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

Average Error: 12.3 → 12.6

Time: 10.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r850494 = x;
        double r850495 = y;
        double r850496 = z;
        double r850497 = r850495 * r850496;
        double r850498 = t;
        double r850499 = a;
        double r850500 = r850498 * r850499;
        double r850501 = r850497 - r850500;
        double r850502 = r850494 * r850501;
        double r850503 = b;
        double r850504 = c;
        double r850505 = r850504 * r850496;
        double r850506 = i;
        double r850507 = r850498 * r850506;
        double r850508 = r850505 - r850507;
        double r850509 = r850503 * r850508;
        double r850510 = r850502 - r850509;
        double r850511 = j;
        double r850512 = r850504 * r850499;
        double r850513 = r850495 * r850506;
        double r850514 = r850512 - r850513;
        double r850515 = r850511 * r850514;
        double r850516 = r850510 + r850515;
        return r850516;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r850517 = x;
        double r850518 = cbrt(r850517);
        double r850519 = r850518 * r850518;
        double r850520 = y;
        double r850521 = z;
        double r850522 = r850520 * r850521;
        double r850523 = t;
        double r850524 = a;
        double r850525 = r850523 * r850524;
        double r850526 = r850522 - r850525;
        double r850527 = r850518 * r850526;
        double r850528 = r850519 * r850527;
        double r850529 = b;
        double r850530 = c;
        double r850531 = r850530 * r850521;
        double r850532 = i;
        double r850533 = r850523 * r850532;
        double r850534 = r850531 - r850533;
        double r850535 = r850529 * r850534;
        double r850536 = r850528 - r850535;
        double r850537 = j;
        double r850538 = r850530 * r850524;
        double r850539 = r850520 * r850532;
        double r850540 = r850538 - r850539;
        double r850541 = r850537 * r850540;
        double r850542 = r850536 + r850541;
        return r850542;
}

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	19.7
Herbie	12.6

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

Derivation

1. Initial program 12.3

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

3. Applied add-cube-cbrt 12.6

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

4. Applied associate-*l* 12.6

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

5. Final simplification 12.6

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

Reproduce

herbie shell --seed 2020036 
(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)))))