Average Error: 12.5 → 10.6
Time: 19.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -1.498929530326490551213697502020984882994 \cdot 10^{66}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;j \le 4.697230347509345127242688915786272829678 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(j \cdot c\right) \cdot a + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - t \cdot i} \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]
\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}\;j \le -1.498929530326490551213697502020984882994 \cdot 10^{66}:\\
\;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\

\mathbf{elif}\;j \le 4.697230347509345127242688915786272829678 \cdot 10^{-7}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(j \cdot c\right) \cdot a + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r615518 = x;
        double r615519 = y;
        double r615520 = z;
        double r615521 = r615519 * r615520;
        double r615522 = t;
        double r615523 = a;
        double r615524 = r615522 * r615523;
        double r615525 = r615521 - r615524;
        double r615526 = r615518 * r615525;
        double r615527 = b;
        double r615528 = c;
        double r615529 = r615528 * r615520;
        double r615530 = i;
        double r615531 = r615522 * r615530;
        double r615532 = r615529 - r615531;
        double r615533 = r615527 * r615532;
        double r615534 = r615526 - r615533;
        double r615535 = j;
        double r615536 = r615528 * r615523;
        double r615537 = r615519 * r615530;
        double r615538 = r615536 - r615537;
        double r615539 = r615535 * r615538;
        double r615540 = r615534 + r615539;
        return r615540;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r615541 = j;
        double r615542 = -1.4989295303264906e+66;
        bool r615543 = r615541 <= r615542;
        double r615544 = 0.0;
        double r615545 = b;
        double r615546 = c;
        double r615547 = z;
        double r615548 = r615546 * r615547;
        double r615549 = t;
        double r615550 = i;
        double r615551 = r615549 * r615550;
        double r615552 = r615548 - r615551;
        double r615553 = r615545 * r615552;
        double r615554 = r615544 - r615553;
        double r615555 = a;
        double r615556 = r615546 * r615555;
        double r615557 = r615541 * r615556;
        double r615558 = y;
        double r615559 = r615558 * r615550;
        double r615560 = -r615559;
        double r615561 = r615541 * r615560;
        double r615562 = r615557 + r615561;
        double r615563 = r615554 + r615562;
        double r615564 = 4.697230347509345e-07;
        bool r615565 = r615541 <= r615564;
        double r615566 = x;
        double r615567 = r615558 * r615547;
        double r615568 = r615549 * r615555;
        double r615569 = r615567 - r615568;
        double r615570 = r615566 * r615569;
        double r615571 = r615570 - r615553;
        double r615572 = r615541 * r615546;
        double r615573 = r615572 * r615555;
        double r615574 = r615541 * r615558;
        double r615575 = -r615550;
        double r615576 = r615574 * r615575;
        double r615577 = r615573 + r615576;
        double r615578 = r615571 + r615577;
        double r615579 = cbrt(r615552);
        double r615580 = r615579 * r615579;
        double r615581 = r615545 * r615580;
        double r615582 = r615581 * r615579;
        double r615583 = r615570 - r615582;
        double r615584 = r615583 + r615562;
        double r615585 = r615565 ? r615578 : r615584;
        double r615586 = r615543 ? r615563 : r615585;
        return r615586;
}

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

Original12.5
Target20.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if j < -1.4989295303264906e+66

    1. Initial program 6.7

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

    3. Applied sub-neg6.7

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

    4. Applied distribute-lft-in6.7

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

    5. Taylor expanded around 0 16.4

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

    if -1.4989295303264906e+66 < j < 4.697230347509345e-07

    1. Initial program 15.2

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

    3. Applied sub-neg15.2

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

    4. Applied distribute-lft-in15.2

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

    6. Applied distribute-rgt-neg-in15.2

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

    7. Applied associate-*r*13.0

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

    9. Applied associate-*r*10.3

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

    if 4.697230347509345e-07 < j

    1. Initial program 7.3

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

    3. Applied sub-neg7.3

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

    4. Applied distribute-lft-in7.3

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

    6. Applied add-cube-cbrt7.6

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

    7. Applied associate-*r*7.6

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.498929530326490551213697502020984882994 \cdot 10^{66}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \mathbf{elif}\;j \le 4.697230347509345127242688915786272829678 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(j \cdot c\right) \cdot a + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - t \cdot i} \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right) + \left(j \cdot \left(c \cdot a\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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 -1.46969429677770502e-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)))))