Average Error: 11.5 → 10.3
Time: 7.2s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4.1190214595724835 \cdot 10^{-49}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(x \cdot \left(-t\right)\right) \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]
\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}\;x \le 4.1190214595724835 \cdot 10^{-49}:\\
\;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(x \cdot \left(-t\right)\right) \cdot a\right) - \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)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\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 r876563 = x;
        double r876564 = y;
        double r876565 = z;
        double r876566 = r876564 * r876565;
        double r876567 = t;
        double r876568 = a;
        double r876569 = r876567 * r876568;
        double r876570 = r876566 - r876569;
        double r876571 = r876563 * r876570;
        double r876572 = b;
        double r876573 = c;
        double r876574 = r876573 * r876565;
        double r876575 = i;
        double r876576 = r876567 * r876575;
        double r876577 = r876574 - r876576;
        double r876578 = r876572 * r876577;
        double r876579 = r876571 - r876578;
        double r876580 = j;
        double r876581 = r876573 * r876568;
        double r876582 = r876564 * r876575;
        double r876583 = r876581 - r876582;
        double r876584 = r876580 * r876583;
        double r876585 = r876579 + r876584;
        return r876585;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r876586 = x;
        double r876587 = 4.1190214595724835e-49;
        bool r876588 = r876586 <= r876587;
        double r876589 = y;
        double r876590 = r876586 * r876589;
        double r876591 = z;
        double r876592 = r876590 * r876591;
        double r876593 = t;
        double r876594 = -r876593;
        double r876595 = r876586 * r876594;
        double r876596 = a;
        double r876597 = r876595 * r876596;
        double r876598 = r876592 + r876597;
        double r876599 = b;
        double r876600 = c;
        double r876601 = r876600 * r876591;
        double r876602 = r876599 * r876601;
        double r876603 = i;
        double r876604 = r876593 * r876603;
        double r876605 = -r876604;
        double r876606 = r876599 * r876605;
        double r876607 = r876602 + r876606;
        double r876608 = r876598 - r876607;
        double r876609 = j;
        double r876610 = r876600 * r876596;
        double r876611 = r876589 * r876603;
        double r876612 = r876610 - r876611;
        double r876613 = r876609 * r876612;
        double r876614 = r876608 + r876613;
        double r876615 = sqrt(r876586);
        double r876616 = r876589 * r876591;
        double r876617 = r876615 * r876616;
        double r876618 = r876615 * r876617;
        double r876619 = r876593 * r876596;
        double r876620 = -r876619;
        double r876621 = r876586 * r876620;
        double r876622 = r876618 + r876621;
        double r876623 = r876601 - r876604;
        double r876624 = r876599 * r876623;
        double r876625 = r876622 - r876624;
        double r876626 = r876625 + r876613;
        double r876627 = r876588 ? r876614 : r876626;
        return r876627;
}

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

Original11.5
Target19.4
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 4.1190214595724835e-49

    1. Initial program 12.8

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

    3. Applied sub-neg12.8

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

    4. Applied distribute-lft-in12.8

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

    6. Applied distribute-lft-neg-in12.8

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

    7. Applied associate-*r*12.0

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

    9. Applied associate-*r*11.1

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

    11. Applied sub-neg11.1

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

    12. Applied distribute-lft-in11.1

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

    if 4.1190214595724835e-49 < x

    1. Initial program 7.5

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

    3. Applied sub-neg7.5

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

    4. Applied distribute-lft-in7.5

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

    6. Applied add-sqr-sqrt7.6

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

    7. Applied associate-*l*7.6

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4.1190214595724835 \cdot 10^{-49}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + \left(x \cdot \left(-t\right)\right) \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))))