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

Average Error: 12.6 → 12.9

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -0.003764833963061511:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-b \cdot \left(i \cdot t\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 3.8053230682781215 \cdot 10^{-186}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot c\right) \cdot b + \left(-t\right) \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-b \cdot \left(i \cdot t\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r41819231 = x;
        double r41819232 = y;
        double r41819233 = z;
        double r41819234 = r41819232 * r41819233;
        double r41819235 = t;
        double r41819236 = a;
        double r41819237 = r41819235 * r41819236;
        double r41819238 = r41819234 - r41819237;
        double r41819239 = r41819231 * r41819238;
        double r41819240 = b;
        double r41819241 = c;
        double r41819242 = r41819241 * r41819233;
        double r41819243 = i;
        double r41819244 = r41819235 * r41819243;
        double r41819245 = r41819242 - r41819244;
        double r41819246 = r41819240 * r41819245;
        double r41819247 = r41819239 - r41819246;
        double r41819248 = j;
        double r41819249 = r41819241 * r41819236;
        double r41819250 = r41819232 * r41819243;
        double r41819251 = r41819249 - r41819250;
        double r41819252 = r41819248 * r41819251;
        double r41819253 = r41819247 + r41819252;
        return r41819253;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r41819254 = a;
        double r41819255 = -0.003764833963061511;
        bool r41819256 = r41819254 <= r41819255;
        double r41819257 = x;
        double r41819258 = y;
        double r41819259 = z;
        double r41819260 = r41819258 * r41819259;
        double r41819261 = t;
        double r41819262 = r41819261 * r41819254;
        double r41819263 = r41819260 - r41819262;
        double r41819264 = r41819257 * r41819263;
        double r41819265 = b;
        double r41819266 = i;
        double r41819267 = r41819266 * r41819261;
        double r41819268 = r41819265 * r41819267;
        double r41819269 = -r41819268;
        double r41819270 = c;
        double r41819271 = r41819265 * r41819259;
        double r41819272 = r41819270 * r41819271;
        double r41819273 = r41819269 + r41819272;
        double r41819274 = r41819264 - r41819273;
        double r41819275 = j;
        double r41819276 = r41819270 * r41819254;
        double r41819277 = r41819258 * r41819266;
        double r41819278 = r41819276 - r41819277;
        double r41819279 = r41819275 * r41819278;
        double r41819280 = r41819274 + r41819279;
        double r41819281 = 3.8053230682781215e-186;
        bool r41819282 = r41819254 <= r41819281;
        double r41819283 = r41819259 * r41819270;
        double r41819284 = r41819283 * r41819265;
        double r41819285 = -r41819261;
        double r41819286 = r41819266 * r41819265;
        double r41819287 = r41819285 * r41819286;
        double r41819288 = r41819284 + r41819287;
        double r41819289 = r41819264 - r41819288;
        double r41819290 = r41819289 + r41819279;
        double r41819291 = r41819282 ? r41819290 : r41819280;
        double r41819292 = r41819256 ? r41819280 : r41819291;
        return r41819292;
}

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	12.6
Target	20.5
Herbie	12.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2.0} - {\left(t \cdot i\right)}^{2.0}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2.0} - {\left(t \cdot i\right)}^{2.0}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -0.003764833963061511 or 3.8053230682781215e-186 < a

   1. Initial program 15.2

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

   3. Applied sub-neg 15.2

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

   4. Applied distribute-rgt-in 15.2

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

   6. Applied associate-*l* 15.8

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

   if -0.003764833963061511 < a < 3.8053230682781215e-186

   1. Initial program 9.1

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

   3. Applied sub-neg 9.1

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

   4. Applied distribute-rgt-in 9.1

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

   6. Applied distribute-lft-neg-in 9.1

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

   7. Applied associate-*l* 8.9

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.003764833963061511:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-b \cdot \left(i \cdot t\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;a \le 3.8053230682781215 \cdot 10^{-186}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot c\right) \cdot b + \left(-t\right) \cdot \left(i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-b \cdot \left(i \cdot t\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))