638 = r1069637 * r1069636;
        double r1069639 = r1069628 * r1069638;
        double r1069640 = c;
        double r1069641 = r1069640 * r1069630;
        double r1069642 = i;
        double r1069643 = r1069632 * r1069642;
        double r1069644 = r1069641 - r1069643;
        double r1069645 = r1069625 * r1069644;
        double r1069646 = r1069639 - r1069645;
        double r1069647 = j;
        double r1069648 = r1069640 * r1069633;
        double r1069649 = r1069629 * r1069642;
        double r1069650 = r1069648 - r1069649;
        double r1069651 = r1069647 * r1069650;
        double r1069652 = r1069646 + r1069651;
        double r1069653 = r1069628 * r1069635;
        double r1069654 = r1069625 * r1069640;
        double r1069655 = r1069654 * r1069630;
        double r1069656 = -1.0;
        double r1069657 = r1069642 * r1069625;
        double r1069658 = r1069632 * r1069657;
        double r1069659 = cbrt(r1069658);
        double r1069660 = r1069659 * r1069659;
        double r1069661 = r1069660 * r1069659;
        double r1069662 = r1069656 * r1069661;
        double r1069663 = r1069655 + r1069662;
        double r1069664 = r1069653 - r1069663;
        double r1069665 = r1069664 + r1069651;
        double r1069666 = r1069627 ? r1069652 : r1069665;
        return r1069666;
}

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

Target

Original12.0
Target19.6
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -1.5693231140139828e+40

    1. Initial program 7.2

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

    3. Applied add-cube-cbrt7.4

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

    if -1.5693231140139828e+40 < b

    1. Initial program 13.0

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

    3. Applied sub-neg13.0

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

    4. Applied distribute-lft-in13.0

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

    5. Taylor expanded around inf 12.5

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + \color{blue}{-1 \cdot \left(t \cdot \left(i \cdot b\right)\right)}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
    6. Using strategy rm

    7. Applied associate-*r*11.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{\left(b \cdot c\right) \cdot z} + -1 \cdot \left(t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
    8. Using strategy rm

    9. Applied add-cube-cbrt11.5

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + -1 \cdot \color{blue}{\left(\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5693231140139828 \cdot 10^{40}:\\ \;\;\;\;\left(x \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + -1 \cdot \left(\left(\sqrt[3]{t \cdot \left(i \cdot b\right)} \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right) \cdot \sqrt[3]{t \cdot \left(i \cdot b\right)}\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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