Average Error: 11.9 → 10.1
Time: 15.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)\right)\right)\right)\\ \end{array}\]
\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}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\
\;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)\right)\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 r987363 = x;
        double r987364 = y;
        double r987365 = z;
        double r987366 = r987364 * r987365;
        double r987367 = t;
        double r987368 = a;
        double r987369 = r987367 * r987368;
        double r987370 = r987366 - r987369;
        double r987371 = r987363 * r987370;
        double r987372 = b;
        double r987373 = c;
        double r987374 = r987373 * r987365;
        double r987375 = i;
        double r987376 = r987367 * r987375;
        double r987377 = r987374 - r987376;
        double r987378 = r987372 * r987377;
        double r987379 = r987371 - r987378;
        double r987380 = j;
        double r987381 = r987373 * r987368;
        double r987382 = r987364 * r987375;
        double r987383 = r987381 - r987382;
        double r987384 = r987380 * r987383;
        double r987385 = r987379 + r987384;
        return r987385;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r987386 = y;
        double r987387 = -8.392736145616377e+98;
        bool r987388 = r987386 <= r987387;
        double r987389 = 3.1039725660684998e+119;
        bool r987390 = r987386 <= r987389;
        double r987391 = !r987390;
        bool r987392 = r987388 || r987391;
        double r987393 = t;
        double r987394 = i;
        double r987395 = r987393 * r987394;
        double r987396 = c;
        double r987397 = z;
        double r987398 = r987396 * r987397;
        double r987399 = r987395 - r987398;
        double r987400 = b;
        double r987401 = x;
        double r987402 = r987401 * r987397;
        double r987403 = j;
        double r987404 = r987394 * r987403;
        double r987405 = r987402 - r987404;
        double r987406 = r987386 * r987405;
        double r987407 = a;
        double r987408 = r987401 * r987393;
        double r987409 = r987407 * r987408;
        double r987410 = r987406 - r987409;
        double r987411 = fma(r987399, r987400, r987410);
        double r987412 = r987396 * r987407;
        double r987413 = r987386 * r987394;
        double r987414 = r987412 - r987413;
        double r987415 = cbrt(r987401);
        double r987416 = r987415 * r987415;
        double r987417 = r987407 * r987393;
        double r987418 = -r987417;
        double r987419 = fma(r987386, r987397, r987418);
        double r987420 = r987415 * r987419;
        double r987421 = r987416 * r987420;
        double r987422 = fma(r987403, r987414, r987421);
        double r987423 = fma(r987399, r987400, r987422);
        double r987424 = r987392 ? r987411 : r987423;
        return r987424;
}

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

Original11.9
Target19.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -8.392736145616377e+98 or 3.1039725660684998e+119 < y

    1. Initial program 21.3

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

    2. Simplified21.3

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

    4. Applied fma-neg21.3

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

    5. Simplified21.3

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

    7. Applied add-cube-cbrt21.5

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

    8. Applied associate-*l*21.5

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

    9. Taylor expanded around inf 23.6

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

    10. Simplified11.9

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

    if -8.392736145616377e+98 < y < 3.1039725660684998e+119

    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

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

    4. Applied fma-neg9.3

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

    5. Simplified9.3

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

    7. Applied add-cube-cbrt9.6

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

    8. Applied associate-*l*9.6

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))