Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 12.3 → 11.8

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r911226 = x;
        double r911227 = y;
        double r911228 = z;
        double r911229 = r911227 * r911228;
        double r911230 = t;
        double r911231 = a;
        double r911232 = r911230 * r911231;
        double r911233 = r911229 - r911232;
        double r911234 = r911226 * r911233;
        double r911235 = b;
        double r911236 = c;
        double r911237 = r911236 * r911228;
        double r911238 = i;
        double r911239 = r911230 * r911238;
        double r911240 = r911237 - r911239;
        double r911241 = r911235 * r911240;
        double r911242 = r911234 - r911241;
        double r911243 = j;
        double r911244 = r911236 * r911231;
        double r911245 = r911227 * r911238;
        double r911246 = r911244 - r911245;
        double r911247 = r911243 * r911246;
        double r911248 = r911242 + r911247;
        return r911248;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r911249 = a;
        double r911250 = -2.9368192681862057e-268;
        bool r911251 = r911249 <= r911250;
        double r911252 = 1.2849751811444222e+121;
        bool r911253 = r911249 <= r911252;
        double r911254 = !r911253;
        bool r911255 = r911251 || r911254;
        double r911256 = x;
        double r911257 = y;
        double r911258 = r911256 * r911257;
        double r911259 = z;
        double r911260 = r911258 * r911259;
        double r911261 = cbrt(r911260);
        double r911262 = r911261 * r911261;
        double r911263 = r911262 * r911261;
        double r911264 = -1.0;
        double r911265 = t;
        double r911266 = r911256 * r911265;
        double r911267 = r911249 * r911266;
        double r911268 = r911264 * r911267;
        double r911269 = r911263 + r911268;
        double r911270 = b;
        double r911271 = c;
        double r911272 = r911271 * r911259;
        double r911273 = i;
        double r911274 = r911265 * r911273;
        double r911275 = r911272 - r911274;
        double r911276 = r911270 * r911275;
        double r911277 = r911269 - r911276;
        double r911278 = j;
        double r911279 = r911271 * r911249;
        double r911280 = r911257 * r911273;
        double r911281 = r911279 - r911280;
        double r911282 = r911278 * r911281;
        double r911283 = r911277 + r911282;
        double r911284 = cbrt(r911256);
        double r911285 = r911284 * r911284;
        double r911286 = r911257 * r911259;
        double r911287 = r911284 * r911286;
        double r911288 = r911285 * r911287;
        double r911289 = r911265 * r911249;
        double r911290 = -r911289;
        double r911291 = r911256 * r911290;
        double r911292 = r911288 + r911291;
        double r911293 = r911292 - r911276;
        double r911294 = r911293 + r911282;
        double r911295 = r911255 ? r911283 : r911294;
        return r911295;
}

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

Original	12.3
Target	20.3
Herbie	11.8

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

Derivation

1. Split input into 2 regimes

2. if a < -2.9368192681862057e-268 or 1.2849751811444222e+121 < a

   1. Initial program 14.0

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

   3. Applied sub-neg 14.0

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

   4. Applied distribute-lft-in 14.0

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

   6. Applied associate-*r* 14.1

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

   7. Taylor expanded around inf 12.9

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 13.0

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}} + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   if -2.9368192681862057e-268 < a < 1.2849751811444222e+121

   1. Initial program 10.1

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

   3. Applied sub-neg 10.1

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

   4. Applied distribute-lft-in 10.1

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

   6. Applied add-cube-cbrt 10.3

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

   7. Applied associate-*l* 10.3

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))))