Average Error: 11.7 → 9.8
Time: 29.3s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.075828090768532 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\ \mathbf{elif}\;z \le 9.741177042073514 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(i \cdot t - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\ \end{array}\]
\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}\;z \le -8.075828090768532 \cdot 10^{+82}:\\
\;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\

\mathbf{elif}\;z \le 9.741177042073514 \cdot 10^{+85}:\\
\;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(i \cdot t - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r24875435 = x;
        double r24875436 = y;
        double r24875437 = z;
        double r24875438 = r24875436 * r24875437;
        double r24875439 = t;
        double r24875440 = a;
        double r24875441 = r24875439 * r24875440;
        double r24875442 = r24875438 - r24875441;
        double r24875443 = r24875435 * r24875442;
        double r24875444 = b;
        double r24875445 = c;
        double r24875446 = r24875445 * r24875437;
        double r24875447 = i;
        double r24875448 = r24875439 * r24875447;
        double r24875449 = r24875446 - r24875448;
        double r24875450 = r24875444 * r24875449;
        double r24875451 = r24875443 - r24875450;
        double r24875452 = j;
        double r24875453 = r24875445 * r24875440;
        double r24875454 = r24875436 * r24875447;
        double r24875455 = r24875453 - r24875454;
        double r24875456 = r24875452 * r24875455;
        double r24875457 = r24875451 + r24875456;
        return r24875457;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r24875458 = z;
        double r24875459 = -8.075828090768532e+82;
        bool r24875460 = r24875458 <= r24875459;
        double r24875461 = a;
        double r24875462 = c;
        double r24875463 = r24875461 * r24875462;
        double r24875464 = y;
        double r24875465 = i;
        double r24875466 = r24875464 * r24875465;
        double r24875467 = r24875463 - r24875466;
        double r24875468 = j;
        double r24875469 = x;
        double r24875470 = r24875469 * r24875464;
        double r24875471 = b;
        double r24875472 = r24875462 * r24875471;
        double r24875473 = r24875470 - r24875472;
        double r24875474 = r24875473 * r24875458;
        double r24875475 = r24875461 * r24875469;
        double r24875476 = t;
        double r24875477 = r24875475 * r24875476;
        double r24875478 = r24875474 - r24875477;
        double r24875479 = fma(r24875467, r24875468, r24875478);
        double r24875480 = 9.741177042073514e+85;
        bool r24875481 = r24875458 <= r24875480;
        double r24875482 = r24875465 * r24875476;
        double r24875483 = r24875462 * r24875458;
        double r24875484 = r24875482 - r24875483;
        double r24875485 = r24875464 * r24875458;
        double r24875486 = r24875461 * r24875476;
        double r24875487 = r24875485 - r24875486;
        double r24875488 = r24875487 * r24875469;
        double r24875489 = fma(r24875484, r24875471, r24875488);
        double r24875490 = fma(r24875467, r24875468, r24875489);
        double r24875491 = r24875481 ? r24875490 : r24875479;
        double r24875492 = r24875460 ? r24875479 : r24875491;
        return r24875492;
}

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

Original11.7
Target18.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.075828090768532e+82 or 9.741177042073514e+85 < z

    1. Initial program 18.8

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

    2. Simplified18.8

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

    3. Taylor expanded around inf 17.9

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

    4. Simplified11.2

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

    if -8.075828090768532e+82 < z < 9.741177042073514e+85

    1. Initial program 9.3

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

    2. Simplified9.3

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.075828090768532 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\ \mathbf{elif}\;z \le 9.741177042073514 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(i \cdot t - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - \left(a \cdot x\right) \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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) (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)))))