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

Average Error: 12.1 → 12.2

Time: 9.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.5533191218213379 \cdot 10^{24}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 1.6388461898592592 \cdot 10^{183}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + b \cdot \left(-t \cdot i\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left(\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(t \cdot i\right) \cdot \left(t \cdot i\right)\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r928192 = x;
        double r928193 = y;
        double r928194 = z;
        double r928195 = r928193 * r928194;
        double r928196 = t;
        double r928197 = a;
        double r928198 = r928196 * r928197;
        double r928199 = r928195 - r928198;
        double r928200 = r928192 * r928199;
        double r928201 = b;
        double r928202 = c;
        double r928203 = r928202 * r928194;
        double r928204 = i;
        double r928205 = r928196 * r928204;
        double r928206 = r928203 - r928205;
        double r928207 = r928201 * r928206;
        double r928208 = r928200 - r928207;
        double r928209 = j;
        double r928210 = r928202 * r928197;
        double r928211 = r928193 * r928204;
        double r928212 = r928210 - r928211;
        double r928213 = r928209 * r928212;
        double r928214 = r928208 + r928213;
        return r928214;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r928215 = b;
        double r928216 = -2.553319121821338e+24;
        bool r928217 = r928215 <= r928216;
        double r928218 = x;
        double r928219 = y;
        double r928220 = z;
        double r928221 = r928219 * r928220;
        double r928222 = t;
        double r928223 = a;
        double r928224 = r928222 * r928223;
        double r928225 = r928221 - r928224;
        double r928226 = cbrt(r928225);
        double r928227 = r928226 * r928226;
        double r928228 = r928218 * r928227;
        double r928229 = r928228 * r928226;
        double r928230 = c;
        double r928231 = r928230 * r928220;
        double r928232 = i;
        double r928233 = r928222 * r928232;
        double r928234 = r928231 - r928233;
        double r928235 = r928215 * r928234;
        double r928236 = r928229 - r928235;
        double r928237 = j;
        double r928238 = r928230 * r928223;
        double r928239 = r928219 * r928232;
        double r928240 = r928238 - r928239;
        double r928241 = r928237 * r928240;
        double r928242 = r928236 + r928241;
        double r928243 = 1.6388461898592592e+183;
        bool r928244 = r928215 <= r928243;
        double r928245 = r928218 * r928225;
        double r928246 = r928215 * r928230;
        double r928247 = r928246 * r928220;
        double r928248 = -r928233;
        double r928249 = r928215 * r928248;
        double r928250 = r928247 + r928249;
        double r928251 = r928245 - r928250;
        double r928252 = cbrt(r928240);
        double r928253 = r928252 * r928252;
        double r928254 = r928237 * r928253;
        double r928255 = r928254 * r928252;
        double r928256 = r928251 + r928255;
        double r928257 = r928231 * r928231;
        double r928258 = r928233 * r928233;
        double r928259 = r928257 - r928258;
        double r928260 = r928215 * r928259;
        double r928261 = r928231 + r928233;
        double r928262 = r928260 / r928261;
        double r928263 = r928245 - r928262;
        double r928264 = r928263 + r928241;
        double r928265 = r928244 ? r928256 : r928264;
        double r928266 = r928217 ? r928242 : r928265;
        return r928266;
}

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	12.1
Target	19.6
Herbie	12.2

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

Derivation

1. Split input into 3 regimes

2. if b < -2.553319121821338e+24

   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. Using strategy rm

   3. Applied add-cube-cbrt 6.9

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

   4. Applied associate-*r* 6.9

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

   if -2.553319121821338e+24 < b < 1.6388461898592592e+183

   1. Initial program 13.8

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

   3. Applied add-cube-cbrt 14.1

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

   4. Applied associate-*r* 14.1

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

   6. Applied sub-neg 14.1

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

   7. Applied distribute-lft-in 14.1

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

   9. Applied associate-*r* 12.8

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

   if 1.6388461898592592e+183 < b

   1. Initial program 6.2

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

   3. Applied flip-- 14.0

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

   4. Applied associate-*r/ 20.8

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

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5533191218213379 \cdot 10^{24}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 1.6388461898592592 \cdot 10^{183}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot c\right) \cdot z + b \cdot \left(-t \cdot i\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left(\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(t \cdot i\right) \cdot \left(t \cdot i\right)\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))))