Average Error: 12.5 → 12.4
Time: 10.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.58303097418706128 \cdot 10^{126}:\\ \;\;\;\;\mathsf{fma}\left(t, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, t \cdot \left(x \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]
\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}\;t \le -2.58303097418706128 \cdot 10^{126}:\\
\;\;\;\;\mathsf{fma}\left(t, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, t \cdot \left(x \cdot a\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1158254 = x;
        double r1158255 = y;
        double r1158256 = z;
        double r1158257 = r1158255 * r1158256;
        double r1158258 = t;
        double r1158259 = a;
        double r1158260 = r1158258 * r1158259;
        double r1158261 = r1158257 - r1158260;
        double r1158262 = r1158254 * r1158261;
        double r1158263 = b;
        double r1158264 = c;
        double r1158265 = r1158264 * r1158256;
        double r1158266 = i;
        double r1158267 = r1158258 * r1158266;
        double r1158268 = r1158265 - r1158267;
        double r1158269 = r1158263 * r1158268;
        double r1158270 = r1158262 - r1158269;
        double r1158271 = j;
        double r1158272 = r1158264 * r1158259;
        double r1158273 = r1158255 * r1158266;
        double r1158274 = r1158272 - r1158273;
        double r1158275 = r1158271 * r1158274;
        double r1158276 = r1158270 + r1158275;
        return r1158276;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1158277 = t;
        double r1158278 = -2.5830309741870613e+126;
        bool r1158279 = r1158277 <= r1158278;
        double r1158280 = i;
        double r1158281 = b;
        double r1158282 = r1158280 * r1158281;
        double r1158283 = z;
        double r1158284 = c;
        double r1158285 = r1158281 * r1158284;
        double r1158286 = x;
        double r1158287 = a;
        double r1158288 = r1158286 * r1158287;
        double r1158289 = r1158277 * r1158288;
        double r1158290 = fma(r1158283, r1158285, r1158289);
        double r1158291 = -r1158290;
        double r1158292 = fma(r1158277, r1158282, r1158291);
        double r1158293 = r1158284 * r1158287;
        double r1158294 = y;
        double r1158295 = r1158294 * r1158280;
        double r1158296 = r1158293 - r1158295;
        double r1158297 = j;
        double r1158298 = cbrt(r1158286);
        double r1158299 = r1158298 * r1158298;
        double r1158300 = r1158294 * r1158283;
        double r1158301 = r1158277 * r1158287;
        double r1158302 = r1158300 - r1158301;
        double r1158303 = r1158298 * r1158302;
        double r1158304 = r1158299 * r1158303;
        double r1158305 = r1158284 * r1158283;
        double r1158306 = r1158277 * r1158280;
        double r1158307 = r1158305 - r1158306;
        double r1158308 = r1158281 * r1158307;
        double r1158309 = r1158304 - r1158308;
        double r1158310 = fma(r1158296, r1158297, r1158309);
        double r1158311 = r1158279 ? r1158292 : r1158310;
        return r1158311;
}

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

Original12.5
Target20.2
Herbie12.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.5830309741870613e+126

    1. Initial program 23.0

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

    2. Simplified23.0

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

    3. Taylor expanded around inf 20.0

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

    4. Simplified20.0

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

    if -2.5830309741870613e+126 < t

    1. Initial program 11.3

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

    2. Simplified11.3

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

    4. Applied add-cube-cbrt11.6

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

    5. Applied associate-*l*11.6

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.58303097418706128 \cdot 10^{126}:\\ \;\;\;\;\mathsf{fma}\left(t, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, t \cdot \left(x \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))))