Average Error: 12.3 → 12.4
Time: 30.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.98155654580347545345036736346737136654 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(c \cdot a - i \cdot y\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 7.975897860710546938442791162719611162024 \cdot 10^{-153}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot t\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(c \cdot a - i \cdot y\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \end{array}\]
\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 -7.98155654580347545345036736346737136654 \cdot 10^{-228}:\\
\;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(c \cdot a - i \cdot y\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\

\mathbf{elif}\;x \le 7.975897860710546938442791162719611162024 \cdot 10^{-153}:\\
\;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot t\right) \cdot \left(-b\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r38824431 = x;
        double r38824432 = y;
        double r38824433 = z;
        double r38824434 = r38824432 * r38824433;
        double r38824435 = t;
        double r38824436 = a;
        double r38824437 = r38824435 * r38824436;
        double r38824438 = r38824434 - r38824437;
        double r38824439 = r38824431 * r38824438;
        double r38824440 = b;
        double r38824441 = c;
        double r38824442 = r38824441 * r38824433;
        double r38824443 = i;
        double r38824444 = r38824435 * r38824443;
        double r38824445 = r38824442 - r38824444;
        double r38824446 = r38824440 * r38824445;
        double r38824447 = r38824439 - r38824446;
        double r38824448 = j;
        double r38824449 = r38824441 * r38824436;
        double r38824450 = r38824432 * r38824443;
        double r38824451 = r38824449 - r38824450;
        double r38824452 = r38824448 * r38824451;
        double r38824453 = r38824447 + r38824452;
        return r38824453;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r38824454 = x;
        double r38824455 = -7.981556545803475e-228;
        bool r38824456 = r38824454 <= r38824455;
        double r38824457 = y;
        double r38824458 = z;
        double r38824459 = r38824457 * r38824458;
        double r38824460 = a;
        double r38824461 = t;
        double r38824462 = r38824460 * r38824461;
        double r38824463 = r38824459 - r38824462;
        double r38824464 = r38824463 * r38824454;
        double r38824465 = b;
        double r38824466 = c;
        double r38824467 = r38824458 * r38824466;
        double r38824468 = i;
        double r38824469 = r38824468 * r38824461;
        double r38824470 = r38824467 - r38824469;
        double r38824471 = r38824465 * r38824470;
        double r38824472 = r38824464 - r38824471;
        double r38824473 = r38824466 * r38824460;
        double r38824474 = r38824468 * r38824457;
        double r38824475 = r38824473 - r38824474;
        double r38824476 = j;
        double r38824477 = cbrt(r38824476);
        double r38824478 = r38824475 * r38824477;
        double r38824479 = r38824477 * r38824477;
        double r38824480 = r38824478 * r38824479;
        double r38824481 = r38824472 + r38824480;
        double r38824482 = 7.975897860710547e-153;
        bool r38824483 = r38824454 <= r38824482;
        double r38824484 = r38824475 * r38824476;
        double r38824485 = -r38824465;
        double r38824486 = r38824470 * r38824485;
        double r38824487 = r38824484 + r38824486;
        double r38824488 = r38824483 ? r38824487 : r38824481;
        double r38824489 = r38824456 ? r38824481 : r38824488;
        return r38824489;
}

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

Original12.3
Target19.7
Herbie12.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.981556545803475e-228 or 7.975897860710547e-153 < x

    1. Initial program 10.5

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

    3. Applied add-cube-cbrt10.8

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

    4. Applied associate-*l*10.8

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

    if -7.981556545803475e-228 < x < 7.975897860710547e-153

    1. Initial program 17.8

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

    2. Taylor expanded around 0 17.5

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.98155654580347545345036736346737136654 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(c \cdot a - i \cdot y\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 7.975897860710546938442791162719611162024 \cdot 10^{-153}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot t\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(c \cdot a - i \cdot y\right) \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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