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

Average Error: 12.2 → 12.4

Time: 30.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r42665352 = x;
        double r42665353 = y;
        double r42665354 = z;
        double r42665355 = r42665353 * r42665354;
        double r42665356 = t;
        double r42665357 = a;
        double r42665358 = r42665356 * r42665357;
        double r42665359 = r42665355 - r42665358;
        double r42665360 = r42665352 * r42665359;
        double r42665361 = b;
        double r42665362 = c;
        double r42665363 = r42665362 * r42665354;
        double r42665364 = i;
        double r42665365 = r42665356 * r42665364;
        double r42665366 = r42665363 - r42665365;
        double r42665367 = r42665361 * r42665366;
        double r42665368 = r42665360 - r42665367;
        double r42665369 = j;
        double r42665370 = r42665362 * r42665357;
        double r42665371 = r42665353 * r42665364;
        double r42665372 = r42665370 - r42665371;
        double r42665373 = r42665369 * r42665372;
        double r42665374 = r42665368 + r42665373;
        return r42665374;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r42665375 = z;
        double r42665376 = y;
        double r42665377 = r42665375 * r42665376;
        double r42665378 = t;
        double r42665379 = a;
        double r42665380 = r42665378 * r42665379;
        double r42665381 = r42665377 - r42665380;
        double r42665382 = x;
        double r42665383 = r42665381 * r42665382;
        double r42665384 = b;
        double r42665385 = cbrt(r42665384);
        double r42665386 = r42665385 * r42665385;
        double r42665387 = c;
        double r42665388 = r42665375 * r42665387;
        double r42665389 = i;
        double r42665390 = r42665389 * r42665378;
        double r42665391 = r42665388 - r42665390;
        double r42665392 = r42665391 * r42665385;
        double r42665393 = r42665386 * r42665392;
        double r42665394 = r42665383 - r42665393;
        double r42665395 = r42665387 * r42665379;
        double r42665396 = r42665376 * r42665389;
        double r42665397 = r42665395 - r42665396;
        double r42665398 = j;
        double r42665399 = r42665397 * r42665398;
        double r42665400 = r42665394 + r42665399;
        return r42665400;
}

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	12.2
Target	19.6
Herbie	12.4

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

Derivation

1. Initial program 12.2

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

3. Applied add-cube-cbrt 12.4

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

4. Applied associate-*l* 12.4

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

5. Final simplification 12.4

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

Reproduce

herbie shell --seed 2019179 
(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)))))