Average Error: 12.1 → 12.2
Time: 9.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2318033429786034 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{elif}\;b \le 9.8019745840013685 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]
\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}\;b \le -5.2318033429786034 \cdot 10^{-166}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\

\mathbf{elif}\;b \le 9.8019745840013685 \cdot 10^{-242}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \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 r847496 = x;
        double r847497 = y;
        double r847498 = z;
        double r847499 = r847497 * r847498;
        double r847500 = t;
        double r847501 = a;
        double r847502 = r847500 * r847501;
        double r847503 = r847499 - r847502;
        double r847504 = r847496 * r847503;
        double r847505 = b;
        double r847506 = c;
        double r847507 = r847506 * r847498;
        double r847508 = i;
        double r847509 = r847500 * r847508;
        double r847510 = r847507 - r847509;
        double r847511 = r847505 * r847510;
        double r847512 = r847504 - r847511;
        double r847513 = j;
        double r847514 = r847506 * r847501;
        double r847515 = r847497 * r847508;
        double r847516 = r847514 - r847515;
        double r847517 = r847513 * r847516;
        double r847518 = r847512 + r847517;
        return r847518;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r847519 = b;
        double r847520 = -5.2318033429786034e-166;
        bool r847521 = r847519 <= r847520;
        double r847522 = c;
        double r847523 = a;
        double r847524 = r847522 * r847523;
        double r847525 = y;
        double r847526 = i;
        double r847527 = r847525 * r847526;
        double r847528 = r847524 - r847527;
        double r847529 = j;
        double r847530 = x;
        double r847531 = cbrt(r847530);
        double r847532 = r847531 * r847531;
        double r847533 = z;
        double r847534 = r847525 * r847533;
        double r847535 = t;
        double r847536 = r847535 * r847523;
        double r847537 = r847534 - r847536;
        double r847538 = r847531 * r847537;
        double r847539 = r847532 * r847538;
        double r847540 = r847522 * r847533;
        double r847541 = r847535 * r847526;
        double r847542 = r847540 - r847541;
        double r847543 = r847519 * r847542;
        double r847544 = r847539 - r847543;
        double r847545 = fma(r847528, r847529, r847544);
        double r847546 = 9.801974584001369e-242;
        bool r847547 = r847519 <= r847546;
        double r847548 = r847530 * r847537;
        double r847549 = 0.0;
        double r847550 = r847548 - r847549;
        double r847551 = fma(r847528, r847529, r847550);
        double r847552 = cbrt(r847528);
        double r847553 = r847552 * r847552;
        double r847554 = r847553 * r847552;
        double r847555 = r847548 - r847543;
        double r847556 = fma(r847554, r847529, r847555);
        double r847557 = r847547 ? r847551 : r847556;
        double r847558 = r847521 ? r847545 : r847557;
        return r847558;
}

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

Original12.1
Target19.6
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.2318033429786034e-166

    1. Initial program 9.9

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

    2. Simplified9.9

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

    4. Applied add-cube-cbrt10.1

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

    5. Applied associate-*l*10.1

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

    if -5.2318033429786034e-166 < b < 9.801974584001369e-242

    1. Initial program 17.6

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

    2. Simplified17.6

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

    3. Taylor expanded around 0 17.0

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

    if 9.801974584001369e-242 < b

    1. Initial program 11.1

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

    2. Simplified11.1

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

    4. Applied add-cube-cbrt11.4

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

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2318033429786034 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \mathbf{elif}\;b \le 9.8019745840013685 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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)))))