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

Average Error: 12.4 → 10.8

Time: 29.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.146405808658658164557833087473723144571 \cdot 10^{120} \lor \neg \left(b \le 3.128365893408555932513215992953215128042 \cdot 10^{138}\right):\\ \;\;\;\;\left(b \cdot \left(t \cdot i - c \cdot z\right) - a \cdot \left(x \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(-i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r568257 = x;
        double r568258 = y;
        double r568259 = z;
        double r568260 = r568258 * r568259;
        double r568261 = t;
        double r568262 = a;
        double r568263 = r568261 * r568262;
        double r568264 = r568260 - r568263;
        double r568265 = r568257 * r568264;
        double r568266 = b;
        double r568267 = c;
        double r568268 = r568267 * r568259;
        double r568269 = i;
        double r568270 = r568261 * r568269;
        double r568271 = r568268 - r568270;
        double r568272 = r568266 * r568271;
        double r568273 = r568265 - r568272;
        double r568274 = j;
        double r568275 = r568267 * r568262;
        double r568276 = r568258 * r568269;
        double r568277 = r568275 - r568276;
        double r568278 = r568274 * r568277;
        double r568279 = r568273 + r568278;
        return r568279;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r568280 = b;
        double r568281 = -1.1464058086586582e+120;
        bool r568282 = r568280 <= r568281;
        double r568283 = 3.128365893408556e+138;
        bool r568284 = r568280 <= r568283;
        double r568285 = !r568284;
        bool r568286 = r568282 || r568285;
        double r568287 = t;
        double r568288 = i;
        double r568289 = r568287 * r568288;
        double r568290 = c;
        double r568291 = z;
        double r568292 = r568290 * r568291;
        double r568293 = r568289 - r568292;
        double r568294 = r568280 * r568293;
        double r568295 = a;
        double r568296 = x;
        double r568297 = r568296 * r568287;
        double r568298 = r568295 * r568297;
        double r568299 = r568294 - r568298;
        double r568300 = j;
        double r568301 = r568290 * r568295;
        double r568302 = y;
        double r568303 = r568302 * r568288;
        double r568304 = r568301 - r568303;
        double r568305 = r568300 * r568304;
        double r568306 = r568299 + r568305;
        double r568307 = r568302 * r568291;
        double r568308 = r568287 * r568295;
        double r568309 = r568307 - r568308;
        double r568310 = r568296 * r568309;
        double r568311 = r568280 * r568290;
        double r568312 = r568291 * r568311;
        double r568313 = r568288 * r568280;
        double r568314 = -r568313;
        double r568315 = r568287 * r568314;
        double r568316 = r568312 + r568315;
        double r568317 = r568310 - r568316;
        double r568318 = r568317 + r568305;
        double r568319 = r568286 ? r568306 : r568318;
        return r568319;
}

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	12.4
Target	19.4
Herbie	10.8

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

Derivation

1. Split input into 2 regimes

2. if b < -1.1464058086586582e+120 or 3.128365893408556e+138 < b

   1. Initial program 6.7

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

   2. Taylor expanded around inf 24.5

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

   3. Simplified 10.8

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

   if -1.1464058086586582e+120 < b < 3.128365893408556e+138

   1. Initial program 13.7

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

   3. Applied sub-neg 13.7

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

   4. Applied distribute-lft-in 13.7

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

   5. Simplified 12.3

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

   6. Simplified 12.3

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

   8. Applied distribute-rgt-neg-in 12.3

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

   9. Applied associate-*l* 10.8

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

   10. Simplified 10.8

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.146405808658658164557833087473723144571 \cdot 10^{120} \lor \neg \left(b \le 3.128365893408555932513215992953215128042 \cdot 10^{138}\right):\\ \;\;\;\;\left(b \cdot \left(t \cdot i - c \cdot z\right) - a \cdot \left(x \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(-i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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 -1.46969429677770502e-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)))))