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

Average Error: 12.3 → 12.5

Time: 7.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.0888365422142503 \cdot 10^{-114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(a \cdot \left(j \cdot c\right) + \left(\left(-j\right) \cdot y\right) \cdot i\right)\\ \mathbf{elif}\;x \le 2.6424334011412275 \cdot 10^{-227}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-j\right) \cdot \left(y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r926516 = x;
        double r926517 = y;
        double r926518 = z;
        double r926519 = r926517 * r926518;
        double r926520 = t;
        double r926521 = a;
        double r926522 = r926520 * r926521;
        double r926523 = r926519 - r926522;
        double r926524 = r926516 * r926523;
        double r926525 = b;
        double r926526 = c;
        double r926527 = r926526 * r926518;
        double r926528 = i;
        double r926529 = r926520 * r926528;
        double r926530 = r926527 - r926529;
        double r926531 = r926525 * r926530;
        double r926532 = r926524 - r926531;
        double r926533 = j;
        double r926534 = r926526 * r926521;
        double r926535 = r926517 * r926528;
        double r926536 = r926534 - r926535;
        double r926537 = r926533 * r926536;
        double r926538 = r926532 + r926537;
        return r926538;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r926539 = x;
        double r926540 = -2.0888365422142503e-114;
        bool r926541 = r926539 <= r926540;
        double r926542 = y;
        double r926543 = z;
        double r926544 = r926542 * r926543;
        double r926545 = t;
        double r926546 = a;
        double r926547 = r926545 * r926546;
        double r926548 = r926544 - r926547;
        double r926549 = r926539 * r926548;
        double r926550 = b;
        double r926551 = c;
        double r926552 = r926551 * r926543;
        double r926553 = i;
        double r926554 = r926545 * r926553;
        double r926555 = r926552 - r926554;
        double r926556 = r926550 * r926555;
        double r926557 = cbrt(r926556);
        double r926558 = r926557 * r926557;
        double r926559 = r926558 * r926557;
        double r926560 = r926549 - r926559;
        double r926561 = j;
        double r926562 = r926561 * r926551;
        double r926563 = r926546 * r926562;
        double r926564 = -r926561;
        double r926565 = r926564 * r926542;
        double r926566 = r926565 * r926553;
        double r926567 = r926563 + r926566;
        double r926568 = r926560 + r926567;
        double r926569 = 2.6424334011412275e-227;
        bool r926570 = r926539 <= r926569;
        double r926571 = 0.0;
        double r926572 = r926571 - r926556;
        double r926573 = r926551 * r926546;
        double r926574 = r926542 * r926553;
        double r926575 = r926573 - r926574;
        double r926576 = r926561 * r926575;
        double r926577 = r926572 + r926576;
        double r926578 = r926549 - r926556;
        double r926579 = r926546 * r926561;
        double r926580 = r926579 * r926551;
        double r926581 = r926564 * r926574;
        double r926582 = r926580 + r926581;
        double r926583 = r926578 + r926582;
        double r926584 = r926570 ? r926577 : r926583;
        double r926585 = r926541 ? r926568 : r926584;
        return r926585;
}

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	12.3
Target	19.9
Herbie	12.5

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

Derivation

1. Split input into 3 regimes

2. if x < -2.0888365422142503e-114

   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 add-cube-cbrt 9.3

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

   4. Applied associate-*l* 9.3

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

   6. Applied sub-neg 9.3

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

   7. Applied distribute-lft-in 9.3

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

   8. Applied distribute-lft-in 9.4

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

   9. Simplified 9.8

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

   10. Simplified 9.7

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

   12. Applied associate-*r* 9.3

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

   14. Applied add-cube-cbrt 9.6

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

   if -2.0888365422142503e-114 < x < 2.6424334011412275e-227

   1. Initial program 16.3

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

   2. Taylor expanded around 0 17.0

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   if 2.6424334011412275e-227 < x

   1. Initial program 11.7

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

   3. Applied add-cube-cbrt 12.0

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

   4. Applied associate-*l* 12.0

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

   6. Applied sub-neg 12.0

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

   7. Applied distribute-lft-in 12.0

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

   8. Applied distribute-lft-in 12.0

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

   9. Simplified 11.8

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

   10. Simplified 11.7

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

   12. Applied associate-*r* 11.5

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.0888365422142503 \cdot 10^{-114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(a \cdot \left(j \cdot c\right) + \left(\left(-j\right) \cdot y\right) \cdot i\right)\\ \mathbf{elif}\;x \le 2.6424334011412275 \cdot 10^{-227}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-j\right) \cdot \left(y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))))