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

Average Error: 11.8 → 9.4

Time: 17.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.1538663063516028 \cdot 10^{27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 1.1956400865377228 \cdot 10^{24}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1120341 = x;
        double r1120342 = y;
        double r1120343 = z;
        double r1120344 = r1120342 * r1120343;
        double r1120345 = t;
        double r1120346 = a;
        double r1120347 = r1120345 * r1120346;
        double r1120348 = r1120344 - r1120347;
        double r1120349 = r1120341 * r1120348;
        double r1120350 = b;
        double r1120351 = c;
        double r1120352 = r1120351 * r1120343;
        double r1120353 = i;
        double r1120354 = r1120345 * r1120353;
        double r1120355 = r1120352 - r1120354;
        double r1120356 = r1120350 * r1120355;
        double r1120357 = r1120349 - r1120356;
        double r1120358 = j;
        double r1120359 = r1120351 * r1120346;
        double r1120360 = r1120342 * r1120353;
        double r1120361 = r1120359 - r1120360;
        double r1120362 = r1120358 * r1120361;
        double r1120363 = r1120357 + r1120362;
        return r1120363;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1120364 = b;
        double r1120365 = -1.1538663063516028e+27;
        bool r1120366 = r1120364 <= r1120365;
        double r1120367 = x;
        double r1120368 = y;
        double r1120369 = z;
        double r1120370 = r1120368 * r1120369;
        double r1120371 = t;
        double r1120372 = a;
        double r1120373 = r1120371 * r1120372;
        double r1120374 = r1120370 - r1120373;
        double r1120375 = r1120367 * r1120374;
        double r1120376 = c;
        double r1120377 = r1120376 * r1120369;
        double r1120378 = i;
        double r1120379 = r1120371 * r1120378;
        double r1120380 = r1120377 - r1120379;
        double r1120381 = r1120364 * r1120380;
        double r1120382 = r1120375 - r1120381;
        double r1120383 = j;
        double r1120384 = r1120376 * r1120372;
        double r1120385 = r1120368 * r1120378;
        double r1120386 = r1120384 - r1120385;
        double r1120387 = r1120383 * r1120386;
        double r1120388 = cbrt(r1120387);
        double r1120389 = r1120388 * r1120388;
        double r1120390 = r1120389 * r1120388;
        double r1120391 = r1120382 + r1120390;
        double r1120392 = 1.1956400865377228e+24;
        bool r1120393 = r1120364 <= r1120392;
        double r1120394 = r1120370 * r1120367;
        double r1120395 = r1120367 * r1120371;
        double r1120396 = r1120372 * r1120395;
        double r1120397 = -r1120396;
        double r1120398 = r1120394 + r1120397;
        double r1120399 = r1120364 * r1120376;
        double r1120400 = r1120369 * r1120399;
        double r1120401 = r1120378 * r1120364;
        double r1120402 = r1120371 * r1120401;
        double r1120403 = -r1120402;
        double r1120404 = r1120400 + r1120403;
        double r1120405 = r1120398 - r1120404;
        double r1120406 = r1120405 + r1120387;
        double r1120407 = sqrt(r1120364);
        double r1120408 = r1120407 * r1120380;
        double r1120409 = r1120407 * r1120408;
        double r1120410 = r1120375 - r1120409;
        double r1120411 = r1120410 + r1120387;
        double r1120412 = r1120393 ? r1120406 : r1120411;
        double r1120413 = r1120366 ? r1120391 : r1120412;
        return r1120413;
}

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	11.8
Target	19.8
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if b < -1.1538663063516028e+27

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.4

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

   if -1.1538663063516028e+27 < b < 1.1956400865377228e+24

   1. Initial program 14.1

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

   3. Applied sub-neg 14.1

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

   4. Applied distribute-lft-in 14.1

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

   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-lft-neg-out 12.3

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

   9. Simplified 9.9

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

   11. Applied sub-neg 9.9

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

   12. Applied distribute-lft-in 9.9

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

   13. Simplified 9.9

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

   14. Simplified 10.4

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

   if 1.1956400865377228e+24 < b

   1. Initial program 7.6

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

   3. Applied add-sqr-sqrt 7.7

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

   4. Applied associate-*l* 7.7

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1538663063516028 \cdot 10^{27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 1.1956400865377228 \cdot 10^{24}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))